From 0aae500ac6ac0f6d80c15601e780ef090d966ee7 Mon Sep 17 00:00:00 2001 From: krahets Date: Thu, 2 Nov 2023 01:39:40 +0800 Subject: [PATCH 01/30] Translate 1.0.0b6 release with the machine learning translator. --- README.md | 40 +- docs-en/assets/covers/chapter_appendix.jpg | Bin 0 -> 116050 bytes .../covers/chapter_array_and_linkedlist.jpg | Bin 0 -> 128202 bytes .../assets/covers/chapter_backtracking.jpg | Bin 0 -> 129346 bytes .../covers/chapter_complexity_analysis.jpg | Bin 0 -> 96178 bytes .../assets/covers/chapter_data_structure.jpg | Bin 0 -> 97946 bytes .../covers/chapter_divide_and_conquer.jpg | Bin 0 -> 118326 bytes .../covers/chapter_dynamic_programming.jpg | Bin 0 -> 167504 bytes docs-en/assets/covers/chapter_graph.jpg | Bin 0 -> 85455 bytes docs-en/assets/covers/chapter_greedy.jpg | Bin 0 -> 138338 bytes docs-en/assets/covers/chapter_hashing.jpg | Bin 0 -> 140215 bytes docs-en/assets/covers/chapter_heap.jpg | Bin 0 -> 111769 bytes .../assets/covers/chapter_introduction.jpg 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100644 docs-en/index.md create mode 100644 mkdocs-en.yml diff --git a/README.md b/README.md index cbd1526006..ec1587ef69 100644 --- a/README.md +++ b/README.md @@ -5,9 +5,9 @@

- hello-algo-typing-svg + hello-algo-typing-svg
- 动画图解、一键运行的数据结构与算法教程 + Data Structures and Algorithms: Animated Illutrations and Off-the-Shelf Code

@@ -24,37 +24,37 @@

-## 关于本书 +## The Book -本项目旨在打造一本开源免费、新手友好的数据结构与算法入门教程。 +This project aims to create an open-source, free, and beginner-friendly tutorial on data structures and algorithms. -- 全书采用动画图解,内容清晰易懂、学习曲线平滑,引导初学者探索数据结构与算法的知识地图。 -- 源代码可一键运行,帮助读者在练习中提升编程技能,了解算法工作原理和数据结构底层实现。 -- 鼓励读者互助学习,提问与评论通常可在两日内得到回复。 +- The book uses animated illustrations to make the content clear, easy to understand, and provides a smooth learning curve, guiding beginners through the knowledge map of data structures and algorithms. +- The source code can be run with a single click, helping readers improve their programming skills, understand the working principles of algorithms, and the underlying implementation of data structures. +- We encourage readers to learn together, ask questions, and comment. Replies are usually received within two days. -若本书对您有所帮助,请在页面右上角点个 Star :star: 支持一下,谢谢! +If this book is helpful to you, please give it a Star :star: in the top right corner of the page to show your support. Thank you! -## 推荐语 +## Praise -> “一本通俗易懂的数据结构与算法入门书,引导读者手脑并用地学习,强烈推荐算法初学者阅读。” +> “An accessible and easy-to-understand introduction to data structures and algorithms, guiding readers to learn actively and hands-on. Highly recommended for beginners in algorithms.” > -> **—— 邓俊辉,清华大学计算机系教授** +> **—— Junhui Deng, Professor of Computer Science at Tsinghua University** -> “如果我当年学数据结构与算法的时候有《Hello 算法》,学起来应该会简单 10 倍!” +> “If I had 'Hello Algorithm' when I was learning data structures and algorithms, it would have been 10 times easier!” > -> **—— 李沐,亚马逊资深首席科学家** +> **—— Mu Li, Senior Principal Scientist at Amazon** -## 贡献 +## Contribution -本开源书仍在持续更新之中,欢迎您参与本项目,一同为读者提供更优质的学习内容。 +This open-source book is continuously being updated, and we welcome your participation in this project to provide better learning content for our readers. -- [内容修正](https://www.hello-algo.com/chapter_appendix/contribution/):请您协助修正或在评论区指出语法错误、内容缺失、文字歧义、无效链接或代码 bug 等问题。 -- [代码转译](https://github.com/krahets/hello-algo/issues/15):期待您贡献各种语言代码,现已支持 Python, Java, C++, Go, JavaScript 等 11 门编程语言。 -- [整书翻译](https://github.com/krahets/hello-algo/tree/en):诚邀您加入我们的中译英小组,成员主要来自计算机相关专业、英语专业和英文母语者。 +- [Content Correction](https://www.hello-algo.com/chapter_appendix/contribution/): Please help us correct or point out issues in the comments section such as grammatical errors, missing content, ambiguities, invalid links, or code bugs. +- [Code Translation](https://github.com/krahets/hello-algo/issues/15): We look forward to your contributions of code in various programming languages. We currently support Python, Java, C++, Go, JavaScript, and 11 other programming languages. +- [Whole Book Translation](https://github.com/krahets/hello-algo/tree/en): We invite you to join our Chinese-to-English translation team. Members are mainly from computer-related majors, English majors, and native English speakers. -欢迎您提出宝贵意见和建议,如有任何问题请提交 Issues 或微信联系 `krahets-jyd` 。 +We welcome your valuable suggestions and feedback. If you have any questions, please submit Issues or contact us via WeChat at `krahets-jyd`. -感谢本开源书的每一位撰稿人,是他们的无私奉献让这本书变得更好,他们是: +Thanks to each contributor of this open-source book. It is their selfless dedication that has made this book better. They are:

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zj2wg}#J;f*73{xzLVj^uHTsoil%*75{iYF3q(FImVqp-9k|}2)b(8$7kt^U#t>KQj zBMe&2V3EvraU4}eK^1TOFRg7P6c#9F5DJHmW zBt&k^? + +![Appendix](../assets/covers/chapter_appendix.jpg){ width="600" } + + diff --git a/docs-en/chapter_appendix/installation.md b/docs-en/chapter_appendix/installation.md new file mode 100644 index 0000000000..c23d6bacf6 --- /dev/null +++ b/docs-en/chapter_appendix/installation.md @@ -0,0 +1,55 @@ +# Programming Environment Installation + +### VSCode + +This book recommends using the open source lightweight VSCode as a local IDE, download and install [VSCode](https://code.visualstudio.com/). + +### The Java Environment + +1. Download and install [OpenJDK](https://jdk.java.net/18/) (version > JDK 9). +2. Search for `java` in VSCode's plugin marketplace and install Extension Pack for Java. + +### C/C++ Environment + +1. Windows systems need to install [MinGW](https://sourceforge.net/projects/mingw-w64/files/) ([Configuration Tutorial](https://blog.csdn.net/qq_33698226/article/details /129031241)), MacOS comes with Clang without installation. +2. Search for `c++` in VSCode's plugin marketplace and install the C/C++ Extension Pack. +3. (Optional) Open the Settings page and search for the `Clang_format_fallback Style` code formatting option set to `{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }`. + +### Python Environment + +1. Download and install [Miniconda3](https://docs.conda.io/en/latest/miniconda.html) . +2. Search for `python` in VSCode's plugin marketplace and install the Python Extension Pack. +3. (Optional) Type `pip install black` at the command line to install the code formatting tool. + +### Go Environment + +1. Download and install [go](https://go.dev/dl/). +2. Search for `go` in VSCode's plugin marketplace and install Go. +3. The shortcut key `Ctrl + Shift + P` calls out the command bar, type go, select `Go: Install/Update Tools`, check all the boxes and install it. + +### JavaScript Environment + +1. Download and install [node.js](https://nodejs.org/en/) . +2. Install JavaScript (ES6) code snippets by searching for `javascript` in VSCode's plugin marketplace. +3. (Optional) Install the code formatting tool by searching for `Prettier` in VSCode's plugin marketplace. + +### C## Environment + +1. Download and install [.Net 6.0](https://dotnet.microsoft.com/en-us/download) . +2. Install the C# Dev Kit by searching for `C# Dev Kit` in VSCode's plugin marketplace ([Configuration Tutorial](https://code.visualstudio.com/docs/csharp/get-started)). +3. Visual Studio is also available ([Installation Tutorial](https://learn.microsoft.com/zh-cn/visualstudio/install/install-visual-studio?view=vs-2022)). + +### Swift Environment + +1. Download and install [Swift](https://www.swift.org/download/). +2. Search for `swift` in VSCode's plugin marketplace and install [Swift for Visual Studio Code](https://marketplace.visualstudio.com/items?itemName=sswg.swift-lang). + +### Dart Environment + +1. Download and install [Dart](https://dart.dev/get-dart) . +2. Search for `dart` in VSCode's plugin marketplace and install [Dart](https:/marketplace.visualstudio.com/items?itemName=Dart-Code.dart-code). + +### Rust Environment + +1. Download and install [Rust](https://www.rust-lang.org/tools/install). +2. Search for `rust` in VSCode's plugin marketplace and install [rust-analyzer](https://marketplace.visualstudio.com/items?itemName=rust-lang.rust-analyzer). diff --git a/docs-en/chapter_appendix/terminology.md b/docs-en/chapter_appendix/terminology.md new file mode 100644 index 0000000000..e3bc3a08b3 --- /dev/null +++ b/docs-en/chapter_appendix/terminology.md @@ -0,0 +1,114 @@ +# Glossary + +The table below lists important terms that appear in the book. It is recommended that you memorize both their Chinese and English names in order to read the English literature. + +

Table   Important Terms for Data Structures and Algorithms

+ +| Chinese | English | +| ------------------------------ | ------------------------------ | +| algorithm | algorithm | +| data structure | data structure | +| asymptotic complexity analysis | asymptotic complexity analysis | +| time complexity | time complexity | +| space complexity | space complexity | +| iteration | iteration | +| recursion | recursion | +| tail recursion | tail recursion | +| recursion tree | recursion tree | +| big $O$ notation | big-$O$ notation | +| asymptotic upper bound | asymptotic upper bound | asymptotic upper bound | asymptotic upper bound | +| true form | +| inverse code | 1's complement code | +| 2's complement code | 2's complement code | +| array | array | +| index | index | +| linked lists | linked list | +| linked list nodes | list node | +| lists | list | +| dynamic arrays | dynamic array | +| stack | stack | +| queues | queue | +| double-ended queue | double-ended queue | double-ended queue | double-ended queue | double-ended queue | double-ended queue | double-ended queue | double-ended queue | double-ended queue | double-ended queue | +| hash table | hash table | +| bucket | bucket | +| hash functions | hash function | +| hash collision | hash collision | +| load factor | load factor | load factor | load factor | +| separate chaining | separate chaining | separate chaining | +| open addressing | open addressing | +| linear probing | linear probing | linear probing | +| lazy deletion | lazy deletion | +| binary tree | binary tree | +| tree nodes | tree node | +| left-child node | left-child node | +| right-child node | right-child node | +| parent node | parent node | +| left subtree | left subtree | +| right subtree | right subtree | +| root node | root node | +| leaf node | leaf node | +| edge | edge | edge | edge | +| level | level | +| degree | degree | degree | degree | degree | +| height | height | +| depth | depth | +| perfect binary tree | perfect binary tree | +| complete binary tree | complete binary tree | +| full binary tree | full binary tree | +| balanced binary trees | balanced binary tree | +| avl tree | avl tree | avl tree | avl tree | +| red-black tree | red-black tree | +| level-order traversal | level-order traversal | level-order traversal | level-order traversal | +| breadth-first traversal | breadth-first traversal | +| depth-first traversal | depth-first traversal | depth-first traversal | +| binary search tree | binary search tree | +| balanced binary search tree | balanced binary search tree | +| balance factor | balance factor | +| heap | heap | heap | heap | heap | heap | heap | heap | heap | heap | heap | +| max heap | max heap | +| min heap | min heap | +| priority queue | priority queue | +| heapify | heapify | heapify | heapify | heapify | heapify | heapify | heapify | heapify | heapify | heapify | heapify | +| graph | graph | +| vertex | vertex | +| undirected graph | undirected graph | +| directed graphs | directed graphs | +| connected graph | connected graph | +| connected graph | disconnected graph | +| weighted graph | weighted graph | weighted graph | weighted graph | +| adjacency | adjacency | +| path | path | +| in-degree | in-degree | in-degree | +| out-degree | out-degree | out-degree | +| adjacency matrix | adjacency matrix | +| adjacency list | adjacency list | adjacency list | adjacency list | +| breadth-first search | breadth-first search | +| depth-first search | depth-first search | +| binary search | binary search | +| searching algorithm | searching algorithm | +| sorting algorithm | sorting algorithm | +| selection sort | selection sort | +| bubble sort | bubble sort | +| insertion sort | insertion sort | +| quick sort | quick sort | +| merge sort | merge sort | +| heap sort | heap sort | +| bucket sort | bucket sort | +| counting sort | counting sort | +| radix sort | radix sort | radix sort | +| divide and conquer | +| hanota problem | hanota problem | +| backtracking algorithm | backtracking algorithm | +| constraints | constraints | +| solution | solution | +| state | state | +| pruning | pruning | +| permutations problem | permutations problem | +| subset-sum problem | subset-sum problem | +| n-queens problem | N-queens problem | +| dynamic programming | dynamic programming | +| initial state | initial state | +| state-trasition equation | state-trasition equation | +| knapsack problem | knapsack problem | +| edit distance problem | edit distance problem | +| greedy algorithm | greedy algorithm | diff --git a/docs-en/chapter_array_and_linkedlist/array.assets/array_definition.png b/docs-en/chapter_array_and_linkedlist/array.assets/array_definition.png new file mode 100644 index 0000000000000000000000000000000000000000..57700f27d22bdc5feaad66c4b5047bf0fe44c062 GIT binary patch literal 21031 zcmd42byQT}|2KNh3`2J}Lx)I7gD}z^0!lN|C`d@>kq{K51XMskLQ0fW1cjkNX%K0K z_)sF<`HY|M_daXgKYsW51|K9!V|6crWa_r4t-9!^V3Yi(`K&(DAU{CQSZR%&YM@87?% 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"C" + + ```c title="array.c" + /* Initialize the array */ + int arr[5] = { 0 }; // { 0, 0, 0, 0, 0 } + int nums[5] = { 1, 3, 2, 5, 4 }; + ``` + +=== "Zig" + + ```zig title="array.zig" + // Initialize the array. + var arr = [_]i32{0} **5; // { 0, 0, 0, 0, 0 } + var nums = [_]i32{ 1, 3, 2, 5, 4 }; + ``` + +### Accessing Elements + +Array elements are stored in contiguous memory space, which means that calculating the memory address of an array element is very easy. Given an array memory address (i.e., the first element memory address) and the index of an element, we can use the formula shown in the figure below to calculate to get the memory address of that element, and thus access this element directly. + +![Memory address calculation for array elements](array.assets/array_memory_location_calculation.png) + +Looking at the figure above, we see that the first element of the array is indexed at $0$, which seems counter-intuitive since it would be more natural to start counting at $1$. However, from the perspective of the address calculation formula, **the meaning of index is essentially the offset of the memory address**. The address offset of the first element is $0$, so it makes sense that it is indexed as $0$. + +Accessing elements in an array is very efficient, we can randomly access any element in the array in $O(1)$ time. + +```src +[file]{array}-[class]{}-[func]{random_access} +``` + +### Insert Element + +The elements of an array are "next to each other" in memory, and there is no room for any more data between them. As shown in the figure below, if you want to insert an element in the middle of an array, you need to move all the elements after that element back one place, and then assign the element to the index. + +![Example of inserting elements into an array](array.assets/array_insert_element.png) + +It is worth noting that since the length of the array is fixed, inserting an element will always result in the "loss" of the last element of the array. We leave the solution to this problem to the section on lists. + +```src +[file]{array}-[class]{}-[func]{insert} +``` + +### Delete Element + +Similarly, as shown in the figure below, if you want to delete the element at index $i$, you need to move all the elements after index $i$ forward by one bit. + +![Example of removing elements from an array](array.assets/array_remove_element.png) + +Note that after the deletion of the element is complete, the element at the end becomes "meaningless", so we don't need to modify it. + +```src +[file]{array}-[class]{}-[func]{remove} +``` + +In general, insertion and deletion operations on arrays have the following disadvantages. + +- **High time complexity**: the average time complexity of both insertion and deletion of arrays is $O(n)$ , where $n$ is the length of the array. +- **Lost Elements**: since the length of the array is immutable, elements out of the range of the length of the array are lost after insertion. +- **Memory Waste**: We can initialize a long array and use only the first part, so that when inserting data, the missing end elements are "meaningless", but this will also result in a partial waste of memory space. + +### Traversing Array + +In most programming languages, we can either traverse an array by index or traverse directly to get each element in the array. + +```src +[file]{array}-[class]{}-[func]{traverse} +``` + +### Find An Element + +Finding the specified element in an array requires traversing the array, determining whether the element values match in each round, and outputting the corresponding index if they do. + +Because arrays are linear data structures, the above lookup operation is called "linear lookup". + +```src +[file]{array}-[class]{}-[func]{find} +``` + +### Expanded Array + +In complex system environments, it is difficult for a program to guarantee that the memory space after an array is available, thus making it impossible to safely extend the array capacity. Therefore in most programming languages, **the length of an array is immutable**. + +If we want to expand the array, we need to recreate a larger array and copy the elements of the original array to the new one in turn. This is a $O(n)$ operation, which is time-consuming when the array is very large. + +```src +[file]{array}-[class]{}-[func]{extend} +``` + +## Arrays Benefits And Limitations + +Arrays are stored in contiguous memory spaces with elements of the same type. This practice contains a wealth of a priori information that can be used by the system to optimize the operational efficiency of the data structure. + +- **Space efficient**: Arrays allocate contiguous blocks of memory for data without additional structural overhead. +- **Supports random access**: Arrays allow any element to be accessed in $O(1)$ time. +- **Cache locality**: When an array element is accessed, the computer not only loads it, but also caches the rest of the data around it, thereby leveraging the cache to speed up the execution of subsequent operations. + +Continuous space storage is a double-edged sword with the following disadvantages. + +- **Insertion and deletion are inefficient**: Insertion and deletion operations require moving a large number of elements when there are many elements in the array. +- **Length immutable**: Arrays have a fixed length after initialization, and expanding an array requires copying all the data to the new array, which has a lot of overhead. +- **Space Waste**: If an array is allocated a larger size than is actually needed, the extra space is wasted. + +## Typical Applications Of Arrays + +Arrays are a basic and common data structure that are frequently used in various types of algorithms as well as for implementing complex data structures. + +- **Random Access**: if we want to randomize some samples, then we can store them in an array and generate a random sequence to achieve random sampling of samples based on the index. +- **Sorting and Searching**: Arrays are the most commonly used data structure for sorting and searching algorithms. Quick sort, merge sort, binary search, etc. are all performed primarily on arrays. +- **Lookup Tables**: When we need to find an element quickly or need to find the correspondence of an element, we can use an array as a lookup table. If we want to implement a character to ASCII mapping, we can use the ASCII value of the character as the index and store the corresponding element in the corresponding position in the array. +- **Machine Learning**: neural networks make heavy use of linear algebra operations between vectors, matrices, and tensors, which are constructed as arrays. Arrays are the most commonly used data structure in neural network programming. +- **Data Structure Implementation**: arrays can be used to implement data structures such as stacks, queues, hash tables, heaps, graphs, and so on. For example, the adjacency matrix representation of a graph is actually a two-dimensional array. diff --git a/docs-en/chapter_array_and_linkedlist/index.md b/docs-en/chapter_array_and_linkedlist/index.md new file mode 100644 index 0000000000..d4b62790cf --- /dev/null +++ b/docs-en/chapter_array_and_linkedlist/index.md @@ -0,0 +1,13 @@ +# Arrays And Linked Lists + +

+ +![arrays and linked lists](../assets/covers/chapter_array_and_linkedlist.jpg){ width="600" } + +
+ +!!! abstract + + The world of data structures is like a thick brick wall. + + The bricks of the array are neatly arranged, tightly fitting one by one. 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z0V7a_^RQ4C3N1ln-c%YFI~_z93uoTFk^{0X$8-NAu{s-?%{OsHo4m6-P=^UB#&`s3 zGz*yLJl^&Vo_g>@aQ~Vv3)g?{k(+^8veH}8)zBEV_E2BqJkFb}xx(&s&0r)FA8T+{ zJc4D2lk5hQ|2$|aGiX&!j8&m}|K=ct5$SQ0nTpEW$EYQ66jydth_87M=x7>hyi{|D F{eO}}(pvxk literal 0 HcmV?d00001 diff --git a/docs-en/chapter_array_and_linkedlist/linked_list.md b/docs-en/chapter_array_and_linkedlist/linked_list.md new file mode 100644 index 0000000000..1c79f0d49f --- /dev/null +++ b/docs-en/chapter_array_and_linkedlist/linked_list.md @@ -0,0 +1,671 @@ +# Linked List + +Memory space is a common resource for all programs, and in a complex system operating environment, free memory space may be scattered all over memory. We know that the memory space for storing an array must be contiguous, and when the array is very large, the memory may not be able to provide such a large contiguous space. This is where the flexibility of linked lists comes into play. + +A "linked list linked list" is a linear data structure in which each element is a node object and each node is connected by a "reference". The reference records the memory address of the next node, which can be accessed from the current node to the next node. + +Linked lists are designed so that individual nodes can be stored scattered throughout memory, and their memory addresses are needlessly contiguous. + +![linked list definitions and storage methods](linked_list.assets/linkedlist_definition.png) + +Looking at the figure above, a linked list is composed of "node node" objects. Each node contains two pieces of data: the node's "value" and a "reference" to the next node. + +- The first node of a linked list is called the "head node" and the last node is called the "tail node". +- The tail node points to "null", which is denoted as $\text{null}$, $\text{nullptr}$, and $\text{None}$ in Java, C++, and Python, respectively. +- In pointer-enabled languages such as C, C++, Go, and Rust, the above "reference" should be replaced with "pointer". + +As shown in the following code, a linked list node `ListNode` holds an additional reference (pointer) in addition to the value. Therefore, for the same amount of data, **linked lists take up more memory space than arrays**. + +=== "Python" + + ```python title="" + class ListNode: + """Class of linked lists nodes"""" + def __init__(self, val: int): + self.val: int = val # node value + self.next: ListNode | None = None # reference to the next node + ``` + +=== "C++" + + ```cpp title="" + /* Linked list node structure */ + struct ListNode { + int val; // node value + ListNode *next; // Pointer to the next node. + ListNode(int x) : val(x), next(nullptr) {} // constructor + }; + ``` + +=== "Java" + + ```java title="" + /* Linked list node class */ + class ListNode { + int val; // node value + ListNode next; // Reference to the next node. + ListNode(int x) { val = x; } // Constructor + } + ``` + +=== "C#" + + ```csharp title="" + /* Linked list node class */ + class ListNode { + int val; // node value + ListNode next; // Reference to the next node. + ListNode(int x) => val = x; //constructor + } + ``` + +=== "Go" + + ```go title="" + /* Linked list node structure */ + type ListNode struct { + Val int // Node value + Next *ListNode // Pointer to the next node. + } + + // The NewListNode constructor, which creates a new linked list. + func NewListNode(val int) *ListNode { + return &ListNode{ + Val: val, + Next: nil, + } + } + ``` + +=== "Swift" + + ```swift title="" + /* Linked list node class */ + class ListNode { + var val: Int // node value + var next: ListNode? // Reference to the next node. + + init(x: Int) { // constructor + val = x + } + } + ``` + +=== "JS" + + ```javascript title="" + /* Linked list node class */ + class ListNode { + constructor(val, next) { + this.val = (val === undefined ? 0 : val); // node value + this.next = (next === undefined ? null : next); // reference to next node + } + } + ``` + +=== "TS" + + ```typescript title="" + /* Linked list node class */ + class ListNode { + val: number; + next: ListNode | null; + constructor(val?: number, next?: ListNode | null) { + this.val = val === undefined ? 0 : val; // node value + this.next = next === undefined ? null : next; // reference to the next node + } + } + ``` + +=== "Dart" + + ```dart title="" + /* Linked list node class */ + class ListNode { + int val; // node value + ListNode? next; // Reference to the next node. + ListNode(this.val, [this.next]); // constructor + } + ``` + +=== "Rust" + + ```rust title="" + use std::rc::Rc; + use std::cell::RefCell; + /* Linked list node class */ + #[derive(Debug)] + struct ListNode { + val: i32, // node value + next: Option>>, // pointer to next node + } + ``` + +=== "C" + + ```c title="" + /* Linked list node structure */ + typedef struct ListNode { + int val; // node value + struct ListNode *next; // Pointer to the next node. + } ListNode; + + /* Constructors */ + ListNode *newListNode(int val) { + ListNode *node, *next; + node = (ListNode *) malloc(sizeof(ListNode)); + node->val = val; + node->next = NULL; + return node; + } + ``` + +=== "Zig" + + ```zig title="" + // Linked list node class + pub fn ListNode(comptime T: type) type { + return struct { + const Self = @This(); + + val: T = 0, // node value + next: ?*Self = null, // pointer to the next node + + // Constructor + pub fn init(self: *Self, x: i32) void { + self.val = x; + self.next = null; + } + }; + } + ``` + +## Common Operations On Linked Lists + +### Initialize A Linked List + +Creating a linked list is a two-step process. The first step is to initialize each node object, and the second step is to build a reference-pointing relationship. After initialization, we can start from the head node of the linked list node, through the reference pointing `next` to access all the nodes in turn. + +=== "Python" + + ```python title="linked_list.py" + # Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 + # Initialize individual nodes + n0 = ListNode(1) + n1 = ListNode(3) + n2 = ListNode(2) + n3 = ListNode(5) + n4 = ListNode(4) + # Construct references to point to + n0.next = n1 + n1.next = n2 + n2.next = n3 + n3.next = n4 + ``` + +=== "C++" + + ```cpp title="linked_list.cpp" + /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ + // Initialize each node + ListNode* n0 = new ListNode(1); + ListNode* n1 = new ListNode(3); + ListNode* n2 = new ListNode(2); + ListNode* n3 = new ListNode(5); + ListNode* n4 = new ListNode(4); + // Construct references to point to + n0->next = n1; + n1->next = n2; + n2->next = n3; + n3->next = n4; + ``` + +=== "Java" + + ```java title="linked_list.java" + /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ + // Initialize each node + ListNode n0 = new ListNode(1); + ListNode n1 = new ListNode(3); + ListNode n2 = new ListNode(2); + ListNode n3 = new ListNode(5); + ListNode n4 = new ListNode(4); + // Construct references to point to + n0.next = n1; + n1.next = n2; + n2.next = n3; + n3.next = n4; + ``` + +=== "C#" + + ```csharp title="linked_list.cs" + /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ + // Initialize each node + ListNode n0 = new(1); + ListNode n1 = new(3); + ListNode n2 = new(2); + ListNode n3 = new(5); + ListNode n4 = new(4); + // Construct references to point to + n0.next = n1; + n1.next = n2; + n2.next = n3; + n3.next = n4; + ``` + +=== "Go" + + ```go title="linked_list.go" + /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ + // Initialize each node + n0 := NewListNode(1) + n1 := NewListNode(3) + n2 := NewListNode(2) + n3 := NewListNode(5) + n4 := NewListNode(4) + // Construct references to point to + n0.Next = n1 + n1.Next = n2 + n2.Next = n3 + n3.Next = n4 + ``` + +=== "Swift" + + ```swift title="linked_list.swift" + /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ + // Initialize each node + let n0 = ListNode(x: 1) + let n1 = ListNode(x: 3) + let n2 = ListNode(x: 2) + let n3 = ListNode(x: 5) + let n4 = ListNode(x: 4) + // Construct references to point to + n0.next = n1 + n1.next = n2 + n2.next = n3 + n3.next = n4 + ``` + +=== "JS" + + ```javascript title="linked_list.js" + /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ + // Initialize each node + const n0 = new ListNode(1); + const n1 = new ListNode(3); + const n2 = new ListNode(2); + const n3 = new ListNode(5); + const n4 = new ListNode(4); + // Construct references to point to + n0.next = n1; + n1.next = n2; + n2.next = n3; + n3.next = n4; + ``` + +=== "TS" + + ```typescript title="linked_list.ts" + /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ + // Initialize each node + const n0 = new ListNode(1); + const n1 = new ListNode(3); + const n2 = new ListNode(2); + const n3 = new ListNode(5); + const n4 = new ListNode(4); + // Construct references to point to + n0.next = n1; + n1.next = n2; + n2.next = n3; + n3.next = n4; + ``` + +=== "Dart" + + ```dart title="linked_list.dart" + /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */\ + // Initialize each node + ListNode n0 = ListNode(1); + ListNode n1 = ListNode(3); + ListNode n2 = ListNode(2); + ListNode n3 = ListNode(5); + ListNode n4 = ListNode(4); + // Construct references to point to + n0.next = n1; + n1.next = n2; + n2.next = n3; + n3.next = n4; + ``` + +=== "Rust" + + ```rust title="linked_list.rs" + /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ + // Initialize each node + let n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None })); + let n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None })); + let n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None })); + let n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None })); + let n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None })); + + // Construct references to point to + n0.borrow_mut().next = Some(n1.clone()); + n1.borrow_mut().next = Some(n2.clone()); + n2.borrow_mut().next = Some(n3.clone()); + n3.borrow_mut().next = Some(n4.clone()); + ``` + +=== "C" + + ```c title="linked_list.c" + /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ + // Initialize each node + ListNode* n0 = newListNode(1); + ListNode* n1 = newListNode(3); + ListNode* n2 = newListNode(2); + ListNode* n3 = newListNode(5); + ListNode* n4 = newListNode(4); + // Construct references to point to + n0->next = n1; + n1->next = n2; + n2->next = n3; + n3->next = n4; + ``` + +=== "Zig" + + ```zig title="linked_list.zig" + // Initialize the linked list. + // Initialize each node + var n0 = inc.ListNode(i32){.val = 1}; + var n1 = inc.ListNode(i32){.val = 3}; + var n2 = inc.ListNode(i32){.val = 2}; + var n3 = inc.ListNode(i32){.val = 5}; + var n4 = inc.ListNode(i32){.val = 4}; + // Construct references to point to + n0.next = &n1; + n1.next = &n2; + n2.next = &n3; + n3.next = &n4; + ``` + +An array as a whole is a variable, e.g., the array `nums` contains the elements `nums[0]` and `nums[1]`, etc., whereas a linked list consists of a number of separate node objects. **We usually take the head node as a surrogate for a linked list**. For example, the linked list in the above code can be notated as a linked list `n0` . + +### Insert Node + +Inserting nodes in a linked list is very easy. As shown in the figure below, suppose we want to insert a new node `P` between two neighboring nodes `n0` and `n1` , **then we just need to change the two node references (pointers)**with a time complexity of $O(1)$ . + +In contrast, inserting elements into an array has a time complexity of $O(n)$ , which is less efficient with large amounts of data. + +![Example of a linked list insert node](linked_list.assets/linkedlist_insert_node.png) + +```src +[file]{linked_list}-[class]{}-[func]{insert} +``` + +### Delete Node + +As shown in the figure below, it is also very easy to delete nodes in a linked list **just by changing the reference (pointer) to a node**. + +Note that although the node `P` still points to `n1` after the deletion operation is completed, it is not actually possible to traverse the linked list to reach `P`, which means that `P` no longer belongs to the linked list. + +![linked list remove node](linked_list.assets/linkedlist_remove_node.png) + +```src +[file]{linked_list}-[class]{}-[func]{remove} +``` + +### Access Nodes + +**It is less efficient to access nodes in a linked list**. As mentioned in the previous section, we can access any element in the array in $O(1)$ time. This is not the case for linked lists, where the program needs to start from the head node and traverse backward one by one until it finds the target node. In other words, accessing the $i$th node of the linked list requires a loop of $i - 1$ rounds, with a time complexity of $O(n)$ . + +```src +[file]{linked_list}-[class]{}-[func]{access} +``` + +### Find The Node + +Traverses the linked list, finds the node whose value is `target`, and outputs the index of the node in the linked list. This process is also a linear lookup. + +```src +[file]{linked_list}-[class]{}-[func]{find} +``` + +## Arrays Vs. Linked Lists + +The table below summarizes and compares the various properties and operational efficiencies of arrays and linked lists. Since they use two opposite storage strategies, the various properties and operational efficiencies are also characterized by opposites. + +

Table   Efficiency of Arrays vs. Linked Lists

+ +| | arrays | linked lists | +| ------------------ | ------------------------------------ | ------------------------ | +| Storage Methods | Continuous Memory Space | Distributed Memory Space | +| Cache Localization | Friendly | Unfriendly | +| Capacity Expansion | Non-variable Length | Flexible Expansion | +| Memory Efficiency | Less memory usage, some space wasted | More memory usage | +| access elements | $O(1)$ | $O(n)$ | +| Add Element | $O(n)$ | $O(1)$ | +| Delete Element | $O(n)$ | $O(1)$ | + +## Common Linked List Types + +As shown in the figure below, there are three common types of linked lists. + +- **Unidirectional linked list**: i.e., the ordinary linked list introduced above. The nodes of a unidirectional linked list contain two pieces of data, the value and a reference to the next node. We will refer to the first node as the head node and the last node as the tail node, which points to null $\text{None}$ . +- **Ring linked list**: If we make the tail node of a unidirectional linked list point to the head node (i.e., the head and tail are connected), we get a ring linked list. In a ring linked list node any node can be considered as the head node. +- **Bidirectional linked lists**: In contrast to unidirectional linked lists, bidirectional linked lists record references in both directions. The node definition of a bi-directional linked list contains references (pointers) to both the successor node (next node) and the predecessor node (previous node). Compared to unidirectional linked lists, bi-directional linked lists are more flexible and can be traversed in both directions, but accordingly require more memory space. + +=== "Python" + + ```python title="" + class ListNode: + """Bidirectional linked list node class"""" + def __init__(self, val: int): + self.val: int = val # node value + self.next: ListNode | None = None # reference to the successor node + self.prev: ListNode | None = None # reference to a predecessor node + ``` + +=== "C++" + + ```cpp title="" + /* Bidirectional linked list node structure */ + struct ListNode { + int val; // node value + ListNode *next; // Pointer to the successor node. + ListNode *prev; // pointer to the predecessor node + ListNode(int x) : val(x), next(nullptr), prev(nullptr) {} // constructor + }; + ``` + +=== "Java" + + ```java title="" + /* Bidirectional linked list node class */ + class ListNode { + int val; // node value + ListNode next; // Reference to the next node. + ListNode prev; // reference to the predecessor node + ListNode(int x) { val = x; } // Constructor + } + ``` + +=== "C#" + + ```csharp title="" + /* Bidirectional linked list node class */ + class ListNode { + int val; // node value + ListNode next; // Reference to the next node. + ListNode prev; // reference to the predecessor node + ListNode(int x) => val = x; // constructor + } + ``` + +=== "Go" + + ```go title="" + /* Bidirectional linked list node structure */ + type DoublyListNode struct { + Val int // Node value + Next *DoublyListNode // Pointer to the successor node. + Prev *DoublyListNode // pointer to the predecessor node + } + + // NewDoublyListNode initialization + func NewDoublyListNode(val int) *DoublyListNode { + return &DoublyListNode{ + Val: val, + Next: nil, + Prev: nil, + } + } + ``` + +=== "Swift" + + ```swift title="" + /* Bidirectional linked list node class */ + class ListNode { + var val: Int // node value + var next: ListNode? // Reference to the next node. + var prev: ListNode? // Reference to the predecessor node. + + init(x: Int) { // constructor + val = x + } + } + ``` + +=== "JS" + + ```javascript title="" + /* Bidirectional linked list node class */ + class ListNode { + constructor(val, next, prev) { + this.val = val === undefined ? 0 : val; // node value + this.next = next === undefined ? null : next; // reference to the successor node + this.prev = prev === undefined ? null : prev; // reference to the predecessor node + } + } + ``` + +=== "TS" + + ```typescript title="" + /* Bidirectional linked list node class */ + class ListNode { + val: number; + next: ListNode | null; + prev: ListNode | null; + constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) { + this.val = val === undefined ? 0 : val; // node value + this.next = next === undefined ? null : next; // reference to the successor node + this.prev = prev === undefined ? null : prev; // reference to the predecessor node + } + } + ``` + +=== "Dart" + + ```dart title="" + /* Bidirectional linked list node class */ + class ListNode { + int val; // node value + ListNode next; // Reference to the next node. + ListNode prev; // reference to the predecessor node + ListNode(this.val, [this.next, this.prev]); // Constructor + } + ``` + +=== "Rust" + + ```rust title="" + use std::rc::Rc; + use std::cell::RefCell; + + /* Bidirectional linked list node type */ + #[derive(Debug)] + struct ListNode { + val: i32, // node value + next: Option>>, // pointer to successor node + prev: Option>>, // pointer to predecessor node + } + + /* Constructors */ + impl ListNode { + fn new(val: i32) -> Self { + ListNode { + val, + next: None, + prev: None, + } + } + } + ``` + +=== "C" + + ```c title="" + /* Bidirectional linked list node structure */ + typedef struct ListNode { + int val; // node value + struct ListNode *next; // Pointer to the successor node. + struct ListNode *prev; // pointer to the predecessor node + } ListNode; + + /* Constructors */ + ListNode *newListNode(int val) { + ListNode *node, *next; + node = (ListNode *) malloc(sizeof(ListNode)); + node->val = val; + node->next = NULL; + node->prev = NULL; + return node; + } + ``` + +=== "Zig" + + ```zig title="" + // Bidirectional linked list node class + pub fn ListNode(comptime T: type) type { + return struct { + const Self = @This(); + + val: T = 0, // node value + next: ?*Self = null, // pointer to the successor node + prev: ?*Self = null, // pointer to the predecessor node + + // Constructor + pub fn init(self: *Self, x: i32) void { + self.val = x; + self.next = null; + self.prev = null; + } + }; + } + ``` + +![Common linked list types](linked_list.assets/linkedlist_common_types.png) + +## Typical Applications Of Linked Lists + +One-way linked lists are commonly used to implement data structures such as stacks, queues, hash tables and graphs. + +- **Stack vs. Queue**: when both insertion and deletion operations are performed at one end of a linked list, it exhibits a FIFO characteristic, corresponding to a stack; when insertion operations are performed at one end of a linked list and deletion operations are performed at the other end of a linked list, it exhibits a FIFO characteristic, corresponding to a queue. +- **Hash table**: the linked address method is one of the dominant solutions to hash collisions, in which all conflicting elements are put into a linked list. +- **Graph**: An adjacency list is a common way of representing a graph, in which each vertex of the graph is associated with a linked list, and each element of the linked list represents the other vertices connected to that vertex. + +Bidirectional linked lists are often used in scenarios where you need to quickly find the previous and next elements. + +- **Advanced Data Structures**: e.g. in red-black tree, B-tree, we need to access the parent node of a node, this can be done by keeping a reference to the parent node in the node, similar to a bidirectional linked list. +- **Browser History**: In web browsers, when a user clicks on the forward or back button, the browser needs to know the previous and next web pages that the user has visited. The property of bi-directional linked lists makes this operation easy. +- **LRU algorithm**: In a cache elimination algorithm (LRU), we need to quickly find the least recently used data as well as support adding and removing nodes quickly. This is where the use of bi-directional linked lists is very appropriate. + +Circular linked lists are often used in scenarios that require periodic operations, such as resource scheduling in operating systems. + +- **Time Slice Rotation Scheduling Algorithm**: In operating systems, the time slice rotation scheduling algorithm is a common CPU scheduling algorithm that involves looping over a set of processes. Each process is given a time slice and when the time slice runs out, the CPU switches to the next process. The operation of such a loop can then be realized by means of a circular linked list. +- **Data Buffers**: In some data buffer implementations, circular linked lists may also be used. For example, in audio and video players, the data stream may be divided into multiple buffer blocks and put into a circular linked list for seamless playback. diff --git a/docs-en/chapter_array_and_linkedlist/list.md b/docs-en/chapter_array_and_linkedlist/list.md new file mode 100644 index 0000000000..45e792c44e --- /dev/null +++ b/docs-en/chapter_array_and_linkedlist/list.md @@ -0,0 +1,872 @@ +# List + +"A list is an abstract data structure that represents an ordered collection of elements and supports operations such as accessing, modifying, adding, deleting, and traversing elements without the need to consider capacity constraints. Lists can be implemented based on linked lists or arrays. + +- A linked list can naturally be viewed as a list, which supports element addition, deletion, checking, and modification operations, and can be flexibly and dynamically expanded. +- Arrays also support adding, deleting, checking, and changing elements, but because their length is immutable, they can only be viewed as a list with a length limit. + +When implementing lists using arrays, **the immutable nature of the length causes lists to be less useful**. This is because we usually cannot determine in advance how much data we need to store, making it difficult to choose an appropriate list length. If the length is too small, it is likely to be unsatisfactory for use; if it is too large, it will result in wasted memory space. + +To solve this problem, we can use "dynamic array dynamic array" to implement the list. It inherits the advantages of arrays and can be dynamically expanded during the program. + +In fact, the standard library in many programming languages provides lists based on dynamic array implementations, such as `list` in Python, `ArrayList` in Java, `vector` in C++, and `List` in C#. For the rest of this discussion, we'll treat "lists" and "dynamic arrays" as equivalent concepts. + +## Common Operations On Lists + +### Initialize The List. + +We usually use the two initialization methods "without initial value" and "with initial value". + +=== "Python" + + ```python title="list.py" + # Initialize the list + # No initial value + nums1: list[int] = [] + # With initial values + nums: list[int] = [1, 3, 2, 5, 4] + ``` + +=== "C++" + + ```cpp title="list.cpp" + /* Initialize list */ + // Note that in C++, vectors are the nums described in this article. + // No initial value + vector nums1; + // With initial values + vector nums = { 1, 3, 2, 5, 4 }; + ``` + +=== "Java" + + ```java title="list.java" + /* Initialize list */ + // No initial value + List nums1 = new ArrayList<>(); + // with initial values (note that the element type of the array needs to be Integer[], the wrapper class for int[]) + Integer[] numbers = new Integer[] { 1, 3, 2, 5, 4 }; + List nums = new ArrayList<>(Arrays.asList(numbers)); + ``` + +=== "C#" + + ```csharp title="list.cs" + /* Initialize list */ + // No initial value + List nums1 = new(); + // With initial values + int[] numbers = new int[] { 1, 3, 2, 5, 4 }; + List nums = numbers.ToList(); + ``` + +=== "Go" + + ```go title="list_test.go" + /* Initialize list */ + // No initial value + nums1 := []int + // With initial values + nums := []int{1, 3, 2, 5, 4} + ``` + +=== "Swift" + + ```swift title="list.swift" + /* Initialize list */ + // No initial value + let nums1: [Int] = [] + // With initial values + var nums = [1, 3, 2, 5, 4] + ``` + +=== "JS" + + ```javascript title="list.js" + /* Initialize list */ + // No initial value + const nums1 = []; + // With initial values + const nums = [1, 3, 2, 5, 4]; + ``` + +=== "TS" + + ```typescript title="list.ts" + /* Initialize list */ + // No initial value + const nums1: number[] = []; + // With initial values + const nums: number[] = [1, 3, 2, 5, 4]; + ``` + +=== "Dart" + + ```dart title="list.dart" + /* Initialize list */ + // No initial value + List nums1 = []; + // With initial values + List nums = [1, 3, 2, 5, 4]; + ``` + +=== "Rust" + + ```rust title="list.rs" + /* Initialize list */ + // No initial value + let nums1: Vec = Vec::new(); + // With initial values + let nums: Vec = vec![1, 3, 2, 5, 4]; + ``` + +=== "C" + + ```c title="list.c" + // C does not provide a built-in dynamic array. + ``` + +=== "Zig" + + ```zig title="list.zig" + // Initialization list + var nums = std.ArrayList(i32).init(std.heap.page_allocator); + defer nums.deinit(); + try nums.appendSlice(&[_]i32{ 1, 3, 2, 5, 4 }); + ``` + +### Accessing Elements + +Lists are essentially arrays, so it is efficient to access and update elements in $O(1)$ time. + +=== "Python" + + ```python title="list.py" + # Access to elements + num: int = nums[1] # access the element at index 1 + + # Update elements + nums[1] = 0 # update element at index 1 to 0 + ``` + +=== "C++" + + ```cpp title="list.cpp" + /* Access to elements */ + int num = nums[1]; // access the element at index 1 + + /* Update elements */ + nums[1] = 0; // update element at index 1 to 0 + ``` + +=== "Java" + + ```java title="list.java" + /* Access to elements */ + int num = nums.get(1); // access the element at index 1 + + /* Update elements */ + nums.set(1, 0); // update the element at index 1 to 0 + ``` + +=== "C#" + + ```csharp title="list.cs" + /* Access to elements */ + int num = nums[1]; // access the element at index 1 + + /* Update elements */ + nums[1] = 0; // update element at index 1 to 0 + ``` + +=== "Go" + + ```go title="list_test.go" + /* Access to elements */ + num := nums[1] // access the element at index 1 + + /* Update elements */ + nums[1] = 0 // update the element at index 1 to 0 + ``` + +=== "Swift" + + ```swift title="list.swift" + /* Access to elements */ + let num = nums[1] // access the element at index 1 + + /* Update elements */ + nums[1] = 0 // update the element at index 1 to 0 + ``` + +=== "JS" + + ```javascript title="list.js" + /* Access to elements */ + const num = nums[1]; // access the element at index 1 + + /* Update elements */ + nums[1] = 0; // update element at index 1 to 0 + ``` + +=== "TS" + + ```typescript title="list.ts" + /* Access to elements */ + const num: number = nums[1]; // access the element at index 1 + + /* Update elements */ + nums[1] = 0; // update element at index 1 to 0 + ``` + +=== "Dart" + + ```dart title="list.dart" + /* Access to elements */ + int num = nums[1]; // access the element at index 1 + + /* Update elements */ + nums[1] = 0; // update element at index 1 to 0 + ``` + +=== "Rust" + + ```rust title="list.rs" + /* Access to elements */ + let num: i32 = nums[1]; // access the element at index 1 + /* Update elements */ + nums[1] = 0; // update element at index 1 to 0 + ``` + +=== "C" + + ```c title="list.c" + // C does not provide a built-in dynamic array. + ``` + +=== "Zig" + + ```zig title="list.zig" + // Accessing elements + var num = nums.items[1]; // access the element at index 1 + + // Update the element + nums.items[1] = 0; // update the element at index 1 to 0 + ``` + +### Inserting And Deleting Elements + +Compared to arrays, lists are free to add and remove elements. Adding elements to the end of the list has a time complexity of $O(1)$ , but inserting and deleting elements is still as efficient as an array with a time complexity of $O(n)$ . + +=== "Python" + + ```python title="list.py" + # Empty the list + nums.clear() + + # Add elements to the end + nums.append(1) + nums.append(3) + nums.append(2) + nums.append(5) + nums.append(4) + + # Insert elements in the middle + nums.insert(3, 6) # insert the number 6 at index 3 + + # Delete element + nums.pop(3) # remove the element at index 3 + ``` + +=== "C++" + + ```cpp title="list.cpp" + /* Empty list */ + nums.clear(); + + /* Add element at the end */ + nums.push_back(1); + nums.push_back(3); + nums.push_back(2); + nums.push_back(5); + nums.push_back(4); + + /* Insertion of elements in the middle */ + nums.insert(nums.begin() + 3, 6); // insert the number 6 at index 3 + + /* Delete element */ + nums.erase(nums.begin() + 3); // remove the element at index 3 + ``` + +=== "Java" + + ```java title="list.java" + /* Empty list */ + nums.clear(); + + /* Add element at the end */ + nums.add(1); + nums.add(3); + nums.add(2); + nums.add(5); + nums.add(4); + + /* Insertion of elements in the middle */ + nums.add(3, 6); // insert the number 6 at index 3 + + /* Delete element */ + nums.remove(3); // Remove the element at index 3. + ``` + +=== "C#" + + ```csharp title="list.cs" + /* Empty list */ + nums.Clear(); + + /* Add element at the end */ + nums.Add(1); + nums.Add(3); + nums.Add(2); + nums.Add(5); + nums.Add(4); + + /* Insertion of elements in the middle */ + nums.Insert(3, 6); + + /* Delete element */ + nums.RemoveAt(3); + ``` + +=== "Go" + + ```go title="list_test.go" + /* Empty list */ + nums = nil + + /* Add element at the end */ + nums = append(nums, 1) + nums = append(nums, 3) + nums = append(nums, 2) + nums = append(nums, 5) + nums = append(nums, 4) + + /* Insertion of elements in the middle */ + nums = append(nums[:3], append([]int{6}, nums[3:]...)...) // Insert the number 6 at index 3. + + /* Delete element */ + nums = append(nums[:3], nums[4:]...) // Delete the element at index 3. + ``` + +=== "Swift" + + ```swift title="list.swift" + /* Empty list */ + nums.removeAll() + + /* Add element at the end */ + nums.append(1) + nums.append(3) + nums.append(2) + nums.append(5) + nums.append(4) + + /* Insertion of elements in the middle */ + nums.insert(6, at: 3) // insert the number 6 at index 3 + + /* Delete element */ + nums.remove(at: 3) // remove the element at index 3 + ``` + +=== "JS" + + ```javascript title="list.js" + /* Empty list */ + nums.length = 0; + + /* Add element at the end */ + nums.push(1); + nums.push(3); + nums.push(2); + nums.push(5); + nums.push(4); + + /* Insertion of elements in the middle */ + nums.splice(3, 0, 6); + + /* Delete element */ + nums.splice(3, 1); + ``` + +=== "TS" + + ```typescript title="list.ts" + /* Empty list */ + nums.length = 0; + + /* Add element at the end */ + nums.push(1); + nums.push(3); + nums.push(2); + nums.push(5); + nums.push(4); + + /* Insertion of elements in the middle */ + nums.splice(3, 0, 6); + + /* Delete element */ + nums.splice(3, 1); + ``` + +=== "Dart" + + ```dart title="list.dart" + /* Empty list */ + nums.clear(); + + /* Add element at the end */ + nums.add(1); + nums.add(3); + nums.add(2); + nums.add(5); + nums.add(4); + + /* Insertion of elements in the middle */ + nums.insert(3, 6); // insert the number 6 at index 3 + + /* Delete element */ + nums.removeAt(3); // Remove the element at index 3. + ``` + +=== "Rust" + + ```rust title="list.rs" + /* Empty list */ + nums.clear(); + + /* Add element at the end */ + nums.push(1); + nums.push(3); + nums.push(2); + nums.push(5); + nums.push(4); + + /* Insertion of elements in the middle */ + nums.insert(3, 6); // insert the number 6 at index 3 + + /* Delete element */ + nums.remove(3); // Remove the element at index 3. + ``` + +=== "C" + + ```c title="list.c" + // C does not provide a built-in dynamic array. + ``` + +=== "Zig" + + ```zig title="list.zig" + // Empty the list. + nums.clearRetainingCapacity(); + + // Add elements to the end + try nums.append(1); + try nums.append(3); + try nums.append(2); + try nums.append(5); + try nums.append(4); + + // Insert elements in the middle + try nums.insert(3, 6); // insert the number 6 at index 3 + + // Delete the element + _ = nums.orderedRemove(3); // remove the element at index 3 + ``` + +### Traverse The List. + +As with arrays, lists can be traversed by index or by traversing the elements directly. + +=== "Python" + + ```python title="list.py" + # Iterate through the list by indexes + count = 0 + for i in range(len(nums)): + count += 1 + + # Iterate over the list elements directly + count = 0 + for num in nums: + count += 1 + ``` + +=== "C++" + + ```cpp title="list.cpp" + /* Iterate through the list by index */ + int count = 0; + for (int i = 0; i < nums.size(); i++) { + count++; + } + + /* Iterate directly over the list elements */ + count = 0; + for (int num : nums) { + count++; + } + ``` + +=== "Java" + + ```java title="list.java" + /* Iterate through the list by index */ + int count = 0; + for (int i = 0; i < nums.size(); i++) { + count++; + } + + /* Iterate directly over the list elements */ + count = 0; + for (int num : nums) { + count++; + } + ``` + +=== "C#" + + ```csharp title="list.cs" + /* Iterate through the list by index */ + int count = 0; + for (int i = 0; i < nums.Count; i++) { + count++; + } + + /* Iterate directly over the list elements */ + count = 0; + foreach (int num in nums) { + count++; + } + ``` + +=== "Go" + + ```go title="list_test.go" + /* Iterate through the list by index */ + count := 0 + for i := 0; i < len(nums); i++ { + count++ + } + + /* Iterate directly over the list elements */ + count = 0 + for range nums { + count++ + } + ``` + +=== "Swift" + + ```swift title="list.swift" + /* Iterate through the list by index */ + var count = 0 + for _ in nums.indices { + count += 1 + } + + /* Iterate directly over the list elements */ + count = 0 + for _ in nums { + count += 1 + } + ``` + +=== "JS" + + ```javascript title="list.js" + /* Iterate through the list by index */ + let count = 0; + for (let i = 0; i < nums.length; i++) { + count++; + } + + /* Iterate directly over the list elements */ + count = 0; + for (const num of nums) { + count++; + } + ``` + +=== "TS" + + ```typescript title="list.ts" + /* Iterate through the list by index */ + let count = 0; + for (let i = 0; i < nums.length; i++) { + count++; + } + + /* Iterate directly over the list elements */ + count = 0; + for (const num of nums) { + count++; + } + ``` + +=== "Dart" + + ```dart title="list.dart" + /* Iterate through the list by index */ + int count = 0; + for (int i = 0; i < nums.length; i++) { + count++; + } + + /* Iterate directly over the list elements */ + count = 0; + for (int num in nums) { + count++; + } + ``` + +=== "Rust" + + ```rust title="list.rs" + /* Iterate through the list by index */ + let mut count = 0; + for (index, value) in nums.iter().enumerate() { + count += 1; + } + + /* Iterate directly over the list elements */ + let mut count = 0; + for value in nums.iter() { + count += 1; + } + ``` + +=== "C" + + ```c title="list.c" + // C does not provide a built-in dynamic array. + ``` + +=== "Zig" + + ```zig title="list.zig" + // Iterate through the list by indexes + var count: i32 = 0; + var i: i32 = 0; + while (i < nums.items.len) : (i += 1) { + count += 1; + } + + // Iterate directly over the list elements. + count = 0; + for (nums.items) |_| { + count += 1; + } + ``` + +### Splice The List + +Given a new list `nums1` , we can splice that list onto the end of the original list. + +=== "Python" + + ```python title="list.py" + # Splicing two lists together + nums1: list[int] = [6, 8, 7, 10, 9] + nums += nums1 # splice list nums1 after nums + ``` + +=== "C++" + + ```cpp title="list.cpp" + /* Splice two lists */ + vector nums1 = { 6, 8, 7, 10, 9 }; + // Splice the list nums1 after nums. + nums.insert(nums.end(), nums1.begin(), nums1.end()); + ``` + +=== "Java" + + ```java title="list.java" + /* Splice two lists */ + List nums1 = new ArrayList<>(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 })); + nums.addAll(nums1); // splice the list nums1 after nums + ``` + +=== "C#" + + ```csharp title="list.cs" + /* Splice two lists */ + List nums1 = new() { 6, 8, 7, 10, 9 }; + nums.AddRange(nums1); // splice the list nums1 after nums + ``` + +=== "Go" + + ```go title="list_test.go" + /* Splice two lists */ + nums1 := []int{6, 8, 7, 10, 9} + nums = append(nums, nums1...) // Append the list nums1 to the list after nums. + ``` + +=== "Swift" + + ```swift title="list.swift" + /* Splice two lists */ + let nums1 = [6, 8, 7, 10, 9] + nums.append(contentsOf: nums1) // splice list nums1 after nums + ``` + +=== "JS" + + ```javascript title="list.js" + /* Splice two lists */ + const nums1 = [6, 8, 7, 10, 9]; + nums.push(...nums1); // splice the list nums1 after nums + ``` + +=== "TS" + + ```typescript title="list.ts" + /* Splice two lists */ + const nums1: number[] = [6, 8, 7, 10, 9]; + nums.push(...nums1); // splice the list nums1 after nums + ``` + +=== "Dart" + + ```dart title="list.dart" + /* Splice two lists */ + List nums1 = [6, 8, 7, 10, 9]; + nums.addAll(nums1); // splice the list nums1 after nums + ``` + +=== "Rust" + + ```rust title="list.rs" + /* Splice two lists */ + let nums1: Vec = vec![6, 8, 7, 10, 9]; + nums.extend(nums1); + ``` + +=== "C" + + ```c title="list.c" + // C does not provide a built-in dynamic array. + ``` + +=== "Zig" + + ```zig title="list.zig" + // Splicing two lists together + var nums1 = std.ArrayList(i32).init(std.heap.page_allocator); + defer nums1.deinit(); + try nums1.appendSlice(&[_]i32{ 6, 8, 7, 10, 9 }); + try nums.insertSlice(nums.items.len, nums1.items); // splice list nums1 after nums + ``` + +### Sort The List + +Once the list is sorting algorithm is complete, we can use the "binary search" and "double pointer" algorithms that are often examined in array-based algorithmic problems. + +=== "Python" + + ```python title="list.py" + # Sort the list + nums.sort() # After sorting, list elements are ordered from smallest to largest + ``` + +=== "C++" + + ```cpp title="list.cpp" + /* Sort list */ + sort(nums.begin(), nums.end()); // after sorting, the list elements are sorted from smallest to largest + ``` + +=== "Java" + + ```java title="list.java" + /* Sort list */ + Collections.sort(nums); // after sorting, the list elements are ordered from smallest to largest + ``` + +=== "C#" + + ```csharp title="list.cs" + /* Sort list */ + nums.Sort(); // After sorting, the list elements are ordered from smallest to largest + ``` + +=== "Go" + + ```go title="list_test.go" + /* Sort list */ + sort.Ints(nums) // After sorting, the list elements are sorted from smallest to largest + ``` + +=== "Swift" + + ```swift title="list.swift" + /* Sort list */ + nums.sort() // After sorting, the list elements are arranged from smallest to largest + ``` + +=== "JS" + + ```javascript title="list.js" + /* Sort list */ + nums.sort((a, b) => a - b); // after sorting, the list elements are ordered from smallest to largest + ``` + +=== "TS" + + ```typescript title="list.ts" + /* Sort list */ + nums.sort((a, b) => a - b); // after sorting, the list elements are ordered from smallest to largest + ``` + +=== "Dart" + + ```dart title="list.dart" + /* Sort list */ + nums.sort(); // after sorting, the list elements are sorted from smallest to largest + ``` + +=== "Rust" + + ```rust title="list.rs" + /* Sort list */ + nums.sort(); // after sorting, the list elements are sorted from smallest to largest + ``` + +=== "C" + + ```c title="list.c" + // C does not provide a built-in dynamic array. + ``` + +=== "Zig" + + ```zig title="list.zig" + // Sort the list + std.sort.sort(i32, nums.items, {}, comptime std.sort.asc(i32)); + ``` + +## List Implementations + +Many programming languages provide built-in lists, such as Java, C++, Python, and so on. Their implementation is more complex, and the setting of each parameter is also very delicate, such as the initial capacity, expansion multiplier and so on. Interested readers can check the source code to learn. + +To deepen the solution of how lists work, let's try to implement a simple version of a list, including the following three key designs. + +- **Initial Capacity**: pick a reasonable initial capacity for the array. In this example, we choose 10 as the initial capacity. +- **Quantity Record**: declares a variable `size` that is used to record the current number of elements in the list and is updated in real time as elements are inserted and deleted. Based on this variable, we can locate the tail of the list, as well as determine if it needs to be expanded. +- **Expansion Mechanism**: If the list capacity is full when an element is inserted, it needs to be expanded. First, a larger array is created based on the expansion factor, and then all the elements of the current array are moved to the new array in turn. In this example, we specify that the array should be expanded to twice its previous size each time. + +```src +[file]{my_list}-[class]{my_list}-[func]{} +``` diff --git a/docs-en/chapter_array_and_linkedlist/summary.md b/docs-en/chapter_array_and_linkedlist/summary.md new file mode 100644 index 0000000000..2f6010195d --- /dev/null +++ b/docs-en/chapter_array_and_linkedlist/summary.md @@ -0,0 +1,78 @@ +# Summary + +### Highlights + +- Arrays and linked lists are two basic data structures that represent two ways of storing data in computer memory: continuous space storage and decentralized space storage, respectively. The characteristics of the two present complementary properties. +- Arrays support random access and take up less memory; however, inserting and deleting elements is inefficient and the length is immutable after initialization. +- The linked list achieves efficient node insertion and deletion by changing references (pointers) and can flexibly adjust its length; however, node access is inefficient and occupies more memory. +- Common types of linked lists include unidirectional linked lists, circular linked lists, and bidirectional linked lists, which have their own application scenarios. +- A list is an ordered collection of elements that supports adding, deleting, checking and changing, usually based on a dynamic array, which retains the advantages of arrays and can be flexibly adjusted in length. +- The advent of lists has dramatically increased the utility of arrays, but may result in some wasted memory space. + +### Q & A + +!!! Question "Does storing an array on the stack versus storing it on the heap have an effect on time efficiency and space efficiency?" + + Arrays stored on both the stack and the heap are stored in contiguous memory space, and the efficiency of data manipulation is essentially the same. However, the stack and heap have their own characteristics that lead to the following differences. + + 1. Allocation and release efficiency: the stack is a smaller piece of memory, and allocation is done automatically by the compiler; whereas heap memory is relatively larger, can be allocated dynamically in code, and is more prone to fragmentation. As a result, allocation and release operations on the heap are usually slower than on the stack. + 2. Size Limit: The stack memory is relatively small, and the size of the heap is generally limited by the available memory. Therefore heap is more suitable for storing large arrays. + 3. Flexibility: the size of an array on the stack needs to be determined at compile time, while the size of an array on the heap can be determined dynamically at runtime. + +!!! question "Why do arrays require elements of the same type, but in linked lists the same type is not emphasized?" + + A linked list consists of nodes, which are connected to each other by references (pointers), and each node can store different types of data, such as int, double, string, object, and so on. + + In contrast, the elements of an array must be of the same type so that the offset can be calculated to obtain the corresponding element position. For example, if the array contains both int and long types, and a single element occupies 4 bytes and 8 bytes respectively, then the following formula cannot be used to calculate the offset, because the array contains elements of both lengths. + + ```shell + # Element memory address = array memory address + element length * element index + ``` + +!!! question "Do I need to set `P.next` to $\text{None}$ after deleting a node?" + + It is possible to leave `P.next` unchanged. From the point of view of the linked list, traversing from the head node to the tail node no longer encounters `P`. This means that the node `P` has been deleted from the linked list, and the node `P` will not be affected by where it points. + + From a garbage collection point of view, for languages like Java, Python, Go, etc. that have automatic garbage collection, whether or not a node `P` is reclaimed depends on whether or not a reference to it still exists, not on the value of `P.next`. In languages like C and C++, we need to manually free node memory. + +!!! question "The time complexity of insertion and deletion operations in a linked list is $O(1)$ . But before adding and deleting, you need to $O(n)$ find the element, so why is the time complexity not $O(n)$?" + + It is true that it is $O(n)$ if the elements are looked up first and then deleted. However, the advantage of $O(1)$ addition and deletion of linked lists can be realized in other applications. For example, a double-ended queue is suitable for implementation using a linked list, where we maintain a pointer variable always pointing to the head node, the tail node, and each insertion and deletion operation is $O(1)$ . + +!!! question "In the figure "Definition and storage of a linked list", does the light blue pointer to the storage node take up a block of memory? Or does it take up half of the memory address and half of the node value?" + + The graphs in the text are only qualitative representations; quantitative representations need to be analyzed on a case-by-case basis. + + - Different types of node values take up different amounts of space, such as int, long, double, and instance objects. + - The amount of memory space occupied by pointer variables depends on the operating system and compilation environment used, mostly 8 bytes or 4 bytes. + +!!! question "Is adding an element to the end of the list $O(1)$ at all times?" + + If you add an element beyond the length of the list, you need to expand the list before adding it. The system will request a new block of memory and carry over all the elements of the original list, at which point the time complexity will be $O(n)$ . + +!!! question ""The advent of lists greatly enhances the utility of arrays, but the side effect is that they cause some memory space to be wasted", where space wasted is the memory taken up by extra added variables such as capacity, length, and expansion multiplier?" + + There are two main implications of wasted space here: on the one hand, lists are set to an initial length, and we don't necessarily need to use that much. On the other hand, in order to prevent frequent expansions, expansions are usually multiplied by a factor, like $\times 1.5$ . This also leaves a lot of empty spaces, and we usually can't fill them completely. + +!!! question "After initializing `n = [1, 2, 3]` in Python, the addresses of these 3 elements are linked, but initializing `m = [2, 1, 3]` reveals that the ids of each of these elements are not contiguous, but are the same as those in `n`. These elements are not contiguous, so is `m` still an array?" + + If we replace the list elements with linked list nodes `n = [n1, n2, n3, n4, n5]` , normally these five node objects are also scattered in memory. However, given a list index, we can still get the node memory address in $O(1)$ time to access the corresponding node. This is because the references to the nodes are stored in the array, not the nodes themselves. + + Unlike many languages, in Python numbers are also wrapped as objects, and what is stored in the list is not the number itself, but a reference to the number. As a result, you'll find two arrays of identical numbers with the same id, and the memory addresses of those numbers are needlessly contiguous. + +!!! question "C++ STL inside `std::list` already implements bi-directional linked lists, but it seems that some algorithm books don't really use this directly, is there some limitation?" + + On the one hand, we tend to favor the use of arrays for implementing algorithms and use linked lists only when necessary for two main reasons. + + - Space overhead: `std::list` usually takes more space than `std::vector` since each element requires two extra pointers (one for the previous element and one for the next). + - Cache-unfriendly: `std::list` has a low utilization of cache because the data is not stored continuously. In general, `std::vector` has better performance. + + On the other hand, the cases where it is necessary to use linked lists are mainly binary trees and graphs. Stacks and queues tend to use the `stack` and `queue` provided by the programming language rather than linked lists. + +!!! question "Does the initialize list `res = [0] * self.size()` operation cause each element of `res` to reference the same address?" + + It won't. But two-dimensional arrays can have this problem, for example initializing a two-dimensional list `res = [[0] * self.size()]` references the same list `[0]` multiple times. + +!!! question "In deleting a node, is it necessary to break the reference pointing between that node and its successor?" + + From a data structure and algorithm (do-it-yourself) point of view, it doesn't matter if you don't disconnect, as long as you make sure the logic of the program is correct. From the point of view of the standard library, it is safer and more logical to disconnect. 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vnL!QjZQ_vfBM>XoODuuh)g+P!58h)SLS+`u2<}J!+eE4" + ![try and backtrack](backtracking_algorithm.assets/preorder_find_paths_step1.png) + +=== "<2>" + ![preorder_find_paths_step2](backtracking_algorithm.assets/preorder_find_paths_step2.png) + +=== "<3>" + ![preorder_find_paths_step3](backtracking_algorithm.assets/preorder_find_paths_step3.png) + +=== "<4>" + ![preorder_find_paths_step4](backtracking_algorithm.assets/preorder_find_paths_step4.png) + +=== "<5>" + ![preorder_find_paths_step5](backtracking_algorithm.assets/preorder_find_paths_step5.png) + +=== "<6>" + ![preorder_find_paths_step6](backtracking_algorithm.assets/preorder_find_paths_step6.png) + +=== "<7>" + ![preorder_find_paths_step7](backtracking_algorithm.assets/preorder_find_paths_step7.png) + +=== "<8>" + ![preorder_find_paths_step8](backtracking_algorithm.assets/preorder_find_paths_step8.png) + +=== "<9>" + ![preorder_find_paths_step9](backtracking_algorithm.assets/preorder_find_paths_step9.png) + +=== "<10>" + ![preorder_find_paths_step10](backtracking_algorithm.assets/preorder_find_paths_step10.png) + +=== "<11>" + ![preorder_find_paths_step11](backtracking_algorithm.assets/preorder_find_paths_step11.png) + +## Pruning + +Complex backtracking problems usually contain one or more constraints, **Constraints can often be used for "pruning"**. + +!!! question "Example 3" + + Search for all nodes in the binary tree with value $7$, return the path from the root node to these nodes, **and require that the path does not contain nodes with value $3$**. + +In order to satisfy the above constraints, **we need to add the pruning operation**: if a node with the value $3$ is encountered during the search process, it is returned early and the search is stopped from continuing. + +```src +[file]{preorder_traversal_iii_compact}-[class]{}-[func]{pre_order} +``` + +Pruning is a very visual term. As shown in the figure below, during the search process, **we "cut off" the search branches that do not satisfy the constraints** to avoid many meaningless attempts, thus improving the search efficiency. + +![pruning based on constraints](backtracking_algorithm.assets/preorder_find_constrained_paths.png) + +## Framework Code + +Next, we try to distill the main framework of "try, fallback, pruning" of backtracking to improve the generality of the code. + +In the following framework code, `state` denotes the current state of the problem, and `choices` denotes the choices that can be made in the current state. + +=== "Python" + + ```python title="" + def backtrack(state: State, choices: list[choice], res: list[state]): + """Backtracking algorithm framework"""" + # Determine if it's a solution + if is_solution(state): + # Recorded solutions + record_solution(state, res) + # Stop continuing the search + return + # Iterate over all choices + for choice in choices: + # Pruning: determining whether a choice is legal or not + if is_valid(state, choice): + # Trying: making choices, updating state + make_choice(state, choice) + backtrack(state, choices, res) + # Fallback: undo the selection and revert to the previous state + undo_choice(state, choice) + ``` + +=== "C++" + + ```cpp title="" + /* Backtracking algorithm framework */ + void backtrack(State *state, vector &choices, vector &res) { + // Determine whether it is a solution or not + if (isSolution(state)) { + // Record the solution + recordSolution(state, res); + // Stop continuing the search + return; + } + // Iterate over all choices + for (Choice choice : choices) { + // Pruning: determining whether a choice is legal or not + if (isValid(state, choice)) { + // Try: make a choice, update the state + makeChoice(state, choice); + backtrack(state, choices, res); + // Fallback: undo the selection and revert to the previous state + undoChoice(state, choice); + } + } + } + ``` + +=== "Java" + + ```java title="" + /* Backtracking algorithm framework */ + void backtrack(State state, List choices, List res) { + // Determine whether it is a solution or not + if (isSolution(state)) { + // Record the solution + recordSolution(state, res); + // Stop continuing the search + return; + } + // Iterate over all choices + for (Choice choice : choices) { + // Pruning: determining whether a choice is legal or not + if (isValid(state, choice)) { + // Try: make a choice, update the state + makeChoice(state, choice); + backtrack(state, choices, res); + // Fallback: undo the selection and revert to the previous state + undoChoice(state, choice); + } + } + } + ``` + +=== "C#" + + ```csharp title="" + /* Backtracking algorithm framework */ + void Backtrack(State state, List choices, List res) { + // Determine whether it is a solution or not + if (IsSolution(state)) { + // Record the solution + RecordSolution(state, res); + // Stop continuing the search + return; + } + // Iterate over all choices + foreach (Choice choice in choices) { + // Pruning: determining whether a choice is legal or not + if (IsValid(state, choice)) { + // Try: make a choice, update the state + MakeChoice(state, choice); + Backtrack(state, choices, res); + // Fallback: undo the selection and revert to the previous state + UndoChoice(state, choice); + } + } + } + ``` + +=== "Go" + + ```go title="" + /* Backtracking algorithm framework */ + func backtrack(state *State, choices []Choice, res *[]State) { + // Determine whether it is a solution or not + if isSolution(state) { + // Record the solution + recordSolution(state, res) + // Stop continuing the search + return + } + // Iterate over all choices + for _, choice := range choices { + // Pruning: determining whether a choice is legal or not + if isValid(state, choice) { + // Try: make a choice, update the state + makeChoice(state, choice) + backtrack(state, choices, res) + // Fallback: undo the selection and revert to the previous state + undoChoice(state, choice) + } + } + } + ``` + +=== "Swift" + + ```swift title="" + /* Backtracking algorithm framework */ + func backtrack(state: inout State, choices: [Choice], res: inout [State]) { + // Determine whether it is a solution or not + if isSolution(state: state) { + // Record the solution + recordSolution(state: state, res: &res) + // Stop continuing the search + return + } + // Iterate over all choices + for choice in choices { + // Pruning: determining whether a choice is legal or not + if isValid(state: state, choice: choice) { + // Try: make a choice, update the state + makeChoice(state: &state, choice: choice) + backtrack(state: &state, choices: choices, res: &res) + // Fallback: undo the selection and revert to the previous state + undoChoice(state: &state, choice: choice) + } + } + } + ``` + +=== "JS" + + ```javascript title="" + /* Backtracking algorithm framework */ + function backtrack(state, choices, res) { + // Determine whether it is a solution or not + if (isSolution(state)) { + // Record the solution + recordSolution(state, res); + // Stop continuing the search + return; + } + // Iterate over all choices + for (let choice of choices) { + // Pruning: determining whether a choice is legal or not + if (isValid(state, choice)) { + // Try: make a choice, update the state + makeChoice(state, choice); + backtrack(state, choices, res); + // Fallback: undo the selection and revert to the previous state + undoChoice(state, choice); + } + } + } + ``` + +=== "TS" + + ```typescript title="" + /* Backtracking algorithm framework */ + function backtrack(state: State, choices: Choice[], res: State[]): void { + // Determine whether it is a solution or not + if (isSolution(state)) { + // Record the solution + recordSolution(state, res); + // Stop continuing the search + return; + } + // Iterate over all choices + for (let choice of choices) { + // Pruning: determining whether a choice is legal or not + if (isValid(state, choice)) { + // Try: make a choice, update the state + makeChoice(state, choice); + backtrack(state, choices, res); + // Fallback: undo the selection and revert to the previous state + undoChoice(state, choice); + } + } + } + ``` + +=== "Dart" + + ```dart title="" + /* Backtracking algorithm framework */ + void backtrack(State state, List, List res) { + // Determine whether it is a solution or not + if (isSolution(state)) { + // Record the solution + recordSolution(state, res); + // Stop continuing the search + return; + } + // Iterate over all choices + for (Choice choice in choices) { + // Pruning: determining whether a choice is legal or not + if (isValid(state, choice)) { + // Try: make a choice, update the state + makeChoice(state, choice); + backtrack(state, choices, res); + // Fallback: undo the selection and revert to the previous state + undoChoice(state, choice); + } + } + } + ``` + +=== "Rust" + + ```rust title="" + /* Backtracking algorithm framework */ + fn backtrack(state: &mut State, choices: &Vec, res: &mut Vec) { + // Determine whether it is a solution or not + if is_solution(state) { + // Record the solution + record_solution(state, res); + // Stop continuing the search + return; + } + // Iterate over all choices + for choice in choices { + // Pruning: determining whether a choice is legal or not + if is_valid(state, choice) { + // Try: make a choice, update the state + make_choice(state, choice); + backtrack(state, choices, res); + // Fallback: undo the selection and revert to the previous state + undo_choice(state, choice); + } + } + } + ``` + +=== "C" + + ```c title="" + /* Backtracking algorithm framework */ + void backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) { + // Determine if it's a solution. + if (isSolution(state)) { + // Record the solution + recordSolution(state, res, numRes); + // Stop continuing the search + return; + } + // Iterate over all choices + for (int i = 0; i < numChoices; i++) { + // Pruning: determining whether a choice is legal or not + if (isValid(state, &choices[i])) { + // Try: make a choice, update the state + makeChoice(state, &choices[i]); + backtrack(state, choices, numChoices, res, numRes); + // Fallback: undo the selection and revert to the previous state + undoChoice(state, &choices[i]); + } + } + } + ``` + +=== "Zig" + + ```zig title="" + + ``` + +Next, we solve example 3 based on the framework code. The state `state` is the node traversal path, the choices `choices` are the left-child nodes and right-child nodes of the current node, and the result `res` is the list of paths. + +```src +[file]{preorder_traversal_iii_template}-[class]{}-[func]{backtrack} +``` + +According to the question, we should continue the search after finding the node with the value $7$, **so we need to remove the `return` statement after recording the solution**. The figure below compares the search process of keeping or removing `return` statements. + +![Comparison of search process for retaining and removing return](backtracking_algorithm.assets/backtrack_remove_return_or_not.png) + +The code implementation based on the backtracking algorithm framework is more general, although it seems verbose, than the code implementation based on the preorder traversal. In fact, **many backtracking problems can be solved under this framework**. We just need to define `state` and `choices` according to the specific problem and implement the methods in the framework. + +## Commonly Used Terms + +In order to analyze the algorithm problem more clearly, we summarize the meaning of the common terms used in the backtracking algorithm and give corresponding examples against Example 3. + +

Table   Common backtracking algorithm terminology

+ +| Nouns | Definitions | Example 3 | +| --------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------- | +| Solution Solution | A solution is an answer that satisfies a specific condition of the problem, there may be one or more | All paths from the root node to the node $7$ that satisfy the constraints | +| Constraint Constraint | A constraint is a condition in a problem that restricts the feasibility of a solution, usually used for pruning | The path does not contain a node $3$ | +| State State | The state indicates the situation of the problem at a given point in time, including the choices that have been made | The path of the nodes that are currently visited, i.e., the list of `path` nodes | +| Attempt | Attempt is the process of exploring the solution space based on the available choices, including making a choice, updating the state, and checking if it is a solution | Recursion accesses the left (right) child node, adds the node to the `path` , and determines if the node's value is $7$ | +| Backtracking | Backtracking means that when a state that does not satisfy the constraints is encountered, the choices made earlier are undone and return to the previous state | The search is terminated when the leaf node is crossed, the end node is accessed, and the node with the value $3$ is encountered, and the function returns | +| Pruning Pruning | Pruning is a method to avoid meaningless search paths based on problem characteristics and constraints, which improves the search efficiency | When a node with value $3$ is encountered, then terminate to continue the search | + +!!! tip + + The concepts of problem, solution, and state are generic and are covered in algorithms such as divide and conquer, backtracking, dynamic programming, and greedy algorithms. + +## Strengths And Limitations + +The backtracking algorithm is essentially a depth-first search algorithm that tries all possible solutions until it finds a solution that satisfies the conditions. The advantage of this method is that it is able to find all possible solutions and is highly efficient with reasonable pruning operations. + +However, when dealing with large-scale or complex problems, **backtracking algorithms may run with unacceptable efficiency**. + +- **Time**: backtracking algorithms usually need to traverse all possibilities of the state space, and the time complexity can be of exponential or multiplicative order. +- **Space**: the need to save the current state (e.g., path, auxiliary variables for pruning, etc.) in recursion calls can make space requirements large when the depth is large. + +Even so, **backtracking algorithms are still the best solution for certain search problems and constraint satisfaction problems**. For these problems, since it is impossible to predict which choices generate valid solutions, we must iterate over all possible choices. In this case, **the key is how to optimize for efficiency**. There are two common efficiency optimization methods. + +- **pruning**: saving time and space by avoiding searching for paths that will definitely not yield a solution. +- **Heuristic search**: some strategy or estimate is introduced into the search process so as to prioritize the search for paths that are most likely to yield valid solutions. + +## Backtracking Through Typical Examples + +Backtracking algorithms can be used to solve many search problems, constraint satisfaction problems and combinatorial optimization problems. + +**Search Problems**: the goal of this type of problem is to find solutions that satisfy specific conditions. + +- Permutations problem: Given a set, find all possible permutations of it. +- Subset-sum problem: Given a set and a target sum, find all subsets of the set whose sum is the target sum. +- Hanota problem: Given three columns and a series of disks of different sizes, it is required to move all the disks from one column to another, one at a time, and without placing the larger disk on top of the smaller disk. + +**Constraint Satisfaction Problems**: the goal of these problems is to find a solution that satisfies all the constraints. + +- $n$ Queens: Place $n$ queens on a $n \times n$ board so that they do not attack each other. +- Sudoku: Fill the grid of $9 \times 9$ with the numbers $1$ ~ $9$ so that the numbers in each row, column and each $3 \times 3$ subgrid are not repeated. +- Graph Coloring Problem: Given an undirected graph, color each vertex of the graph with the least number of colors such that adjacent vertices are different colors. + +**Combinatorial Optimization Problems**: the goal of these problems is to find an optimal solution in a combinatorial space that satisfies certain conditions. + +- 0-1 Knapsack problem: Given a set of items and a knapsack, each item has a certain value and weight, and it is required to select the items such that the total value is maximized within the capacity constraints of the knapsack. +- Traveler's problem: In a graph, find the shortest path from a point that visits all other points exactly once and then returns to the starting point. +- Maximum Group Problem: Given an undirected graph, find the largest complete subgraph where any two vertices in the subgraph are connected by an edge. + +Note that for many combinatorial optimization problems, backtracking is not an optimal solution. + +- 0-1 Knapsack problems are often solved using dynamic programming to achieve greater time efficiency. +- The traveler is a well-known NP-Hard problem, and commonly used solutions include genetic algorithms and ant colony algorithms. +- The maximal group problem is a classical problem in graph theory and can be solved with heuristic algorithms such as greedy algorithms. diff --git a/docs-en/chapter_backtracking/index.md b/docs-en/chapter_backtracking/index.md new file mode 100644 index 0000000000..21a3e24f07 --- /dev/null +++ b/docs-en/chapter_backtracking/index.md @@ -0,0 +1,13 @@ +# Backtracking + +
+ +![backtracking](../assets/covers/chapter_backtracking.jpg){ width="600" } + +
+ +!!! abstract + + Like explorers in a labyrinth, we may encounter difficulties along the way. + + The power of retrospection allows us to start over, keep trying, and eventually find the exit to the light. diff --git a/docs-en/chapter_backtracking/n_queens_problem.assets/n_queens_cols_diagonals.png b/docs-en/chapter_backtracking/n_queens_problem.assets/n_queens_cols_diagonals.png new file mode 100644 index 0000000000000000000000000000000000000000..cd290ea08555046593cf49c0527719d6de9eaf35 GIT binary patch literal 36176 zcmb??_d^q1(C%(R=)FskDn&#P1W^P6i1Z=?QlyCVB8c>2f^<+pI)Z?7DbkD7fFQji zy-RPQ_aqnI@4NRexcMa~dv>0A=9!sf_v}fSmWCo3F%vNW0A$KaPqYC5b~S_y5JIos zPBR59u0EV=sp%--aJc>beR%ffo1i)Ey3)%_3N2IHfI;n=o8H1 z=HA-IZdl6F!V)?uDe3GI+c~i1{&8V&Y1g)TXJ%%`KW?#Sad%>7r+0Wut8i;&d#|iv zJ+o|WdH=HL*G5p{znTg3-<{3%_4W5XZNIxW6LMFkrl!#Q7v-bqv8f%u65p}0vHri? zqbs{|Ihzjo_G@cvX+Kv(Tetu8ZPv7`cg*jNOl<2FZx0O(t!|#buH1fEw*7DQ>|4)P z-pqJ!Z*Rf)apa%v&d$!!t@HQMilpO-%abM&~Y8xTj-0j#+~Bg;u-YC*v#?J#8J=4;lGvLskOb%<+GZa 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Table   Full Arrangement Example

+ +| Input Array | All Arrangements | +| :---------- | :----------------------------------------------------------------- | +| $[1]$ | $[1]$ | +| $[1, 2]$ | $[1, 2], [2, 1]$ | +| $[1, 2, 3]$ | $[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]$ | + +## The Case Of Unequal Elements + +!!! question + + Inputs an array of integers, the array contains no duplicate elements and returns all possible permutations. + +From the backtracking algorithm point of view, **we can imagine the process of generating an arrangement as the result of a series of choices**. Suppose the input array is $[1, 2, 3]$ , and if we first select $1$, then $3$, and finally $2$ , we obtain the arrangement $[1, 3, 2]$ . Fallback means to undo a choice and continue to try other choices later. + +From the point of view of the backtracking code, the candidate set `choices` is all the elements in the input array, and the state `state` is the elements that have been selected until now. Note that each element is only allowed to be selected once, **so all elements in `state` should be unique**. + +As shown in the figure below, we can expand the search process into a recursion tree where each tree node represents the current state `state` . Starting from the root node, we reach the leaf nodes after three rounds of selection, and each leaf node corresponds to an arrangement. + +![fully-aligned recursion tree](permutations_problem.assets/permutations_i.png) + +### Repeat Selection Pruning + +In order to realize that each element is selected only once, we consider introducing a Boolean array `selected` , where `selected[i]` indicates whether `choices[i]` has been selected or not, and implement the following pruning operation based on it. + +- After making the selection `choice[i]`, we assign `selected[i]` to $\text{True}$ to represent that it has been selected. +- When traversing the choices list `choices`, skip all nodes that have already been selected, i.e. pruning. + +As shown in the figure below, suppose we choose 1 in the first round, 3 in the second round, and 2 in the third round, then we need to cut off the branch of element 1 in the second round, and the branches of element 1 and element 3 in the third round. + +![Example of fully-aligned pruning](permutations_problem.assets/permutations_i_pruning.png) + +Observing the figure above, we find that this pruning operation reduces the search space size from $O(n^n)$ to $O(n!)$ . + +### Code Implementation + +With the above information in mind, we can do the "fill-in-the-blanks" in the framework code. In order to reduce the number of lines of code, instead of implementing each function in the framework code separately, we will expand them in the `backtrack()` function. + +```src +[file]{permutations_i}-[class]{}-[func]{permutations_i} +``` + +## Consider The Case Of Equal Elements + +!!! question + + Input an array of integers, **The array may contain duplicate elements**. Return all non-duplicate permutations. + +Suppose the input array is $[1, 1, 2]$ . In order to easily distinguish between the two repeated elements $1$ , we denote the second $1$ as $\hat{1}$ . + +As shown in the figure below, half of the alignments generated by the above method are duplicates. + +![duplicate alignment](permutations_problem.assets/permutations_ii.png) + +So how do you remove duplicate alignments? Most directly, consider de-duplicating the alignments directly with the help of a hash table. However, this is not elegant enough, **because the search branch that generates the duplicate alignments is unnecessary and should be recognized and pruned in advance**, which further improves the efficiency of the algorithm. + +### Pruning Of Equivalent Elements + +Observe the figure below, in the first round, choosing $1$ or choosing $\hat{1}$ is equivalent, and all alignments generated under these two choices are duplicates. Therefore $\hat{1}$ should be pruned. + +Similarly, after selecting $2$ in the first round, $1$ and $\hat{1}$ in the second round of selection will also generate duplicate branches, so $\hat{1}$ in the second round should also be pruned. + +Essentially, **our goal is to ensure that multiple equal elements are selected only once in a given round of selection**. + +![duplicate alignment pruning](permutations_problem.assets/permutations_ii_pruning.png) + +### Code Implementation + +Building on the code from the previous question, we consider opening a hash table `duplicated` in each round of selection to keep track of the elements that have been tried in that round and pruning duplicates. + +```src +[file]{permutations_ii}-[class]{}-[func]{permutations_ii} +``` + +Assuming that the elements are not the same as each other two by two, there are $n$ a total of $n!$ permutations (factorials) of the elements; when recording the result, a list of length $n$ needs to be copied, using $O(n)$ time. **Therefore the time complexity is $O(n!n)$** . + +Maximum recursion depth is $n$, using $O(n)$ stack frame space. `selected` uses $O(n)$ space. There are at most $n$ `duplicated` at the same moment, using $O(n^2)$ space. **Therefore the space complexity is $O(n^2)$** . + +### Two Pruning Comparisons + +Note that although both `selected` and `duplicated` are used for pruning, they have different goals. + +- **Repeated selection pruning**: there is only one `selected` for the entire search. It records which elements are contained in the current state, and serves to prevent any of the elements in `choices` from recurring in `state`. +- **Equal element pruning**: each round of selection (i.e., each call to the `backtrack` function) contains a `duplicated` . It records which elements have been selected in the current round of traversal (i.e., the `for` loop), and serves to ensure that equal elements are selected only once. + +The figure below illustrates the range of validity of the two pruning conditions. Note that each node in the tree represents a choice, and the individual nodes on the path from the root node to the leaf nodes form an arrangement. + +![Scope of the two pruning conditions](permutations_problem.assets/permutations_ii_pruning_summary.png) diff --git a/docs-en/chapter_backtracking/subset_sum_problem.assets/subset_sum_i.png b/docs-en/chapter_backtracking/subset_sum_problem.assets/subset_sum_i.png new file mode 100644 index 0000000000000000000000000000000000000000..396e8b1f854a31e2ae6943139531140bcc352275 GIT binary patch literal 40044 zcmX`S1z1$i_dk9&=)0*=dmRytK4*Zf;I0G4k^A61}`!^y!nly!`3e zSzKHkY5{d%Z@0g{pOuxhwzgI{RPgWUsJpA{dSaX^Wbg*P^5x5ynVH$j%F2R*g5cob zgoFe;J3I8j^0J^(Mtb_*-kzeO;=sVbvuDpzQc^ZHHYO)0fBpJ3G&Gd?rNP+Pczk@q z!ongfEbQaQk4Z^MwY9ZF(~CMfI>W;w13!NB_4VcD=I;F2Sz26-kB<)t3Hf(+*3sU6 zQD48ay}h-))zZ>ZUtizY*r=+ix+<))w6v_RuYbC~H$FBtIy!niJ##a^(9+aYTwHwn zXIDULc-z*?+1c66&23pq>+0krZ{qLm#^%+)KwDeej?)`oU*EmImkSFE8C~hytE(|> 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Recursion + +In data structures and algorithms, it is common to repeat a task, which is closely related to the complexity of the algorithm. There are two basic program structures that we usually use to repeat a task: iteration and recursion. + +## Iteration + +An "iteration iteration" is a control structure that repeats a task. In iteration, a program repeats the execution of a piece of code until the condition is no longer satisfied. + +### For Loops + +`for` loops are one of the most common forms of iteration, **suitable when the number of iterations is known in advance**. + +The following function implements the summation $1 + 2 + \dots + n$ based on a `for` loop, and the result is recorded using the variable `res`. Note that `range(a, b)` in Python corresponds to a "left-closed-right-open" interval, which is traversed in the range $a, a + 1, \dots, b-1$. + +```src +[file]{iteration}-[class]{}-[func]{for_loop} +``` + +The figure below shows the flow block diagram of this summation function. + +![Flow block diagram of the summation function](iteration_and_recursion.assets/iteration.png) + +The number of operations in this summation function is proportional to the size of the input data $n$, or a "linear relationship". In fact, **time complexity describes this "linear relationship"**. This is described in more detail in the next section. + +### While Loop + +Similar to a `for` loop, a `while` loop is a way to implement iteration. In a `while` loop, the program first checks the condition at each turn, and if the condition is true, it continues, otherwise it ends the loop. + +Below, we use a `while` loop to realize the summation $1 + 2 + \dots + n$ . + +```src +[file]{iteration}-[class]{}-[func]{while_loop} +``` + +In `while` loops, since the steps of initializing and updating condition variables are independent of the loop structure, **it has more degrees of freedom than `for` loops**. + +For example, in the following code, the condition variable $i$ is updated twice per round, which is not convenient to implement with a `for` loop. + +```src +[file]{iteration}-[class]{}-[func]{while_loop_ii} +``` + +Overall, **`for` loops have more compact code and `while` loops are more flexible**, and both can implement iteration structures. The choice of which one to use should be based on the needs of the particular problem. + +### Nested Loops + +We can nest one loop structure inside another, using the `for` loop as an example: + +```src +[file]{iteration}-[class]{}-[func]{nested_for_loop} +``` + +The figure below gives the block diagram of the flow of this nested loop. + +![Block diagram of the flow of nested loops](iteration_and_recursion.assets/nested_iteration.png) + +In this case, the number of operations of the function is proportional to $n^2$, or the algorithm's running time is "squared" to the size of the input data $n$. + +We can continue to add nested loops, and each nest is a "dimension up", which will increase the time complexity to "cubic relations", "quadratic relations", and so on. + +## Recursion + + "Recursion recursion is an algorithmic strategy to solve a problem by calling the function itself. It consists of two main phases. + +1. **recursive**: the program calls itself deeper and deeper, usually passing smaller or simpler arguments, until a "termination condition" is reached. +2. **Recursion**: After the "termination condition" is triggered, the program returns from the deepest level of the recursion function, level by level, aggregating the results of each level. + +And from an implementation point of view, recursion code contains three main elements. + +1. **Termination condition**: used to decide when to switch from "recursive" to "inductive". +2. **Recursion call**: corresponds to "recursion", where the function calls itself, usually with smaller or more simplified input parameters. +3. **return result**: corresponds to "return", returning the result of the current recursion level to the previous one. + +Observe the following code, we only need to call the function `recur(n)` , and the calculation of $1 + 2 + \dots + n$ is done: + +```src +[file]{recursion}-[class]{}-[func]{recur} +``` + +The figure below shows the recursion of the function. + +![Recursion process for the summation function](iteration_and_recursion.assets/recursion_sum.png) + +Although iteration and recursion can yield the same results from a computational point of view, **they represent two completely different paradigms for thinking about and solving problems**. + +- **Iteration**: solving problems "from the bottom up". Start with the most basic steps and repeat or add to them until the task is completed. +- **Recursion**: solving problems "from the top down". The original problem is broken down into smaller subproblems that have the same form as the original problem. Next, the subproblem continues to be broken down into smaller subproblems until it stops at the base case (the solution to the base case is known). + +As an example of the above summation function, set the problem $f(n) = 1 + 2 + \dots + n$ . + +- **Iteration**: the summation process is simulated in a loop, iterating from $1$ to $n$ and executing the summation operation in each round to find $f(n)$. +- **Recursion**: decompose the problem into subproblems $f(n) = n + f(n-1)$ and keep (recursively) decomposing until the base case $f(1) = 1$ terminates. + +### Call The Stack + +Each time a recursion function calls itself, the system allocates memory for the newly opened function to store local variables, call addresses, other information, and so on. This results in two things. + +- The context data for a function is stored in an area of memory called "stack frame space" and is not freed until the function returns. As a result, **recursion is usually more memory-intensive than iteration**. +- Recursion calls to functions incur additional overhead. **Therefore recursion is usually less time efficient than loops**. + +As shown in the figure below, before the termination condition is triggered, there are $n$ unreturned recursion functions at the same time, **with a recursion depth of $n$** . + +![Recursion call depth](iteration_and_recursion.assets/recursion_sum_depth.png) + +In practice, the depth of recursion allowed by a programming language is usually limited, and too deep a recursion may result in a stack overflow error. + +### Tail Recursion + +Interestingly, **if a function makes a recursion call only at the last step before returning**, the function can be optimized by the compiler or interpreter to be comparable to iteration in terms of space efficiency. This situation is called "tail recursion tail recursion". + +- **Ordinary recursion**: when a function returns to a function at a higher level, it needs to continue executing the code, so the system needs to save the context of the previous call. +- **tail recursion**: the recursion call is the last operation before the function returns, which means that the function does not need to continue with other operations after returning to the previous level, so the system does not need to save the context of the previous function. + +In the case of calculating $1 + 2 + \dots + n$, for example, we can implement tail recursion by setting the result variable `res` as a function parameter. + +```src +[file]{recursion}-[class]{}-[func]{tail_recur} +``` + +The execution of tail recursion is shown in the figure below. Comparing normal recursion and tail recursion, the execution point of the summation operation is different. + +- **Ordinary recursion**: the summing operation is performed during the "return" process, and the summing operation is performed again after returning from each level. +- **Tail recursion**: the summing operation is performed in a "recursion" process, the "recursion" process simply returns in levels. + +![tail recursion process](iteration_and_recursion.assets/tail_recursion_sum.png) + +!!! tip + + Note that many compilers or interpreters do not support tail recursion optimization. For example, Python does not support tail recursion optimization by default, so even if a function is tail recursive, you may still encounter stack overflow problems. + +### Recursion Tree + +When dealing with algorithmic problems related to divide and conquer, recursion is often more intuitive and easier to read than iteration. Take the Fibonacci sequence as an example. + +!!! question + + Given a Fibonacci series $0, 1, 1, 2, 3, 5, 8, 13, \dots$ , find the $n$th number of the series. + +Let the $n$th number of the Fibonacci series be $f(n)$ , which leads to two easy conclusions. + +- The first two numbers of the series are $f(1) = 0$ and $f(2) = 1$. +- Each number in the series is the sum of the previous two numbers, i.e. $f(n) = f(n - 1) + f(n - 2)$ . + +Recursion code can be written by making recursion calls according to the recursion relationship, using the first two numbers as termination conditions. Call `fib(n)` to get the $n$th number of the Fibonacci series. + +```src +[file]{recursion}-[class]{}-[func]{fib} +``` + +Looking at the above code, we have recursively called two functions within a function, **this means that from one call, two call branches are created**. As shown in the figure below, this recursion will result in a recursion tree with the number of levels $n$. + +![Recursion tree for Fibonacci series](iteration_and_recursion.assets/recursion_tree.png) + +Essentially, recursion embodies the paradigm of "breaking down a problem into smaller sub-problems", and this divide and conquer strategy is essential. + +- From an algorithmic point of view, many important algorithmic strategies such as searching, sorting algorithm, backtracking, divide and conquer, dynamic programming, etc. directly or indirectly apply this way of thinking. +- From a data structure point of view, recursion is naturally suited to problems related to linked lists, trees and graphs because they are well suited to be analyzed with the idea of partitioning. + +## Compare The Two + +To summarize the above, as shown in the table below, iteration and recursion differ in implementation, performance and applicability. + +

Table   Comparison of iteration and recursion features

+ +| | iteration | recursion | +| ------------------- | ------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------- | +| implementation | circular structure | function call itself | +| time-efficient | typically efficient, no function call overhead | overhead on every function call | +| Memory Usage | Usually uses a fixed size of memory space | Cumulative function calls may use a lot of stack frame space | +| Applicable Problems | For simple cyclic tasks, code is intuitive and readable | For sub-problem decomposition, such as trees, graphs, divide and conquer, backtracking, etc., the code structure is concise and clear | + +!!! tip + + If you find the following solutions difficult to understand, you can review them after reading the "Stack" chapter. + +So what is the intrinsic connection between iteration and recursion? In the case of the recursive function described above, the summing operation takes place in the "return" phase of the recursion. This means that the function that is initially called is actually the last to complete its summing operation, **This mechanism works in the same way as the stack's "first in, last out" principle**. + +In fact, recursion terms like "call stack" and "stack frame space" already imply a close relationship between recursion and the stack. + +1. **Recursive**: When a function is called, the system allocates a new stack frame on the "call stack" for the function, which is used to store the function's local variables, parameters, return address, and other data. +2. **Return to**: When a function completes execution and returns, the corresponding stack frame is removed from the "call stack", restoring the function's previous execution environment. + +Thus, **we can use an explicit stack to model the behavior of the call stack**, thus transforming recursion into an iteration form: + +```src +[file]{recursion}-[class]{}-[func]{for_loop_recur} +``` + +Observing the code above, it becomes more complex when recursion is converted to iteration. Although iteration and recursion can be converted to each other in many cases, it is not always worth doing so for two reasons. + +- The transformed code may be more difficult to understand and less readable. +- For some complex problems, simulating the behavior of the system call stack can be very difficult. + +In short, **the choice of iteration or recursion depends on the nature of the particular problem**. In programming practice, it is crucial to weigh the advantages and disadvantages of both and choose the appropriate method based on the context. diff --git a/docs-en/chapter_computational_complexity/performance_evaluation.md b/docs-en/chapter_computational_complexity/performance_evaluation.md new file mode 100644 index 0000000000..c1237b3705 --- /dev/null +++ b/docs-en/chapter_computational_complexity/performance_evaluation.md @@ -0,0 +1,48 @@ +# Evaluation Of Algorithm Efficiency + +In the algorithm design, we pursue the following two levels successively. + +1. **Finding the solution to the problem**: the algorithm needs to reliably find the correct solution to the problem within a specified input range. +2. **Seek optimal solutions**: there may be multiple solutions to the same problem and we want to find the most efficient algorithm possible. + +That is to say, under the premise of being able to solve the problem, algorithm efficiency has become the main evaluation index for measuring the merits of algorithms, which includes the following two dimensions. + +- **Time efficiency**: how fast or slow the algorithm runs. +- **Space efficiency**: the amount of memory space occupied by the algorithm. + +In short, our goal is to design data structures and algorithms that are "fast and frugal". Evaluating the efficiency of algorithms effectively is crucial because only then can we compare various algorithms to guide the algorithm design and optimization process. + +There are two main types of efficiency assessment methods: practical testing, and theoretical estimation. + +## Practical Tests + +Suppose we now have algorithm `A` and algorithm `B` , both of which solve the same problem, and we now need to compare the efficiency of these two algorithms. The most direct way is to find a computer, run these two algorithms, and monitor and record their running time and memory usage. This type of evaluation reflects the real situation, but also has major limitations. + +On the one hand, **it is difficult to exclude disturbing factors in the test environment**. Hardware configuration affects the performance of algorithms. For example, in a certain computer, the running time of algorithm `A` is shorter than that of algorithm `B`; however, in another computer with a different configuration, we may get the opposite test results. This means that we need to test on a variety of machines and count the average efficiency, which is unrealistic. + +On the other hand, **it is very resource-consuming to carry out a full test**. As the amount of input data varies, the algorithms show different efficiencies. For example, when the amount of input data is small, the running time of algorithm `A` is less than that of algorithm `B`; while the test results may be opposite when the amount of input data is large. Therefore, in order to obtain convincing conclusions, we need to test various sizes of input data, which requires a lot of computational resources. + +## Theoretical Estimates + +Since practical tests have large limitations, we can consider evaluating the efficiency of an algorithm by only a few calculations. This estimation method is called "asymptotic complexity analysis", or "complexity analysis" for short. + +The complexity analysis reflects the relationship between the time (space) resources required to run the algorithm and the size of the input data. **It describes the growing trend in the time and space required for the execution of an algorithm as the size of the input data increases**. This definition is a bit of a mouthful, and we can understand it by dividing it into three key points. + +- The terms "time and space resources" correspond to "time complexity" and "space complexity" respectively. +- "As the size of the input data increases" means that the complexity reflects the relationship between the efficiency of the algorithm operation and the volume of the input data. +- "Trends in the growth of time and space" means that the analysis of complexity is not concerned with specific values of running time or occupied space, but rather with the "speed" of the growth of time or space. + +**Complexity analysis overcomes the drawbacks of practical testing methods in two ways**. + +- It is independent of the test environment and the analysis results are applicable to all operating platforms. +- It can reflect the efficiency of algorithms under different data volumes, especially the performance of algorithms under large data volumes. + +!!! tip + + If you're still confused about the concept of complexity, don't worry, we'll cover it in detail in subsequent chapters. + +Complexity analysis provides a "yardstick" for evaluating the efficiency of an algorithm, allowing us to measure the time and space resources required to execute an algorithm and to compare the efficiency of different algorithms. + +Complexity is a mathematical concept that may be abstract and relatively difficult for beginners to learn. From this point of view, complexity analysis may not be very suitable as the first introduction. 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zfJF4T2>1brH-pFs8(IOzEu8>lCz}i)c0({gqcpw(x}bEKv8Du+Gu$2_t{RZq=NtuC zW#t7(ME}UON?dgP)DQm8u=o3$EmNlUW!JmQBlgJGioXQ;#lGDeHw?o76b5r3DC>?H zAqA%J3fX_5h<}poK84Bm;oP}?2ZfVl_I*Z={np>2S6 z1uz+hAOM!2J%FhNaCh#(09eL(2wdL*`u^Y;7{DTKw_sdE(guTOP)y7 zf094|Q_=GQxG#VvRo*>~0ya{948t%}OpDsRf64bOO{gniE#N@`U`jEz{?!FANat4p zNg#kZ#h3oM44U^=z>&aiCQs{M;{v-JX+ zg~J8_a|yM6*88`~i literal 0 HcmV?d00001 diff --git a/docs-en/chapter_data_structure/classification_of_data_structure.md b/docs-en/chapter_data_structure/classification_of_data_structure.md new file mode 100644 index 0000000000..012e21e6db --- /dev/null +++ b/docs-en/chapter_data_structure/classification_of_data_structure.md @@ -0,0 +1,47 @@ +# Classification Of Data Structures + +Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs, which can be categorized in terms of both "logical structure" and "physical structure" dimensions. + +## Logical Structures: Linear And Nonlinear + +**Logical structures reveal logical relationships between data elements**. In arrays and linked lists, data are arranged sequentially in order, reflecting the linear relationship between data; while in trees, data are arranged hierarchically from the top down, showing the derived relationship between ancestors and descendants; and graphs are composed of nodes and edges, reflecting the complex network relationship. + +As shown in the figure below, logical structures can be categorized into "linear" and "non-linear". Linear structures are more intuitive, meaning that the data are arranged linearly in terms of logical relationships; non-linear structures, on the other hand, are arranged non-linearly. + +- **Linear data structures**: arrays, linked lists, stacks, queues, hash tables. +- **Nonlinear data structures**: trees, heaps, graphs, hash tables. + +![Linear and nonlinear data structures](classification_of_data_structure.assets/classification_logic_structure.png) + +Nonlinear data structures can be further categorized into tree and graph structures. + +- **Linear structures**: arrays, linked lists, queues, stacks, hash tables, with one-to-one sequential relationships between elements. +- **Tree structure**: tree, heap, hash table with one-to-many relationships between elements. +- **Graph**: graph with many-to-many relationships between elements. + +## Physical Structure: Continuous Vs. Dispersed + +In computers, memory and hard disk are the two main storage hardware devices. Hard disks are used primarily for long-term data storage, and are larger (usually up to the terabyte degree) and slower. Memory is used for temporary storage of data while running a program, and is faster, but has a smaller capacity (usually at the GB level). + +**During the operation of the algorithm, the relevant data are stored in memory**. The figure below shows a computer memory stick where each black square contains a block of memory space. We can think of the memory as a giant Excel sheet in which each cell can store data of a certain size, and in which all data is stored while the algorithm is running. + +**The system accesses the data at the target location by means of a memory address**. As shown in the figure below, the computer assigns numbers to each cell in the table according to specific rules, ensuring that each memory space has a unique memory address. With these addresses, the program can access the data in memory. + +![memory_strip, memory_space, memory_address](classification_of_data_structure.assets/computer_memory_location.png) + +Memory is a shared resource for all programs, and when a block of memory is occupied by one program, it cannot be used by other programs at the same time. **Therefore, memory resources are an important consideration in the design of data structures and algorithms**. For example, the peak memory occupied by an algorithm should not exceed the remaining free memory of the system; if there is a lack of consecutive large chunks of memory space, then the data structure chosen must be able to be stored in a decentralized memory space. + +As shown in the figure below, **Physical structure reflects the way data is stored in computer memory and can be categorized into continuous space storage (arrays) and distributed space storage (linked lists)**. The physical structure determines from the bottom up the methods of accessing, updating, adding, deleting, and other operations on the data, while presenting complementary characteristics in terms of time efficiency and space efficiency. + +![continuous vs. decentralized spatial storage](classification_of_data_structure.assets/classification_phisical_structure.png) + +**It is worth stating that all data structures are implemented based on arrays, linked lists, or a combination of the two**. For example, stacks and queues can be implemented using both arrays and linked lists; and implementations of hash tables may contain both arrays and linked lists. + +- **Based on arrays can be realized**: stacks, queues, hash tables, trees, heaps, graphs, matrices, tensors (arrays of dimension $\geq 3$), and so on. +- **Can be implemented based on linked lists**: stacks, queues, hash tables, trees, heaps, graphs, etc. + +Data structures based on arrays are also known as "static data structures", which means that they are not changeable in length after initialization. In contrast, data structures based on linked lists are called "dynamic data structures", meaning that their length can be adjusted during program execution after initialization. + +!!! tip + + If you find it difficult to understand the physical structure, it is recommended that you read the next chapter, "Arrays and Linked Lists," before reviewing this section. diff --git a/docs-en/chapter_data_structure/index.md b/docs-en/chapter_data_structure/index.md new file mode 100644 index 0000000000..7a655a2c86 --- /dev/null +++ b/docs-en/chapter_data_structure/index.md @@ -0,0 +1,13 @@ +# Data Structure + +
+ +![data structure](../assets/covers/chapter_data_structure.jpg){ width="600" } + +
+ +!!! abstract + + Data structures are like a solid and varied framework. + + It provides a blueprint for the orderly organization of data upon which algorithms can come alive. diff --git a/docs-en/chapter_data_structure/number_encoding.assets/1s_2s_complement.png b/docs-en/chapter_data_structure/number_encoding.assets/1s_2s_complement.png new file mode 100644 index 0000000000000000000000000000000000000000..587b8cd2afd889240dd2ec45a04496979ac6ab1e GIT binary patch literal 21585 zcmcG#bzD?mxG%i-Fm#8cNT-ybARsVwDbgWbf`D{~3=8q-7yN1(p@ricc<_8 zi}U&1bMO6}b3X6;&L4a470-7)wbx$rto3|SRguTXp~e9K0AEo-MjZgqQIC*8EHu>Z zJe%7BRk%@osww;X_wViP?N6UR-Q3<@Uth1Tu3lbU9v>g??d@%DZXO;Uo}Ha7EG%dx zY2FFwEM<5WRqoZwYZQS95Z>s{LqM}YuPp77)tg>HTTwI)=pU1_;)z;Py z4h|+IC3SaqH#9UXFE1k(kSnVzdMUcSy}et?|xQQRe9AcOs-FkjEvOO z)QG+q{QUXzkJ)w=n=u));oaTcj*gD_`1qNbnS+A^9jA%m;o-B(o4&rjy|b(C#iQ)L zt?B7$KGnW}xP=F5eJw35%j@TzgNGRz8Q;HuPfJU4EpmCCZ&y}U79JkHb+i?mvtr~v z5fc;hNNXUcD?2?s-R!ez#ZblU?5tMyBwb+N(*DK4>Ar6yGOax=Ha2!-`lM;R5xIlx z@9+Ql^Jo9q5$b1NFyr*qQNnOY!+t!jHCFZWB=e_!uM*Fm`JRM?goYoxq4gmX6BC@F z1Br==A8P}r=1;%(?mGs~3dRiAj?@ke44hw`ADtcgH!N9LSPU%ozj^a!e0;oR_JCi# ze|`JHuiQ7TW4*Gn(z)W7iPuz7QITEgT;b2cqmygo%2~zOZt=)YNyC=3v~*z0^0z63 zd+6Ng%BX(cw5_de-Q-?N_uki<&6L8mH_^x!u9Kp1BgMtV!6^%qlaunPW5?&$;h9So zzEkbv1+ z7bjn;Ha{dT=Jn)`uAdAo4Qgv^e;wLR_?9p_KWb!T^e%43)6=taqSMjQ@#y@h3b8Xc zzp;OKRob-OFtQv_JM+DJMLB&!B4KoL^R#<%wX(0`fqMVI%JJ~bdV251-16q^_E~z- zd`1Vdpm%X*V|r?Kqi?3~bKB?Hh10O_EAijv%NiCN+m?ryheI0Y5?c}`7B)KicavI@ z+J`>20e~8zqKu@b=k)e;Ur7NUS>OUS5w0sVJm?d)0BP#bF9@1_SFeO^wP*O+9>i|m ziZC=#%Q&}^-+YN_IjyR^tc2M? zs#KE48zVi{cU1@5qzUpBa}kr?wgGU1C;>XWdutxRRse`aZ?1JovQLZ9qcBnMpr1%K zZUwsZMbPr}BQtRaKe)#Lo28l9^zfoD=#vUStzaD)jD|M=6M{wWK#gE9LYrZNueomx zLvs9q!&4*p+<1BfFwJv|6L<@D%^ip)_P3yU&Od}d0-=#&*e1o`?Ct}&#uJDH1q#8R z8nHqZfzZIoWPt-c`vjRccqRkHlrV6Sxb8O`qLpGfSyd-3;d9>@HS#&oZjnL*Db20`$ zfulWIARh3f=X=+_soa&ause9&;>bOdzot2h&l>(JY%loAhQauOorYCQKUuMo1>nN7 zn#`%9g!QBlKiYZ%F$!)^X*TPZlIJ`aerylgCGD@%+_4p9({+3vmuGIOvd4kL%nwJ$ zH#YpCfE{31ElIMGk7tn!zh{0pnhdNK=S3v~?hBRXhKQm7mL0=NSuQ;;WN;3&Y;eb2 z;`+-cDz+tnul73W!gTGITT0*}5c>PMbzKkvT@Lq$Sm*mBnR4%?)IN}5j|QZuVq!3x zm`IzM!-ZgTl#BulZcjAi(h=G$rT2=u9NNlBg%XlOpz6M}NBS@7X;nV@R{QZ>)X9UR zNhXpn74G0rE_E)ulEdaw%#49GDcG4|`VB!Y{ ze*}%a&W!q zY)q(80XOzw0?T>0la7*&_v{kL?9C&P6a3Iy)waE?@N^)mJ{n%iT5TTcA-E0onxLB^ z@(MU`W?+7UDPtDp?w`Y2&zc>XF@(1rERYKUw;z~FG$OfW+9Mu_sH=bM;4XxDRO42@ zQOm|D{!ywKHh7h#Ns9GtBH#G4TxwiWq~J=;pA)ZM`2RHSd}hb;Nl!>M2=bBRM16WvteeZ+gxg0T<%T zi0C9mzHgS9v}R{0w0+Y12k&U37*MbNKYK@t!`3tgRT+mF5w3oB>rSG@!A zY_fa;Z73d1<%-$pBJWV-_`2cr{7X4Q<@j&``8_g|`YK(dc7BZoXN$#WG zbEyI|eR!VW5%)bB*re)LS&2MQqSjgmFcz3Zs||2c;u2h+k6vE%mE9KX@Nsh1!pm%G zt8=T?-qeAA>}>w>Z7bE7<>UoQ!CPj`%?m#}U%pq;4i6C-|M>dAJ-G-Cv+_}q;= z#1IeKHaj+&(^wu63X zQ=Vxp<4ztWm5pFn3l|eyMoV^Fn6R+|W+}z+YE!gLpE2 z>xpsor$Z<+R($S0>1`JMW%fmwNXvqad5rhs6x#UZwJB9ucRQvIdh{TWi z$Z0q$Pmt3onA593PebX0w>oCkp8y+0v8`9?-*)w! 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zpPAdA*P`%)!IQp8VxO=9cqqTNeg22d6lKZlPw*kd#;lS`UI49B;rgI8b*)>onKHN{ rQh6|k4g}0iaoe+*tK$t|yvPp*WU5y2`^-@R%mbjIqN7}`2z&k?<&K|_ literal 0 HcmV?d00001 diff --git a/docs-en/chapter_data_structure/number_encoding.md b/docs-en/chapter_data_structure/number_encoding.md new file mode 100644 index 0000000000..9155d4dfe9 --- /dev/null +++ b/docs-en/chapter_data_structure/number_encoding.md @@ -0,0 +1,150 @@ +# Numeric Code * + +!!! note + + Throughout the book, the chapters with the * symbol in the title are optional. If you have limited time or find it difficult to understand, you can skip them and then attack them individually when you have finished the required chapters. + +## Integer Encoding + +In the table in the previous section, we found that all integer types are capable of representing negative numbers one more than positive numbers, e.g., the range of values for `byte` is $[-128, 127]$ . This phenomenon is rather counter-intuitive, and its intrinsic cause involves knowledge of true forms, inverse codes, and 2's complement codes. + +First of all, **it should be noted that numbers are stored in the computer as in the form of 2's complement code**. Before analyzing the reasons for this, we first give the definitions of the three. + +- **True form**: we consider the highest bit of the binary representation of a number as the sign bit, where $0$ denotes a positive number, $1$ denotes a negative number, and the rest of the bits denote the value of the number. +- **Inverse Code**: the inverse code of a positive number is the same as its true form, and the inverse code of a negative number is the inverse of all bits of its true form except the sign bit. +- **Complement code**: The complement code of a positive number is the same as its true form, and the complement code of a negative number is its inverse code plus $1$ . + +The figure below shows how to convert between true form, 1's complement code and 2's complement code. + +![Interconversion between true form, 1's complement code and 2's complement code](number_encoding.assets/1s_2s_complement.png) + +Although "true form" is the most intuitive, it has some limitations. On the one hand, **the true form of negative numbers cannot be used directly in arithmetic**. For example, if you calculate $1 + (-2)$ in true form, the result will be $-3$, which is obviously incorrect. + +$$ +\begin{aligned} +& 1 + (-2) \newline +& \rightarrow 0000 \; 0001 + 1000 \; 0010 \newline +& = 1000 \; 0011 \newline +& \rightarrow -3 +\end{aligned} +$$ + +In order to solve this problem, computers have introduced "1's complement code". If we first convert the original code to the inverse code, and calculate $1 + (-2)$ under the inverse code, and finally convert the result from the inverse code back to the original code, then we can get the true form $-1$. + +$$ +\begin{aligned} +& 1 + (-2) \newline +& \rightarrow 0000 \; 0001 \; \text{(true form)} + 1000 \; 0010 \; \text{(true form)} \newline +& = 0000 \; 0001 \; \text{(complement code)} + 1111 \; 1101 \; \text{(complement code)} \newline +& = 1111 \; 1110 \; \text{(1's complement code)} \newline +& = 1000 \; 0001 \; \text{(true form)} \newline +& \rightarrow -1 +\end{aligned} +$$ + +On the other hand, **the true form of the number zero has two representations $0$ and $-0$**. This means that the number zero corresponds to two different binary codes, and its may bring ambiguity. For example, in conditional judgment, if there is no distinction between positive and negative zeros, it may lead to wrong judgment results. If we want to deal with positive and negative zero ambiguities, we need to introduce additional judgment operations, which may reduce the computational efficiency of the computer. + +$$ +\begin{aligned} ++0 & \rightarrow 0000 \; 0000 \newline +-0 & \rightarrow 1000 \; 0000 +\end{aligned} +$$ + +Like the true form, the inverse code also has the problem of positive and negative zero ambiguity, so the computer further introduces the "complement code 2's complement code". Let's first observe the conversion process of negative-zero true form, 1's complement code, and 2's complement code: + +$$ +\begin{aligned} +-0 \rightarrow \; & 1000 \; 0000 \; \text{(true form)} \newline += \; & 1111 \; 1111 \; \text{(complement code)} \newline += 1 \; & 0000 \; 0000 \; \text{(complement code)} \newline +\end{aligned} +$$ + +Adding $1$ to the 1's complement code of negative zeros results in a rounding, but the length of the `byte` type is only 8 bits, so the $1$ that overflows to bit 9 is discarded. In other words, **the complement code for negative zeros is $0000 \; 0000$, which is the same as the complement code for positive zeros**. This means that there is only one zero in the 2's complement code representation, and the positive and negative zero ambiguity is thus resolved. + +There is one last puzzle: the range of the `byte` type is $[-128, 127]$, how do we get the extra negative number $-128$? We notice that all integers in the range $[-127, +127]$ have their true form, 1's complement code and 2's complement code, and the true form and 2's complement code can be converted to each other. + +However, **complement code $1000 \; 0000$ is an exception which does not have a corresponding true form code**. According to the conversion method, we get the true form of this complement code as $0000 \; 0000$ . This is clearly a contradiction, since that true form represents the number $0$ and its 2's complement code should be itself. The computer specifies that this particular 2's complement code $1000 \; 0000$ represents $-128$ . In fact, $(-1) + (-127)$ under the 2's complement code results in $-128$ . + +$$ +\begin{aligned} +& (-127) + (-1) \newline +& \rightarrow 1111 \; 1111 \; \text{(true form)} + 1000 \; 0001 \; \text{(true form)} \newline +& = 1000 \; 0000 \; \text{(complement code)} + 1111 \; 1110 \; \text{(complement code)} \newline +& = 1000 \; 0001 \; \text{(complement code)} + 1111 \; 1111 \; \text{(complement code)} \newline +& = 1000 \; 0000 \; \text{(complement code)} \newline +& \rightarrow -128 +\end{aligned} +$$ + +You may have noticed that all of the above calculations are additive operations. This implies an important fact:**The hardware circuits inside computers are designed primarily based on addition operations**. This is because addition operations are simpler to implement in hardware, easier to parallelize, and faster to perform compared to other operations such as multiplication, division, and subtraction. + +Note that this does not mean that computers can only do addition. **By combining addition with some basic logical operations, computers are able to perform a variety of other mathematical operations**. For example, calculating subtraction $a - b$ can be converted to calculating addition $a + (-b)$; calculating multiplication and division can be converted to calculating multiple additions or subtractions. + +We can now summarize the reasons why computers use complement codes: based on the 2's complement code representation, computers can use the same circuits and operations to handle addition of positive and negative numbers, do not need to design special hardware circuits to handle subtraction, and do not have to deal specifically with the ambiguity of positive and negative zeros. This greatly simplifies hardware design and improves arithmetic efficiency. + +The design of 2's complement code is very subtle, because of space relations we will introduce here, we recommend interested readers to further depth. + +## Floating Point Encoding + +You may notice that `int` and `float` are both 4 bytes long, but why is the range of `float` much larger than that of `int`? This is very counter-intuitive, because `float` needs to represent decimals, so its range should be smaller. + +In fact, **this is because floating-point numbers `float` use a different representation**. Remember that a 32-bit length binary number is: + +$$ +b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0 +$$ + +According to the IEEE 754 standard, a 32-bit length `float` consists of the following three parts. + +- Symbol bit $\mathrm{S}$ : Occupies 1 bit and corresponds to $b_{31}$. +- Exponential bits $\mathrm{E}$ : occupies 8 bits, corresponding to $b_{30} b_{29} \ldots b_{23}$. +- Fractional bits $\mathrm{N}$ : occupies 23 bits, corresponding to $b_{22} b_{21} \ldots b_0$. + +Calculates the value corresponding to a binary number `float`: + +$$ +\text {val} = (-1)^{b_{31}} \times 2^{\left(b_{30} b_{29} \ldots b_{23}\right)_2-127} \times\left(1 . b_{22} b_{21} \ldots b_0\right)_2 +$$ + +Converts to a formula in decimal: + +$$ +\text {val}=(-1)^{\mathrm{S}} \times 2^{\mathrm{E} -127} \times (1 + \mathrm{N}) +$$ + +The range of values of each of these: + +$$ +\begin{aligned} +\mathrm{S} \in & \{ 0, 1\}, \quad \mathrm{E} \in \{ 1, 2, \dots, 254 \} \newline +(1 + \mathrm{N}) = & (1 + \sum_{i=1}^{23} b_{23-i} 2^{-i}) \subset [1, 2 - 2^{-23}] +\end{aligned} +$$ + +![Calculation example of float under IEEE 754 standard](number_encoding.assets/ieee_754_float.png) + +Looking at the figure above, given a sample data $\mathrm{S} = 0$ , $\mathrm{E} = 124$ , $\mathrm{N} = 2^{-2} + 2^{-3} = 0.375$ , we have: + +$$ +\text { val } = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875 +$$ + +Now we can answer the original question:**The representation of `float` contains exponent bits, resulting in a much larger range of values than `int`**. Based on the above calculations, the largest positive number that can be represented by `float` is $2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}$ , and switching the sign bit will give us the smallest negative number. + +**Although floating-point numbers `float` extend the range of values, they have the side effect of sacrificing precision**. The integer type `int` dedicates all 32 bits to representing numbers, which are evenly distributed; whereas, the larger the value of a floating-point number `float`, the greater the difference between two neighboring numbers tends to be, due to the exponent bit. + +As shown in the table below, the exponent bits $E = 0$ and $E = 255$ have special meanings and **are used to denote zero, infinity, $\mathrm{NaN}$ and so on**. + +

Table   Meaning of Index Bits

+ +| Exponential E | Score $\mathrm{N} = 0$ | Score $\mathrm{N} \ne 0$ | Calculation formula | +| ------------------ | ---------------------- | ------------------------ | ---------------------------------------------------------------------- | +| $0$ | $\pm 0$ | Subregular number | $(-1)^{\mathrm{S}} \times 2^{-126} \times (0.\mathrm{N})$ | +| $1, 2, \dots, 254$ | Regular number | Regular number | $(-1)^{\mathrm{S}} \times 2^{(\mathrm{E} -127)} \times (1.\mathrm{N})$ | +| $255$ | $\pm \infty$ | $\mathrm{NaN}$ | | + +It is worth stating that the subregular numbers significantly improve the precision of floating point numbers. The smallest positive regular number is $2^{-126}$ and the smallest positive subregular number is $2^{-126} \times 2^{-23}$. + +Double-precision `double` also uses a similar representation to `float`, which will not be repeated here. diff --git a/docs-en/chapter_data_structure/summary.md b/docs-en/chapter_data_structure/summary.md new file mode 100644 index 0000000000..a9631d6365 --- /dev/null +++ b/docs-en/chapter_data_structure/summary.md @@ -0,0 +1,33 @@ +# Summary + +### Highlights + +- Data structures can be categorized in terms of both logical and physical structures. Logical structure describes the logical relationships between data elements, while physical structure describes how data is stored in computer memory. +- Common logical structures include linear, tree and net. Usually we categorize data structures into linear (arrays, linked lists, stacks, queues) and non-linear (trees, graphs, heaps) based on their logical structure. Implementations of hash tables may contain both linear and nonlinear structures. +- When a program runs, data is stored in computer memory. Each memory space has a corresponding memory address through which the program accesses the data. +- Physical structures are mainly categorized into continuous space storage (arrays) and distributed space storage (linked lists). All data structures are implemented as arrays, linked lists, or a combination of both. +- The basic data types in computers include integers `byte`, `short`, `int`, `long`, floats `float`, `double`, characters `char`, and booleans `boolean`. The range of their values depends on the amount of space occupied and the representation. +- True form, 1's complement code and 2's complement code are the three methods of coding numbers in a computer and they are interconvertible. The highest bit of the true form of an integer is the sign bit and the remaining bits are the value of the number. +- Integers are stored in computers as 2's complement code. Under the 2's complement code representation, computers can treat addition of positive and negative numbers equally, do not need to design separate special hardware circuits for subtraction operations, and do not have the problem of positive and negative zero ambiguity. +- Floating point numbers are encoded with 1 sign bit, 8 exponent bits, and 23 fractional bits. Because of the exponent bits, floating-point numbers have a much wider range of values than integers, at the expense of precision. +- ASCII is the earliest English character set, with a length of 1 byte and a total of 127 characters; GBK is the most commonly used Chinese character set, with a total of more than 20,000 Chinese characters; Unicode is committed to providing a complete character set standard, including characters of various languages in the world, so as to solve the problem of garbled characters caused by inconsistent character encoding methods. +- UTF-8 is the most popular Unicode encoding method, which is very versatile. It is a variable-length encoding method with good scalability, which effectively improves the efficiency of storage space usage.UTF-16 and UTF-32 are equal-length encoding methods, and UTF-16 occupies less space than UTF-8 when encoding Chinese. When encoding Chinese, UTF-16 takes less space than UTF-8. Programming languages like Java and C# use UTF-16 encoding by default. + +### Q & A + +!!! question "Why does a hash table contain both a linear data structure and a nonlinear data structure?" + + Hash table underlying the array, and in order to solve hash collision, we may use the "chained address" (subsequent hash table chapter will talk about): each bucket in the array to point to a chained list, when the length of the chained list exceeds a certain threshold, and may be transformed into a tree (usually a red-black tree). + From a storage point of view, the underlying level of a hash table is an array, where each bucket slot bit may contain a value, a linked list or a tree. Thus, hash tables may contain both linear (arrays, linked lists) and nonlinear (trees) data structures. + +!!! question "Is the length of the `char` type 1 byte?" + + The length of the `char` type is determined by the encoding method used by the programming language. For example, Java, JS, TS, C# all use UTF-16 encoding (save Unicode code point), so the length of char type is 2 bytes. + +!!! Question "Is it ambiguous that data structures implemented based on arrays are also called "static data structures"? Because the stack can also perform operations such as stack out and stack in, which are "dynamic"." + + Stacks do allow dynamic data manipulation, but the data structure is still "static" (immutable in length). Although data structures based on arrays can dynamically add or remove elements, their capacity is fixed. If the amount of data exceeds the pre-allocated size, a new, larger array needs to be created and the contents of the old array copied into the new one. + +!!! question "Why are stacks (queues) "static data structures" when their size is not specified at the time of construction?" + + In high-level programming languages, we do not have to manually specify the initial size of the stack (queue), this is done automatically from within the class. For example, Java's ArrayList usually has an initial capacity of 10. In addition, the expansion operation is also automatic. 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When the subproblem is solved, the original problem is solved at the same time. + +Partitioning can improve search efficiency, essentially because brute-force search can only eliminate one option per round, **while partitioned search can eliminate half of the options per round**. + +### Realize The Dichotomy Based On The Divide And Conquer + +In the previous sections, binary search was implemented based on recursion (iteration). Now we implement it based on divide and conquer (recursion). + +!!! question + + Given an ordered array `nums` of length $n$ and all elements in the array are unique, find the element `target`. + +From the divide and conquer perspective, we denote the subproblem corresponding to the search interval $[i, j]$ as $f(i, j)$ . + +Starting from the original problem $f(0, n-1)$, perform a binary search by the following steps. + +1. Calculate the midpoint $m$ of the search interval $[i, j]$ and exclude half of the search interval based on it. +2. The recursion solves the subproblem that halves in size, possibly $f(i, m-1)$ or $f(m+1, j)$ . +3. Loop through steps `1.` and `2.` until `target` is found or the interval is empty. + +The figure below shows the process of dividing and conquering a binary search for the element $6$ in an array. + +![The divide and conquer process for binary search](binary_search_recur.assets/binary_search_recur.png) + +In the implementation code, we declare a recursion function `dfs()` to solve the problem $f(i, j)$ . + +```src +[file]{binary_search_recur}-[class]{}-[func]{binary_search} +``` diff --git a/docs-en/chapter_divide_and_conquer/build_binary_tree_problem.assets/build_tree_division_pointers.png b/docs-en/chapter_divide_and_conquer/build_binary_tree_problem.assets/build_tree_division_pointers.png new file mode 100644 index 0000000000000000000000000000000000000000..52b769e2fbf64fdf38565f3f40020ccdad84cca9 GIT binary patch literal 17892 zcmd42XH*nH7dF^EfgDDXAUTLgRzyG$21UskBqxcIa}EO{Nis-Aq69@iGLi;Raug(o zL2^cNX2iv2sy<1xeDT&XB%ytui!v41yzb$xw#d8yX$4f$zvVPRo^fB*da{OIWDk%I&C$`( 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+ +| | index of root node in `preorder` | index interval of subtree in `inorder` | +| ------------- | -------------------------------- | -------------------------------------- | +| Current Tree | $i$ | $[l, r]$ | +| left subtree | $i + 1$ | $[l, m-1]$ | +| right subtree | $i + 1 + (m - l)$ | $[m+1, r]$ | + +Please note that $(m-l)$ in the index of the root node of the right-child node means "the number of nodes in the left subtree", and it is recommended to understand it with the figure below. + +![Index interval representation of root node and left and right subtrees](build_binary_tree_problem.assets/build_tree_division_pointers.png) + +### Code Implementation + +To improve the efficiency of the query $m$, we resort to a hash table `hmap` to store the mapping of the elements of the array `inorder` to the indexes. + +```src +[file]{build_tree}-[class]{}-[func]{build_tree} +``` + +The figure below illustrates the recursion process of building a binary tree, where nodes are built in a downward "recursion" process and edges (i.e., references) are built in an upward "recursion" process. + +=== "<1>" + ![Recursion process for building a binary tree](build_binary_tree_problem.assets/built_tree_step1.png) + +=== "<2>" + ![built_tree_step2](build_binary_tree_problem.assets/built_tree_step2.png) + +=== "<3>" + ![built_tree_step3](build_binary_tree_problem.assets/built_tree_step3.png) + +=== "<4>" + ![built_tree_step4](build_binary_tree_problem.assets/built_tree_step4.png) + +=== "<5>" + ![built_tree_step5](build_binary_tree_problem.assets/built_tree_step5.png) + +=== "<6>" + ![built_tree_step6](build_binary_tree_problem.assets/built_tree_step6.png) + +=== "<7>" + ![built_tree_step7](build_binary_tree_problem.assets/built_tree_step7.png) + +=== "<8>" + ![built_tree_step8](build_binary_tree_problem.assets/built_tree_step8.png) + +=== "<9>" + ![built_tree_step9](build_binary_tree_problem.assets/built_tree_step9.png) + +The result of the division of the preorder traversal `preorder` and the middle order traversal `inorder` within each recursion function is shown in the figure below. + +![Division results in each recursion function](build_binary_tree_problem.assets/built_tree_overall.png) + +Let the number of nodes in the tree be $n$ , initializing each node (executing a recursion function `dfs()` ) uses $O(1)$ time. **Therefore the overall time complexity is $O(n)$** . + +A hash table stores the mapping of `inorder` elements to indexes with a space complexity of $O(n)$ . 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Divide and conquer is usually implemented based on recursion and consists of two steps: "divide" and "conquer". + +1. **Division (dividing stage)**: recursionally decomposes the original problem into two or more subproblems until it terminates when it reaches the smallest subproblem. +2. The **cure (merging phase)**: Starting from the smallest subproblem with a known solution, the solutions of the subproblems are merged from the bottom to the top to construct a solution to the original problem. + +As shown in the figure below, "merge sort" is one of the typical applications of divide and conquer strategy. + +1. **Divide**: recursively divide the original array (original problem) into two subarrays (subproblems) until only one element of the subarray remains (smallest subproblem). +2. **cure**: merge ordered subarrays (solutions to sub-problems) from bottom to top to obtain an ordered original array (solutions to the original problem). + +![Divide and conquer strategy for merge sort](divide_and_conquer.assets/divide_and_conquer_merge_sort.png) + +## How To Determine The Divide And Conquer Problem + +Whether a problem is suitable for solution using divide and conquer can usually be judged by referring to the following bases. + +1. **Problems can be decomposed**: the original problem can be decomposed into smaller, similar sub-problems, as well as being able to be divided recursively in the same way. +2. **Subproblems are independent**: there is no overlap between subproblems, no dependence on each other, and can be solved independently. +3. **The solutions of subproblems can be combined**: the solution of the original problem is obtained by combining the solutions of the subproblems. + +Clearly, merge sort is satisfying the above three bases of judgment. + +1. The **problem can be decomposed by**: recursively dividing the array (original problem) into two subarrays (subproblems). +2. **Subproblems are independent**: each subarray can be sorted independently (subproblems can be solved independently). +3. **Solutions of subproblems can be merged**: two ordered subarrays (solutions of subproblems) can be merged into one ordered array (solutions of the original problem). + +## Enhancing Efficiency Through Divide And Conquer + +Not only does divide and conquer effectively solve algorithmic problems, **it often leads to an increase in algorithmic efficiency**. In sorting algorithms, quick sort, merge sort, and heap sort are faster compared to selection, bubble, and insertion sort because they apply a partition strategy. + +So, we can't help but ask:**Why does dividing and conquering improve algorithm efficiency, and what is its underlying logic**? In other words, why are the steps of decomposing a large problem into multiple subproblems, solving the subproblems, and combining the solutions of the subproblems into the solution of the original problem more efficient than solving the original problem directly? This question can be discussed in terms of the number of operations and parallel computation. + +### Operation Number Optimization + +Take "bubble sort" for example, it takes $O(n^2)$ time to process an array of length $n$. Assuming we divide the array into two subarrays from the midpoint as shown in the figure below, it takes $O(n)$ time to divide, $O((n / 2)^2)$ time to sort each subarray, and $O(n)$ time to merge the two subarrays, with an overall time complexity of: + +$$ +O(n + (\frac{n}{2})^2 \times 2 + n) = O(\frac{n^2}{2} + 2n) +$$ + +![Bubble sort before and after dividing the array](divide_and_conquer.assets/divide_and_conquer_bubble_sort.png) + +Next, we compute the following inequality, whose left and right edges are the total number of operations before and after division, respectively: + +$$ +\begin{aligned} +n^2 & > \frac{n^2}{2} + 2n \newline +n^2 - \frac{n^2}{2} - 2n & > 0 \newline +n(n - 4) & > 0 +\end{aligned} +$$ + +**This means that when $n > 4$, the number of operations after division is smaller and the sorting should be more efficient**. Note that the time complexity after division is still of squared order $O(n^2)$ , just that the constant term in the complexity becomes smaller. + +Thinking further, **what if we keep dividing the subarray into two more subarrays from the midpoint until there is only one element left in the subarray**? This idea is actually a "merge sort" with a time complexity of $O(n \log n)$ . + +On second thought, **what if we set a few more division points and divide the original array into $k$ subarrays evenly**? This situation is very similar to "bucket sort", which is ideal for sorting large amounts of data, and has a theoretical time complexity of $O(n + k)$ . + +### Parallel Computing Optimization + +We know that the subproblems generated by partitioning are independent of each other, **and thus can usually be solved in parallel**. In other words, divide and conquer not only reduces the time complexity of the algorithm, **but also facilitates the parallel optimization of the operating system**. + +Parallel optimization is particularly effective in multi-core or multi-processor environments, as the system can process multiple sub-problems at the same time, making fuller use of computational resources and thus significantly reducing overall runtime. + +For example, in the "bucket sort" shown in the figure below, we will distribute the huge amount of data evenly into each bucket, then the sorting task of all the buckets can be distributed to each computing unit, and then merge the results after completion. + +![Parallel computing for bucket sorting](divide_and_conquer.assets/divide_and_conquer_parallel_computing.png) + +## Common Applications Of Divide And Conquer + +On the one hand, divide and conquer can be used to solve many classical algorithmic problems. + +- **Finding Nearest Point Pairs**: the algorithm first divides the set of points into two parts, then finds the nearest point pairs in each of the two parts, and finally finds the nearest point pairs that span the two parts. +- **Large integer multiplication**: e.g. the Karatsuba algorithm, which is a solution to large integer multiplication by breaking it down into multiplication and addition of several smaller integers. +- **Matrix multiplication**: for example, the Strassen algorithm, which is a solution to multiply a large matrix into multiple multiplications and additions of smaller matrices. +- **Hanota problem**: The Hanota problem can be considered as a typical divide and conquer strategy, solved by recursion. +- **Solving Inverse Pairs**: In a sequence, if the preceding number is greater than the following number, then these two numbers form an inverse pair. The problem of solving inverse order pairs can be solved by the idea of divide and conquer with the help of merge sort. + +On the other hand, divide and conquer is widely used in the design of algorithms and data structures. + +- **Binary Search**: binary search is to divide an ordered array into two parts from its midpoint index, then decide which half of the interval to exclude based on the result of comparing the target value with the value of the middle element, and then perform the same binary operation on the remaining interval. +- **Merge sort**: introduced at the beginning of the article and will not be repeated. +- **Quick Sort**: Quick sort is the process of picking a base value and then dividing the array into two subarrays, one with elements smaller than the base value and the other with elements larger than the base value, and then performing the same division operation on both parts until there is only one element left in the subarray. +- **Bucket Sort**: The basic idea of bucket sort is to spread the data into multiple buckets, then sort the elements in each bucket, and finally remove the elements from each bucket in turn to get an ordered array. +- **Trees**: e.g., binary search trees, AVL trees, red-black trees, B-trees, B+-trees, etc., whose operations of finding, inserting, and deleting can be considered as applications of divide and conquer. +- **Heap**: A heap is a special kind of complete binary tree whose various operations, such as insertion, deletion, and heapifying, actually imply the idea of partitioning. +- **Hash Tables**: although hash tables come do not directly apply divide and conquer, certain hash collision resolution strategies apply divide and conquer indirectly, e.g., long linked lists in chained addresses are transformed into red-black trees to improve query efficiency. + +It can be seen that **divide and conquer is a "silent" algorithmic idea**, implicit in various algorithms and data structures. diff --git 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For the Hanota problem, however, we use a different solution strategy. + +!!! question + + Given three columns, denoted `A`, `B` and `C` . In the starting state, column `A` is covered with $n$ disks, which are arranged from top to bottom in the order of smallest to largest. Our task is to move these $n$ disks to the column `C` and keep their original order. The following rules need to be observed while moving the disks. + + 1. Discs can only be taken out of the top of one post and put in from the top of another. + 2. Only one disk can be moved at a time. + 3. The smaller disk must be located above the larger disk at all times. + +![Hanota problem example](hanota_problem.assets/hanota_example.png) + +**We denote a Hanota problem of size $i$ as $f(i)$** . For example $f(3)$ represents the Hanota problem of moving $3$ disks from `A` to `C` . + +### Consider The Basics + +As shown in the figure below, for the problem $f(1)$ , i.e., when there is only one disk, we can just move it directly from `A` to `C` . + +=== "<1>" + ![Solution for problem of size 1](hanota_problem.assets/hanota_f1_step1.png) + +=== "<2>" + ![hanota_f1_step2](hanota_problem.assets/hanota_f1_step2.png) + +As shown in the figure below, for the problem $f(2)$ , i.e., when there are two discs, **Since it is necessary to satisfy at all times that the smaller disc is on top of the larger disc, the movement of is accomplished with the help of `B`** . + +1. First, move the top disk from `A` to `B`. +2. Then move the large disk from `A` to `C` . +3. Finally, move the disk from `B` to `C`. + +=== "<1>" + ![Solution for problem of size 2](hanota_problem.assets/hanota_f2_step1.png) + +=== "<2>" + ![hanota_f2_step2](hanota_problem.assets/hanota_f2_step2.png) + +=== "<3>" + ![hanota_f2_step3](hanota_problem.assets/hanota_f2_step3.png) + +=== "<4>" + ![hanota_f2_step4](hanota_problem.assets/hanota_f2_step4.png) + +The solution $f(2)$ can be summarized as follows:**Move two discs from `A` to `C`** with the help of `B` . Where `C` is called the target column and `B` is called the buffer column. + +### Sub-Problem Decomposition + +For the problem $f(3)$ , i.e., when there are three discs, the situation becomes slightly more complicated. + +Since the solutions of $f(1)$ and $f(2)$ are known, we can think in terms of dividing and conquer, **Consider the two disks at the top of `A` as a whole**. Perform the steps shown in the figure below. Thus the three disks are smoothly moved from `A` to `C`. + +1. With `B` as the target column and `C` as the buffer column, move the two disks from `A` to `B`. +2. Move one of the remaining disks in `A` directly from `A` to `C`. +3. With `C` as the target column and `A` as the buffer column, move the two disks from `B` to `C`. + +=== "<1>" + ![Solution for problem of size 3](hanota_problem.assets/hanota_f3_step1.png) + +=== "<2>" + ![hanota_f3_step2](hanota_problem.assets/hanota_f3_step2.png) + +=== "<3>" + ![hanota_f3_step3](hanota_problem.assets/hanota_f3_step3.png) + +=== "<4>" + ![hanota_f3_step4](hanota_problem.assets/hanota_f3_step4.png) + +Essentially, **we divide the problem $f(3)$ into two subproblems $f(2)$ and subproblem $f(1)$** . After solving these three subproblems in order, the original problem is then solved. This shows that the subproblems are independent and that the solutions can be combined. + +At this point, we can summarize the solution strategy for the Hanota problem shown in the figure below: divide and conquer the original problem $f(n)$ into two subproblems $f(n-1)$ and one subproblem $f(1)$ and solve these three subproblems in the following order. + +1. Move $n-1$ discs from `A` to `B` with the help of `C`. +2. Move the remaining $1$ discs directly from `A` to `C`. +3. Move $n-1$ discs from `B` to `C` with the help of `A`. + +For both subproblems $f(n-1)$ , **can be divided recursionally in the same way until the smallest subproblem $f(1)$ is reached**. And the solution of $f(1)$ is known with only one move operation. + +![Divide and conquer strategy for the hanota problem](hanota_problem.assets/hanota_divide_and_conquer.png) + +### Code Implementation + +In the code, we declare a recursion function `dfs(i, src, buf, tar)` that moves the $i$ discs at the top of the column `src` to the target column `tar` with the help of the buffer column `buf` . + +```src +[file]{hanota}-[class]{}-[func]{solve_hanota} +``` + +As shown in the figure below, the Hanota problem forms a recursion tree of height $n$, where each node represents a sub-problem, corresponding to an open `dfs()` function, **and thus has a time complexity of $O(2^n)$ and a space complexity of $O(n)$**. + +![Recursion tree for the hanota problem](hanota_problem.assets/hanota_recursive_tree.png) + +!!! quote + + The hanota problem stems from an ancient legendary story. In an ancient Indian temple, the monks had three tall diamond pillars and $64$ gold disks of various sizes. The monks were constantly moving the original disks, and they believed that the world would end the moment the last disk was placed correctly. + + However, even if the monks moved once every second, it would take a total of about $2^{64} \approx 1.84×10^{19}$ seconds, a contract of $585$ billion years, which is far more than the current estimate of the age of the universe. So, if this legend is true, we should not need to worry about the end of the world. diff --git a/docs-en/chapter_divide_and_conquer/index.md b/docs-en/chapter_divide_and_conquer/index.md new file mode 100644 index 0000000000..a208715f38 --- /dev/null +++ b/docs-en/chapter_divide_and_conquer/index.md @@ -0,0 +1,13 @@ +# Divide And Conquer + +

+ +!!! abstract + + The puzzle is broken down layer by layer, each time making it easier. + + Divide and conquer reveals an important truth: start simple and nothing is complicated anymore. diff --git a/docs-en/chapter_divide_and_conquer/summary.md b/docs-en/chapter_divide_and_conquer/summary.md new file mode 100644 index 0000000000..ab9aae9348 --- /dev/null +++ b/docs-en/chapter_divide_and_conquer/summary.md @@ -0,0 +1,11 @@ +# Summary + +- Partitioning algorithm is a common algorithm design strategy that consists of two phases, divide (divide) and conquer (merge), and is usually implemented based on recursion. +- The basis for determining whether a problem is a divide and conquer algorithm problem includes whether the problem can be divided and conquer, whether the subproblems are independent, and whether the subproblems can be merged. +- Merge sort is a typical application of the partition strategy, which recursively divides and conquers an array into two subarrays of equal length until only one element remains and starts merging them level by level, thus completing the sort. +- Introducing a divide and conquer strategy often leads to an increase in algorithm efficiency. On the one hand, the divide and conquer strategy reduces the number of operations; on the other hand, the partition facilitates the parallel optimization of the system. +- Divide and conquer is both a solution to many algorithmic problems and is widely used everywhere in data structure and algorithm design. +- Compared to brute-force search, adaptive search is more efficient. Searching algorithms with a time complexity of $O(\log n)$ are usually implemented based on a divide and conquer strategy. +- Binary search is another typical application of the divide and conquer strategy that does not include the step of combining the solutions of the subproblems. We can implement binary search by recursion division and conquer. +- In the constructed binary tree problem, the constructed tree (original problem) can be divided into constructed left subtrees and right subtrees (subproblems), which can be realized by dividing the index intervals of the preorder traversal and the middle order traversal. +- In the Hanota problem, a problem of size $n$ can be divided into two subproblems of size $n-1$ and one subproblem of size $1$. 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Programming Problems + +In the previous section, we learned how dynamic programming solves problems by subproblem decomposition. In fact, subproblem decomposition is a generalized algorithmic idea with a different focus in divide and conquer, dynamic programming, and backtracking algorithm. + +- The backtracking algorithm recursively divides and conquers the original problem into a number of mutually independent subproblems up to the smallest subproblem and merges the solutions of the subproblems in backtracking to finally obtain a solution to the original problem. +- Dynamic programming also performs a recursion decomposition of the problem, but the main difference with the divide and conquer algorithm is that the subproblems in dynamic programming are interdependent, and many overlapping subproblems appear during the decomposition process. +- The backtracking algorithm exhausts all possible solutions in try and backtrack and avoids unnecessary search branches by pruning. The solution to the original problem consists of a series of decision steps, and we can view the subsequence preceding each decision step as a subproblem. + +In fact, dynamic programming is often used to solve optimization problems, which not only contain overlapping subproblems, but also have two other major properties: optimal substructure, and no posteriority. + +## Optimal Substructure + +We modify the stair-climbing problem slightly to make it more suitable for demonstrating the optimal substructure concept. + +!!! question "Minimal cost of stair climbing" + + Given a staircase where you can go up either $1$ or $2$ steps per step, each step of the staircase is labeled with a non-negative integer indicating the cost you need to pay at that step. Given an array of non-negative integers $cost$ , where $cost[i]$ denotes the cost you need to pay at the $i$th step, and $cost[0]$ is the ground level starting point. Please calculate the minimum cost to reach the top? + +As shown in the figure below, if the costs of the $1$, $2$, and $3$ steps are $1$, $10$, and $1$, respectively, then the minimum cost of climbing from the ground to the $3$ step is $2$. + +![Minimum cost to climb to 3rd order](dp_problem_features.assets/min_cost_cs_example.png) + +Let $dp[i]$ be the cumulative cost of climbing to the $i$th step. Since the $i$th step can only come from the $i - 1$th step or the $i - 2$th step, $dp[i]$ can only be equal to either $dp[i - 1] + cost[i]$ or $dp[i - 2] + cost[i]$. To minimize the cost, we should choose the smaller of the two: + +$$ +dp[i] = \min(dp[i-1], dp[i-2]) + cost[i] +$$ + +This leads to the meaning of optimal substructure:**the optimal solution of the original problem is constructed from the optimal solution of the subproblem**. + +This problem clearly has an optimal substructure: we pick the better of the two subproblem optimal solutions $dp[i-1]$ and $dp[i-2]$ and use it to construct an optimal solution to the original problem $dp[i]$. + +So, is there an optimal substructure for the stair-climbing problem in the previous section? Its goal is to solve for the number of solutions, which may seem like a counting problem, but if it is asked in a different way, "Solve for the maximum number of solutions." We unexpectedly find that **although the questions are equivalent before and after the modification, the optimal substructure surfaces**: the number of maximal solutions of order $n$ is equal to the sum of the number of maximal solutions of order $n-1$ and order $n-2$. So, the optimal substructure is interpreted in a flexible way and will have different meanings in different problems. + +Based on the state-trasition equation, and the initial states $dp[1] = cost[1]$ and $dp[2] = cost[2]$ , we get the dynamic programming code. + +```src +[file]{min_cost_climbing_stairs_dp}-[class]{}-[func]{min_cost_climbing_stairs_dp} +``` + +The figure below shows the dynamic programming process of the above code. + +![Dynamic programming process for minimizing the cost of stair climbing](dp_problem_features.assets/min_cost_cs_dp.png) + +This problem can also be space optimized by compressing one dimension to zero dimension, which makes the space complexity decrease from $O(n)$ to $O(1)$ . + +```src +[file]{min_cost_climbing_stairs_dp}-[class]{}-[func]{min_cost_climbing_stairs_dp_comp} +``` + +## No After-Effects + +The absence of posteriority is one of the important properties of dynamic programming that enables effective problem solving and is defined as: **Given a definite state, its future development is related only to the current state and not to all the states that the current state has experienced in the past**. + +Taking the stair climbing problem as an example, given the state $i$, it will develop the states $i+1$ and $i+2$, which correspond to jumping $1$ and $2$ steps respectively. When making these two choices, we do not need to consider the states before state $i$, they have no effect on the future of state $i$. + +However, if we add a constraint to the stair climbing problem, the situation is different. + +!!! question "climbing stairs with constraints" + + Given a staircase with a total of $n$ steps, you can go up either $1$ steps or $2$ steps per step, **but you can't jump $1$ steps in two consecutive rounds**. How many scenarios are there to get to the top of the building. + +For example, in the figure below, there are only $2$ feasible solutions left to climb up to the $3$ stage, among which the solution with three consecutive jumps of $1$ stage does not satisfy the constraints, and is therefore discarded. + +![Number of scenarios with constraints to climb to 3rd degree](dp_problem_features.assets/climbing_stairs_constraint_example.png) + +In that problem, if the previous round came up by jumping $1$ steps, then the next round must jump $2$ steps. **This means that the choice of next step cannot be independently determined by the current state (current stair step), but is also related to the previous state (previous round stair step)**. + +It is easy to find that this problem does not satisfy the no-laterality, and the state-trasition equation $dp[i] = dp[i-1] + dp[i-2]$ also fails because $dp[i-1]$ stands for the current round of jumping $1$ steps, but it contains many "previous round of jumping $1$ steps up" scenarios. In order to satisfy the constraint, we cannot directly count $dp[i-1]$ into $dp[i]$. + +To do so, we need to extend the state definition:**The state $[i, j]$ denotes that it is at the $i$th order and has jumped the $j$th order in the previous round**, where $j \in \{1, 2\}$ . This state definition effectively distinguishes whether the last round has jumped $1$ or $2$ steps, and we can use it to determine where the current state comes from. + +- When $1$ step was jumped in the previous round, the previous round can only choose to jump $2$ step, i.e. $dp[i, 1]$ can only be transferred from $dp[i-1, 2]$. +- When $2$ step is jumped in the previous round, the previous round can choose to jump $1$ step or $2$ step, i.e. $dp[i, 2]$ can be transferred from $dp[i-2, 1]$ or $dp[i-2, 2]$. + +As shown in the figure below, under this definition, $dp[i, j]$ denotes the number of programs corresponding to the state $[i, j]$. At this time, the state-transition equation is: + +$$ +\begin{cases} +dp[i, 1] = dp[i-1, 2] \\ +dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] +\end{cases} +$$ + +![Consider recursive relationships under constraints](dp_problem_features.assets/climbing_stairs_constraint_state_transfer.png) + +In the end, it is enough to return $dp[n, 1] + dp[n, 2]$, the sum of which represents the total number of scenarios that have climbed to the $n$th step. + +```src +[file]{climbing_stairs_constraint_dp}-[class]{}-[func]{climbing_stairs_constraint_dp} +``` + +In the above case, by considering only one more previous state, we can still extend the definition of the state to make the problem satisfy no posteriority again. However, some problems have very serious "consequences". + +!!! question "Stair Climbing and Obstacle Generation" + + Given a staircase with a total of $n$ steps, you can go up either $1$ steps or $2$ steps per step. **states that when climbing to the $i$th step, the system will automatically put obstacles on the $2i$th step, and all rounds after that are not allowed to jump to the $2i$th step**. For example, if the first two rounds are jumped to the $2$ and $3$ steps, you are not allowed to jump to the $4$ and $6$ steps after that. Ask how many options there are to climb to the top of the building. + +In this problem, the next jump depends on all past states, since each jump puts obstacles on higher rungs of the ladder and affects future jumps. Dynamic programming is often difficult to solve for this type of problem. + +In fact, many complex combinatorial optimization problems (e.g., the traveler's problem) do not satisfy no posteriority. 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zGVXpp6AGx3E(z?`TXrcwULBLSq3v$g%X?yN=y@O5uFrnfXu}W&qDy+nvXyqID|o=1 zmw3#yl9*q$0q@>bbmxbeAJ~WcTdapnC$(>mbzJm}(@B^q+NP=M;lQ~?rfh4^*~+=f zrVd*}g~yBL@5cbP=+orxZ0^DS_{y?rf>C7pL%=K~y?at+U$DI>*SnIF5doy+E0!JS d06-T>!" + ![edit dynamic programming process for distance](edit_distance_problem.assets/edit_distance_dp_step1.png) + +=== "<2>" + ![edit_distance_dp_step2](edit_distance_problem.assets/edit_distance_dp_step2.png) + +=== "<3>" + ![edit_distance_dp_step3](edit_distance_problem.assets/edit_distance_dp_step3.png) + +=== "<4>" + ![edit_distance_dp_step4](edit_distance_problem.assets/edit_distance_dp_step4.png) + +=== "<5>" + ![edit_distance_dp_step5](edit_distance_problem.assets/edit_distance_dp_step5.png) + +=== "<6>" + ![edit_distance_dp_step6](edit_distance_problem.assets/edit_distance_dp_step6.png) + +=== "<7>" + ![edit_distance_dp_step7](edit_distance_problem.assets/edit_distance_dp_step7.png) + +=== "<8>" + ![edit_distance_dp_step8](edit_distance_problem.assets/edit_distance_dp_step8.png) + +=== "<9>" + ![edit_distance_dp_step9](edit_distance_problem.assets/edit_distance_dp_step9.png) + +=== "<10>" + ![edit_distance_dp_step10](edit_distance_problem.assets/edit_distance_dp_step10.png) + +=== "<11>" + ![edit_distance_dp_step11](edit_distance_problem.assets/edit_distance_dp_step11.png) + +=== "<12>" + ![edit_distance_dp_step12](edit_distance_problem.assets/edit_distance_dp_step12.png) + +=== "<13>" + ![edit_distance_dp_step13](edit_distance_problem.assets/edit_distance_dp_step13.png) + +=== "<14>" + ![edit_distance_dp_step14](edit_distance_problem.assets/edit_distance_dp_step14.png) + +=== "<15>" + ![edit_distance_dp_step15](edit_distance_problem.assets/edit_distance_dp_step15.png) + +### Space Optimization + +Since $dp[i,j]$ is transferred from the upper $dp[i-1, j]$, the left $dp[i, j-1]$, and the upper-left state $dp[i-1, j-1]$, and the orthogonal traversal loses the upper-left $dp[i-1, j-1]$ , and the inverted traversal can not build $dp[i, j-1]$ in advance, both traversal orders are not desirable. + +To do this, we can use a variable `leftup` to temporarily store the top-left solution $dp[i-1, j-1]$ so that only the left and top solutions need to be considered. The situation is the same as the complete knapsack problem, and can be traversed using positive order. + +```src +[file]{edit_distance}-[class]{}-[func]{edit_distance_dp_comp} +``` diff --git a/docs-en/chapter_dynamic_programming/index.md b/docs-en/chapter_dynamic_programming/index.md new file mode 100644 index 0000000000..4064c34549 --- /dev/null +++ b/docs-en/chapter_dynamic_programming/index.md @@ -0,0 +1,13 @@ +# Dynamic Programming + +

+ +!!! abstract + + Streams merge into rivers and rivers merge into the sea. + + Dynamic programming brings together solutions to small problems into answers to large ones, leading us step by step to the solution of the problem. diff --git a/docs-en/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_dfs_memo_tree.png b/docs-en/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_dfs_memo_tree.png new file mode 100644 index 0000000000000000000000000000000000000000..bccc8424cde2ac2529a61cf23d09b6e9f086d0c8 GIT binary patch literal 28461 zcmb4qbyQVf@b9_z(hXA5ARygHNeD;?2m;b64HD8Iap?{T=|&m^K|u1NbR*r}ozlF+ z_x;}c`#sjWi*xoqd*(BY;0UzUmqSG{`>b& zL_{PkEbQ`k+vU~e-rnAUxmjy#>&?wgO?CD5_V&WULUeSrl$2C=clY%4bbel5M@L6S zdPe`md|hpAWo2bjQc_CE&$YERUtiyXf`Z=O-o}Q8#Kc5#aq(aE_5S|;t#hrnXJ>PB za}f~{73Jj%9MV}?S)nZh>nSNIm&eD$!^0sVAqNKsU%q^?va)JxYdbqT zi;s`@^zUY zv;FBe2sHmvL>FMT~rZrBPw6wJS^ZlZt 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zbj@CWR})^(^Y&K-IAmHEJX)3eB?AG3dU>=B_Wtn>9jFKwuAj7p6;3OcnER5izR4vae+*1c@fk< zmYlb1#YLdG2Je!+7*9;43zqt8 lq%+jlFGL4=IJrKz94?5)l05sm0uEfYv4MsDYdzBK{{kQ44N3q2 literal 0 HcmV?d00001 diff --git a/docs-en/chapter_dynamic_programming/knapsack_problem.md b/docs-en/chapter_dynamic_programming/knapsack_problem.md new file mode 100644 index 0000000000..7277e457f3 --- /dev/null +++ b/docs-en/chapter_dynamic_programming/knapsack_problem.md @@ -0,0 +1,168 @@ +# 0-1 Knapsack Problem + +The knapsack problem is a great introductory topic to dynamic programming and is the most common form of the problem in dynamic programming. It has many variants, such as the 0-1 knapsack problem, the complete knapsack problem, and the multiple knapsack problem. + +In this section, we start by solving the most common 0-1 knapsack problem. + +!!! question + + Given $n$ items, the weight of the first $i$ item is $wgt[i-1]$, the value is $val[i-1]$ , and a backpack with capacity $cap$. Each item can be selected only once, and the maximum value of the item that can be put into the backpack without exceeding its capacity is asked. + +Observing the figure below, since the item number $i$ is counted from $1$ and the array index is counted from $0$, the item $i$ corresponds to the weight $wgt[i-1]$ and value $val[i-1]$ . + +![0-1 Example data for backpacks](knapsack_problem.assets/knapsack_example.png) + +We can view the 0-1 knapsack problem as a process consisting of $n$ rounds of decisions, with two decisions for each object: not to put it in and to put it in, and thus the problem is satisfying the decision tree model. + +The goal of the problem is to solve for the "maximum value given the capacity of the knapsack", so with a high probability it is a dynamic programming problem. + +**Step 1: Think about the decisions made in each round and define the state that leads to the $dp$ table** + +For each item, if it is not put into the backpack, the backpack capacity remains unchanged; if it is put into the backpack, the backpack capacity decreases. This leads to the state definition: the current item number $i$ and the remaining backpack capacity $c$ , denoted as $[i, c]$ . + +**The subproblem corresponding to the state $[i, c]$ is**: The maximum value of the previous $i$ items in the knapsack with remaining capacity $c$, denoted as $dp[i, c]$ . + +The solution to be solved is $dp[n, cap]$ , so a two-dimensional $dp$ table of size $(n+1) \times (cap+1)$ is needed. + +**Step 2: Find the optimal substructure, and thus derive the state-transition equation** + +When we make a decision for item $i$, what remains is the decision for the first $i-1$ items, which can be categorized into the following two cases. + +- **Do not put in items $i$** : The backpack capacity remains unchanged and the state changes to $[i-1, c]$ . +- **Put in item $i$** : Backpack capacity decreases $wgt[i-1]$ , value increases $val[i-1]$ , state changes to $[i-1, c-wgt[i-1]]$ . + +The above analysis reveals to us the optimal substructure of the problem: **The maximum value $dp[i, c]$ equals the one** that has the greater value of the two alternatives of leaving items out $i$ and putting items in $i$. This leads to the state-trasition equation: + +$$ +dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) +$$ + +Note that if the current item weight $wgt[i - 1]$ exceeds the remaining backpack capacity $c$, you can only choose not to put it into the backpack. + +**Step 3: Determine boundary conditions and state transfer order** + +The maximum value when there are no items or no remaining backpack capacity is $0$ , i.e. the first column $dp[i, 0]$ and the first row $dp[0, c]$ are both equal to $0$ . + +The current state $[i, c]$ is transferred from the top state $[i-1, c]$ and the top-left state $[i-1, c-wgt[i-1]]$, so it is sufficient to traverse the entire $dp$ table in positive order with a two-level loop. + +Based on the above analysis, we next implement the brute-force search, memorized search, and dynamic programming solutions in order. + +### Method One: Brute Force Search + +The search code contains the following elements. + +- **recursion parameter**: state $[i, c]$ . +- **Return value**: solution to the subproblem $dp[i, c]$ . +- **Termination condition**: when the item number crosses the boundary $i = 0$ or the remaining capacity of the backpack is $0$, the recursion is terminated and the value $0$ is returned. +- **Pruning**: if the weight of the current item exceeds the remaining capacity of the backpack, it can only be left out of the backpack. + +```src +[file]{knapsack}-[class]{}-[func]{knapsack_dfs} +``` + +As shown in the figure below, the time complexity is $O(2^n)$ since each item generates two search branches, unselected and selected. + +Observing the recursion tree, it is easy to find that there are overlapping subproblems in it, such as $dp[1, 10]$ and so on. And when there are more items and the knapsack capacity is large, especially when there are more items of the same weight, the number of overlapping subproblems will increase dramatically. + +![0-1 Backpacking's brute-force search recursion tree](knapsack_problem.assets/knapsack_dfs.png) + +### Method 2: Memorized Search + +To ensure that the overlapping subproblems are computed only once, we record the solutions of the subproblems with the help of a memoized list `mem` , where `mem[i][c]` corresponds to $dp[i, c]$ . + +With the introduction of memoization, **the time complexity depends on the number of subproblems**, which is $O(n \times cap)$ . + +```src +[file]{knapsack}-[class]{}-[func]{knapsack_dfs_mem} +``` + +The figure below shows the search branches that were clipped in the memorized recursion. + +![0-1 Memorized search recursion tree for backpacks](knapsack_problem.assets/knapsack_dfs_mem.png) + +### Method 3: Dynamic Programming + +Dynamic programming is essentially the process of populating the $dp$ table in a state transfer, as shown in the code below. + +```src +[file]{knapsack}-[class]{}-[func]{knapsack_dp} +``` + +As shown in the figure below, both time complexity and space complexity are determined by the size of the array `dp` , i.e. $O(n \times cap)$ . + +=== "<1>" + ![0-1 Dynamic programming process for backpacks](knapsack_problem.assets/knapsack_dp_step1.png) + +=== "<2>" + ![knapsack_dp_step2](knapsack_problem.assets/knapsack_dp_step2.png) + +=== "<3>" + ![knapsack_dp_step3](knapsack_problem.assets/knapsack_dp_step3.png) + +=== "<4>" + ![knapsack_dp_step4](knapsack_problem.assets/knapsack_dp_step4.png) + +=== "<5>" + ![knapsack_dp_step5](knapsack_problem.assets/knapsack_dp_step5.png) + +=== "<6>" + ![knapsack_dp_step6](knapsack_problem.assets/knapsack_dp_step6.png) + +=== "<7>" + ![knapsack_dp_step7](knapsack_problem.assets/knapsack_dp_step7.png) + +=== "<8>" + ![knapsack_dp_step8](knapsack_problem.assets/knapsack_dp_step8.png) + +=== "<9>" + ![knapsack_dp_step9](knapsack_problem.assets/knapsack_dp_step9.png) + +=== "<10>" + ![knapsack_dp_step10](knapsack_problem.assets/knapsack_dp_step10.png) + +=== "<11>" + ![knapsack_dp_step11](knapsack_problem.assets/knapsack_dp_step11.png) + +=== "<12>" + ![knapsack_dp_step12](knapsack_problem.assets/knapsack_dp_step12.png) + +=== "<13>" + ![knapsack_dp_step13](knapsack_problem.assets/knapsack_dp_step13.png) + +=== "<14>" + ![knapsack_dp_step14](knapsack_problem.assets/knapsack_dp_step14.png) + +### Space Optimization + +Since each state is only related to the state of its previous row, we can use two arrays to scroll forward, reducing the space complexity from $O(n^2)$ to $O(n)$ . + +Thinking further, can we achieve spatial optimization using only an array? Observation shows that each state is transferred from the lattice directly above or to the upper left. Assuming there is only one array, when starting to traverse the $i$th row, the array stores still the state of the $i-1$th row. + +- If we take positive order traversal, then when traversing to $dp[i, j]$, the top left $dp[i-1, 1]$ ~ $dp[i-1, j-1]$ values may have been overwritten, and then we cannot get the correct state transfer result. +- If an inverted traversal is taken, the overwrite problem does not occur and the state transfer can be performed correctly. + +The figure below shows the conversion from row $i = 1$ to row $i = 2$ under a single array. Consider the difference between an orthogonal traversal and an inverted traversal. + +=== "<1>" + ![0-1 Dynamic programming process after spatial optimization of backpacks](knapsack_problem.assets/knapsack_dp_comp_step1.png) + +=== "<2>" + ![knapsack_dp_comp_step2](knapsack_problem.assets/knapsack_dp_comp_step2.png) + +=== "<3>" + ![knapsack_dp_comp_step3](knapsack_problem.assets/knapsack_dp_comp_step3.png) + +=== "<4>" + ![knapsack_dp_comp_step4](knapsack_problem.assets/knapsack_dp_comp_step4.png) + +=== "<5>" + ![knapsack_dp_comp_step5](knapsack_problem.assets/knapsack_dp_comp_step5.png) + +=== "<6>" + ![knapsack_dp_comp_step6](knapsack_problem.assets/knapsack_dp_comp_step6.png) + +In the code implementation, we simply remove the first dimension $i$ of the array `dp` and change the inner loop to an inverted traversal. + +```src +[file]{knapsack}-[class]{}-[func]{knapsack_dp_comp} +``` diff --git a/docs-en/chapter_dynamic_programming/summary.md b/docs-en/chapter_dynamic_programming/summary.md new file mode 100644 index 0000000000..feaf457088 --- /dev/null +++ b/docs-en/chapter_dynamic_programming/summary.md @@ -0,0 +1,23 @@ +# Summary + +- Dynamic programming decomposes the problem and achieves efficient computational efficiency by storing solutions to subproblems to circumvent repeated computations. +- All dynamic programming problems can be solved using backtracking (brute force search) without taking time into account, but there are a large number of overlapping subproblems in the recursion tree, which is extremely inefficient. By introducing a memorized list, the solutions of all computed subproblems can be stored, thus ensuring that overlapping subproblems are computed only once. +- Memorized recursion is a top-to-bottom recursion, while its counterpart, dynamic programming, is a bottom-to-top recursion, which is like "filling in a table". Since the current state only depends on some local states, we can eliminate one dimension of the $dp$ table, thus reducing the space complexity. +- Subproblem decomposition is a generalized algorithmic idea with different properties in divide and conquer, dynamic programming, and backtracking algorithm. +- Three main properties of dynamic programming problems: overlapping subproblems, optimal substructure, and no posteriority. +- An original problem has an optimal substructure if its optimal solution can be constructed from the optimal solution of the subproblem. +- A posteriority-free approach means that for a state, its future development is related only to that state, independent of all past states it has experienced. Many combinatorial optimization problems are not posteriority free and cannot be solved quickly using dynamic programming. + +**Knapsack problem** + +- The knapsack problem is the most typical dynamic programming topic with variants of the problem such as 0-1 knapsack, complete knapsack, and multiple knapsack. +- 0-1 The state of the backpack is defined as the maximum value of the first $i$ items in the backpack with remaining capacity $c$. Based on the two decisions of not putting into the backpack and putting into the backpack, the optimal substructure can be obtained and the state-trasition equation is constructed. In spatial optimization, since each state depends on the states directly above and above left, it is necessary to traverse the list in inverse order to avoid the state above left being covered. +- A full backpack has an unlimited number of selections per item, so the state transfer for selecting items to put into it is different from that of a 0-1 backpack. Since the state depends on the states directly above and directly to the left, it should be traversed in positive order in spatial optimization. +- The change exchange problem is a variant of the knapsack problem. It changes from finding the "maximum" value to finding the "minimum" number of coins, so $\max()$ in the state-transition equation should be changed to $\min()$. From "not exceeding" the backpack capacity to "exactly" the target amount, so $amt + 1$ is used to denote an invalid solution that "fails to reach the target amount". +- The Change Exchange II problem was changed from "Minimum number of coins" to "Number of combinations of coins", and the state-transition equation was changed from $\min()$ to a summation operator. + +**Edit distance problem** + +- The edit distance (Levenshtein distance) is used to measure the similarity between two strings, which is defined as the minimum number of editing steps from one string to another, and the editing operations include add, delete, and replace. +- The state of the edit distance problem is defined as the minimum number of edit steps required to change the first $i$ characters of $s$ to the first $j$ characters of $t$. When $s[i] \ne t[j]$ has three decisions: add, delete, and replace, they all have corresponding remaining subproblems. Accordingly, the optimal substructure can be found with the construction of state-trasition equation. And when $s[i] = t[j]$, there is no need to edit the current character. +- In edit distance, the state depends on the states directly above, directly to the left, and above left of it, so that neither positive-order nor inverted-order traversal after spatial optimization can correctly transfer the state. 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+1,209 @@ +# Complete Knapsack Problem + +In this section, we start by solving another common knapsack problem: complete knapsacks, and then learn about a special case of it: change exchange. + +## Totally Backpacked + +!!! question + + Given $n$ items, the weight of the $i$th item is $wgt[i-1]$, the value is $val[i-1]$ , and a backpack with capacity $cap$. **Each item can be selected repeatedly**. Please return the maximum value of an item that can be put into the backpack without exceeding its capacity. + +![Example data for the complete knapsack problem](unbounded_knapsack_problem.assets/unbounded_knapsack_example.png) + +### Ideas For Dynamic Programming + +The full knapsack is very similar to the 0-1 knapsack problem, **differing only in not limiting the number of times an item can be selected**. + +- In 0-1 backpacks, there is only one item per backpack, so when you put an item $i$ into your backpack, you can only choose from the first $i-1$ items. +- In a Full Backpack, there are an infinite number of each item, so after placing an item $i$ into the backpack, **can still select from the previous $i$ items**. + +Under the full backpacking provision, changes in the state $[i, c]$ are categorized into two cases. + +- **Do not put items $i$** : Same as 0-1 backpack, transfer to $[i-1, c]$ . +- **Put in item $i$** : Transfer to $[i, c-wgt[i-1]]$ unlike 0-1 backpack. + +Thus the state-trasition equation becomes: + +$$ +dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) +$$ + +### Code Implementation + +Comparing the code of the two questions, one of the state transfers changes from $i-1$ to $i$, and the rest are identical. + +```src +[file]{unbounded_knapsack}-[class]{}-[func]{unbounded_knapsack_dp} +``` + +### Space Optimization + +Since the current state is transferred from the state on the left and above, **the spatial optimization should take a positive-order traversal of each row in the $dp$ table**. + +This traversal order is the opposite of 0-1 backpack. Please take the help of the figure below to understand the difference. + +=== "<1>" + ![Dynamic programming process after spatial optimization of a complete knapsack](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step1.png) + +=== "<2>" + ![unbounded_knapsack_dp_comp_step2](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step2.png) + +=== "<3>" + ![unbounded_knapsack_dp_comp_step3](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step3.png) + +=== "<4>" + ![unbounded_knapsack_dp_comp_step4](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step4.png) + +=== "<5>" + ![unbounded_knapsack_dp_comp_step5](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step5.png) + +=== "<6>" + ![unbounded_knapsack_dp_comp_step6](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step6.png) + +The code is relatively simple to implement, simply removing the first dimension of the array `dp`. + +```src +[file]{unbounded_knapsack}-[class]{}-[func]{unbounded_knapsack_dp_comp} +``` + +## Change Exchange Problems + +The knapsack problem is representative of a large class of dynamic programming problems that have many variants, such as the change exchange problem. + +!!! question + + Given $n$ coins, the face value of the first $i$ coin is $coins[i - 1]$ and the target amount is $amt$ , **each coin can be selected repeatedly**, ask the minimum number of coins that can make the target amount. Return $-1$ if the target amount cannot be rounded up. + +![Example data for the coin exchange problem](unbounded_knapsack_problem.assets/coin_change_example.png) + +### Ideas For Dynamic Programming + +**Change exchange can be seen as a special case of full backpacking**. The two have the following connections and differences. + +- The two questions can be interchanged, with "item" corresponding to "coin", "item weight" corresponding to "coin denomination "Item" corresponds to "Coin", "Item Weight" corresponds to "Coin Face Value", and "Backpack Capacity" corresponds to "Target Amount". +- The optimization goals are reversed; the knapsack problem is to maximize the value of items and the change exchange problem is to minimize the number of coins. +- The knapsack problem is to find a solution that "does not exceed" the capacity of the knapsack, and the change exchange is to find a solution that "just happens" to be the target amount. + +**Step 1: Think about the decisions made in each round and define the state that leads to the $dp$ table** + +The subproblem corresponding to the state $[i, a]$ is: **The minimum number of coins that the first $i$ coins can come up with the amount $a$**, denoted as $dp[i, a]$ . + +The dimensions of the 2D $dp$ table are $(n+1) \times (amt+1)$ . + +**Step 2: Find the optimal substructure, and hence derive the state-transition equation** + +There are two differences between this problem and the state-trasition equation for a complete backpack, as follows. + +- This question requires the minimum value, so you need to change the operator $\max()$ to $\min()$. +- The subject of optimization is the number of coins, not the value of the item, so just execute $+1$ when you select the coins. + +$$ +dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1) +$$ + +**Step 3: Determine boundary conditions and state transfer order** + +When the target amount is $0$, the minimum number of coins to make it is $0$, i.e. all $dp[i, 0]$ in the first column are equal to $0$. + +When there are no coins, **cannot come up with any $> 0$ target amount**, i.e., it is an invalid solution. In order for the $\min()$ function in the state-transition equation to recognize and filter invalid solutions, we consider using $+ \infty$ to represent them, i.e., making all $dp[0, a]$ in the first row equal to $+ \infty$ . + +### Code Implementation + +Most programming languages do not provide $+ \infty$ variables, and the maximum value of the integer `int` can be used instead. This, in turn, can lead to large number overruns: the $+ 1$ operation in the state-transition equation can overflow. + +For this reason, we use the number $amt + 1$ to denote the null solution, since the maximum number of coins that make up $amt$ is $amt$. + +Before the final return, determine if $dp[n, amt]$ is equal to $amt + 1$, if it is, then return $-1$, which means the target amount cannot be reached. + +```src +[file]{coin_change}-[class]{}-[func]{coin_change_dp} +``` + +The figure below shows the dynamic programming process for change exchange, very similar to a full backpack. + +=== "<1>" + ![Dynamic programming process for the change exchange problem](unbounded_knapsack_problem.assets/coin_change_dp_step1.png) + +=== "<2>" + ![coin_change_dp_step2](unbounded_knapsack_problem.assets/coin_change_dp_step2.png) + +=== "<3>" + ![coin_change_dp_step3](unbounded_knapsack_problem.assets/coin_change_dp_step3.png) + +=== "<4>" + ![coin_change_dp_step4](unbounded_knapsack_problem.assets/coin_change_dp_step4.png) + +=== "<5>" + ![coin_change_dp_step5](unbounded_knapsack_problem.assets/coin_change_dp_step5.png) + +=== "<6>" + ![coin_change_dp_step6](unbounded_knapsack_problem.assets/coin_change_dp_step6.png) + +=== "<7>" + ![coin_change_dp_step7](unbounded_knapsack_problem.assets/coin_change_dp_step7.png) + +=== "<8>" + ![coin_change_dp_step8](unbounded_knapsack_problem.assets/coin_change_dp_step8.png) + +=== "<9>" + ![coin_change_dp_step9](unbounded_knapsack_problem.assets/coin_change_dp_step9.png) + +=== "<10>" + ![coin_change_dp_step10](unbounded_knapsack_problem.assets/coin_change_dp_step10.png) + +=== "<11>" + ![coin_change_dp_step11](unbounded_knapsack_problem.assets/coin_change_dp_step11.png) + +=== "<12>" + ![coin_change_dp_step12](unbounded_knapsack_problem.assets/coin_change_dp_step12.png) + +=== "<13>" + ![coin_change_dp_step13](unbounded_knapsack_problem.assets/coin_change_dp_step13.png) + +=== "<14>" + ![coin_change_dp_step14](unbounded_knapsack_problem.assets/coin_change_dp_step14.png) + +=== "<15>" + ![coin_change_dp_step15](unbounded_knapsack_problem.assets/coin_change_dp_step15.png) + +### Space Optimization + +The space-optimized handling of change redemption is consistent with a full backpack. + +```src +[file]{coin_change}-[class]{}-[func]{coin_change_dp_comp} +``` + +## Change Exchange Problem II + +!!! question + + Given $n$ coins, the face value of the first $i$ coin is $coins[i - 1]$ , and the target amount is $amt$ , and each coin can be selected repeatedly, **Please return the number of coin combinations** that are in the process of rounding up to the target amount. + +![Example data for Coin Exchange Problem II](unbounded_knapsack_problem.assets/coin_change_ii_example.png) + +### Ideas For Dynamic Programming + +Compared to the previous question, the goal of this question is the number of combinations, so the sub-problem becomes: **the number of combinations** of the first $i$ coins that can come up with the amount $a$. And the $dp$ table is still a 2D matrix of size $(n+1) \times (amt + 1)$. + +The number of combinations of the current state is equal to the sum of the number of combinations of the two decisions of not choosing the current coin and choosing the current coin. The state-transition equation is: + +$$ +dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] +$$ + +When the target amount is $0$, no coins need to be selected to get the target amount, so all $dp[i, 0]$ in the first row should be initialized to $1$ . When there are no coins, it is impossible to get any $>0$ target amount, so all $dp[0, a]$ in the first row are equal to $0$. + +### Code Implementation + +```src +[file]{coin_change_ii}-[class]{}-[func]{coin_change_ii_dp} +``` + +### Space Optimization + +Spatial optimization is handled in the same way, by removing the coin dimension. + +```src +[file]{coin_change_ii}-[class]{}-[func]{coin_change_ii_dp_comp} +``` diff --git a/docs-en/chapter_graph/graph.assets/adjacency_list.png b/docs-en/chapter_graph/graph.assets/adjacency_list.png new file mode 100644 index 0000000000000000000000000000000000000000..2c91ed0f05d4226bddf1d6cb3d53fb5cf4315f91 GIT binary patch literal 20549 zcmc$_XHXPD8!kFCyGzblNehy5k_@g0k_AzM#1&BxBq&G@gNT6S9EDZMNt6rQ?R6O!eFS%=sHm8ooxOVX>W?2kI%hkaoSb@ldlwfMZES4n>+27W4>mS7ii?Z2wY7(r zh7wy7BAZ`6Zw$%H%X4>kx3#t1-Q9I_b2BzJj){r!^z{7n>C?OZaz8)6*RNmSxN&1- zWTd~pe|>#@XJ==Ab527;V|jUbY;0_3XsBttDXt|pr21LyNKH#iOJ-)Ks;Vjs!`;8S zYierly|#Vz>Q!K1;GH{njEs!Z($ah@eA2$8hQ1En+}|8u8SkC%tsMIB;>8Qx*PK6x ze>}^cKFo7?@ZdpsQ`o@5Kngzj*|TS5Wo6yn-SP49i`$Dy$w^rqnNw?%6B83rQBe&e z4aI#$k&%(b-9_PzVS(=h^1AY@vMqahdh&bnDhDcEi(FDllEV_iLdruvbbPoyd+{{@ z+<>*!uio|^Uz<>JS(Tu{+*ErOn527?Z?w?%wK;hlxDjOF`r^Z{k28?kbSt-3+_9~E z6(xYqRC_X2^T>i2_%h1l^h)=x)cFumt=5s{J^-uL=_%IL^<@N+NgZe(+Uj))xq`DD zJ0^JyZ=V}JXu+QCdqmEQ^R0fG=?zG!>;<-3*A?s+#U54)>Ojl~GC?f0`a6rK^~9F} zjO4R%gU!pq6S5gIY`O(UKaMHi%`JgAfsXBqRlyl}m(}lbZ})qT@2rj-+^cy>3j|gf zbRAo>%KN=Tv3*Zn^FKjXP@cfjvHMiXT+ZQW!9Y5xY7MhKxB}!g zK%>g0>|j8{2ENE|RGMv0(dtzC1rieYJ|x?(x9a`;hiXHTDSSX##}x#;+`EBwp9$Wf zt8njM=WH=VEw|hFy=Z2;C0&M#mYNp14{N@q_gS(n6&W;nFMSMKQ#pwPe>-|Fu?t9sk0{( z%FT$wiobFE7y=8o1El=g{$rFOKmO@2S{BUFbWX^H1tmCe1dv72V77VSXd<_R17sm+ zNs5O5JOoDe+)*)}d@!)$M(kFzU5Q|;hY*FBCMed2SPK9LLjkCe2}Bx-LL@kI0zx$j z8g`_`07Kj`P^^@uf7{S@W2#byf*54ElB+f4Ta`a#SZzkTW@JNplkM59i}B`cmGVPFU9sYAAL-%- zstoHE<=DHPu$^$G5nTU2{0~k(^=AIS8)LUW<&qeEW5znSJm=Hm{y@Bt>+V6`BIP`1 zfFQ+-^{IL`5Gh-{nX;SfR*)^@d8CkvHeG%$*Gx`M3|Bxvh2% z4@Hb?*@H4}{QP@I^NF;I&D@j5Z~!XWy1a+u-OkT>tlOOXAU{4crbm6z$>lcE?ZSMy zh8^4TTOR+tJloR5Pdq4ti%eGX9tGR_xy6_4JqC30HE4T6Im(_WK;br?8;f!eC1U0d z`8}_Ba2z59qEKrCr191QOgbj&w`wCjxDdYcpM#_0>e>>6nzjsa_tUgq5$*X{HO%~7 z(9ZT6E||z|?hr+oxcBfhiC%C;FtNZ+KvbB}m* z?{&&_Uw`|v-&rIYefY@s^)=DpWOp&|hQzB@A8&6o5#ty_ImU!t3jg-_VC(>1LANZO zJdqfqBJnh(yclPu5uI?s0?9X|Tm|lNZWs1d+to7x^OtT+{sV|9gL^N=fQ^9Ws}l#O z$Oj&=&&x@0k0-tsezTtNS`OpWL1hly{OChZugNpX7N1S#oZ*~;7*d#yk zM@w+(9@UgtXXFp%}k7YfDWsjc7Ds(ASNL3|+do=Y?kz(Hc8W9mUd>^^; 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We can abstractly represent a graph $G$ as a set of vertices $V$ and a set of edges $E$. The following example shows a graph with 5 vertices and 7 edges. + +$$ +\begin{aligned} +V & = \{ 1, 2, 3, 4, 5 \} \newline +E & = \{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \} \newline +G & = \{ V, E \} \newline +\end{aligned} +$$ + +If we consider the vertices as nodes and the edges as references (pointers) connecting the individual nodes, we can think of a graph as a data structure expanded from a linked list. As shown in the figure below, **compared to linear relationships (linked lists) and divide and conquer relationships (trees), network relationships (graphs) have more degrees of freedom and thus are more complex**. + +![Relationship between linked lists, trees, and graphs](graph.assets/linkedlist_tree_graph.png) + +## Common Types Of Graphs And Terminology + +According to whether the edges are directed or not, it can be categorized into "undirected graph" and "directed graph" as shown in the figure below. + +- In an undirected graph, edges represent "bi-directional" connections between two vertices, such as "friend relationships" in WeChat or QQ. +- In a directed graph, edges are directional, i.e. $A \rightarrow B$ and $A \leftarrow B$ edges are independent of each other, e.g., the relationship between "following" and "being followed" on Weibo or Jitterbug. + +![Directed vs. undirected graphs](graph.assets/directed_graph.png) + +According to whether all the vertices are connected or not, it can be categorized into "connected graph" and "disconnected graph" as shown in the figure below. + +- For a connected graph, starting from a vertex, you can reach any of the remaining vertices. +- For a disconnected graph, at least one vertex is unreachable from some vertex. + +![Connected vs. disconnected graphs](graph.assets/connected_graph.png) + +We can also add "weight" variables to the edges to get a "weighted graph" as shown in the figure below. For example, in handheld games such as Honor of Kings, the system will calculate the "intimacy" between players based on the common playing time, and this kind of intimacy network can be represented by a weighted graph. + +![Weighted vs. unweighted graphs](graph.assets/weighted_graph.png) + +The graph data structure contains the following common terms. + +- "adjacency adjacency": two vertices are said to be "adjacency" when there are edges connecting them. In the figure above, the adjacency of vertex 1 is vertex 2, 3 and 5. +- "path": The sequence of edges passing from vertex A to vertex B is called a "path" from A to B. In the figure above, the edge sequence 1-5-2-4 is a path from vertex 1 to vertex 4. In the figure above, the edge sequence 1-5-2-4 is a path from vertex 1 to vertex 4. +- "degree": the number of edges a vertex has. For a directed graph, "in-degree" means how many edges are directed to the vertex, and "out-degree" means how many edges are directed away from the vertex. + +## Representation Of Graphs + +Common representations of graphs include "adjacency matrix" and "adjacency list". The following is an example of an undirected graph. + +### Adjacency Matrix + +Let the number of vertices of a graph be $n$, the "adjacency matrix" uses a matrix of size $n \times n$ to represent the graph, each row (column) represents a vertex, and the elements of the matrix represent edges, and $1$ or $0$ denotes the existence of an edge between two vertices. + +As shown in the figure below, let the adjacency matrix be $M$ and the list of vertices be $V$, then the matrix element $M[i, j] = 1$ means that there exists an edge between the vertex $V[i]$ to the vertex $V[j]$ and vice versa $M[i, j] = 0$ means that there is no edge between the two vertices. + +![Adjacency matrix representation of the graph](graph.assets/adjacency_matrix.png) + +The adjacency matrix has the following properties. + +- A vertex cannot be connected to itself, so the main diagonal elements of the adjacency matrix are meaningless. +- For undirected graphs, the edges in both directions are equivalent, when the adjacency matrix is symmetric about the main diagonal. +- By replacing the elements of the adjacency matrix from $1$ and $0$ to weights, the weighted graph can be represented. + +When using an adjacency matrix to represent a graph, we can directly access the matrix elements to get the edges, so the addition, deletion, and checking operations are very efficient with a time complexity of $O(1)$ . However, the space complexity of the matrix is $O(n^2)$ , which is more memory intensive. + +### Adjacency List + +"The adjacency list represents a graph using $n$ linked lists, where the nodes of the linked lists represent the vertices. The $i$th linked list corresponds to vertex $i$, which stores all the adjacency vertices (i.e., vertices connected to that vertex) of that vertex. The figure below shows an example of a graph stored using an adjacency list. + +![Adjacency list representation of the graph](graph.assets/adjacency_list.png) + +An adjacency list stores only the edges that actually exist, and the total number of edges is usually much less than $n^2$, so it is more space efficient. However, you need to find edges in an adjacency list by traversing the linked list, so it is not as time efficient as an adjacency matrix. + +Observing the figure above, **the structure of the adjacency list is very similar to the "separate chaining" of addresses in hash tables, so we can use a similar method to optimize the efficiency**. For example, when the linked list is long, it can be converted into an avl tree or a red-black tree to optimize the time efficiency from $O(n)$ to $O(\log n)$; or it can be converted into a hash table to reduce the time complexity to $O(1)$. + +## Graphs Common Applications + +As shown in the table below, many realistic systems can be modeled by graphs, and the corresponding problems can be approximated as graph computation problems. + +

    Table   Common Real-Life Graphs

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    Table   Adjacency matrix vs. adjacency list

    + +| | Adjacency matrix | Adjacency list (linked list) | Adjacency table (hash table) | +| --------------------------- | ---------------- | ---------------------------- | ---------------------------- | +| determine whether adjacency | $O(1)$ | $O(m)$ | $O(1)$ | +| Add Edge | $O(1)$ | $O(1)$ $O(1)$ $O(1)$ | +| Delete Edge | $O(1)$ | $O(m)$ | $O(1)$ | +| Add vertex | $O(n)$ | $O(1)$ | $O(1)$ | +| Delete vertex | $O(n^2)$ | $O(n + m)$ | $O(n)$ | +| Memory Space Occupancy | $O(n^2)$ | $O(n + m)$ | $O(n + m)$ | + +Observing the above table, it seems that the adjacency list (hash table) is optimal in terms of time and space efficiency. But in fact, the efficiency of operating edges in the adjacency matrix is higher, only one array access or assignment operation is needed. 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zs83-61c1Nbbr=T6Fd20$7|WNdxE;WhHu|E)_>0AeGWl1^8as-axs6(TV!E>(v$a`& z3r;&G=u?;g0pO3zh*J5dW5HNJWV8clZIi=!5n1Yru9t31z2Hfh8OhdSjSVeQpAuK^ zv6AHC`43pLS$_-W(PjQVw3}0ZW&+$FdCwq#0RK=<y8db5=xj zP?b9*M0snGhD4W~X|YEXy{B1g&&l^z3C)_#`dcu|7h85tbIPI(yNCdWKQRGr?q2{Y zWB|}pmUZ)pV>T=6%LV^>^Ms2(mfkbV#LA_W-rW8crrc7W`sq7CHZp*j3?Prc9l+6B z-Y;(vK!E>*pJaCNitc{vy}&V-0n9el1bAu!*lka|PcQ%i{5R}~1kd`N_X4W`o~j9O z^e~kHte*mR3g7l7fyd+9Jp+LMg=wG4nEgKLXMi$*slEuJ1K2GLr#-RXj~D>Jue=)! zZ(jvI)fYib1^_T%rg^;ze7XkEP*;It(LWRr00zxe1&q~IVB;FVQ(3DtHrd6s- zAZ2kjMl&Xj19crZcjD@$Fm(h00ER^*E{>gCT>}orRUCtjajptDu@qBbIB?*=F^EI5 zlu6aUv#|+Grj%j?d{D>&^0;ZB1pqJ*P5ojn`&T>=17OK2rV-4tR29&z6DXNDaNxi( zfJ5icRsBn|Szxf56^yC^+Kxydk3|II$z&ov007);v5J}a5Oa|~$FBM}3M|%eWdR?g z=3j!hVbfhY{?z?g%>@(|an7HE&Fw1Y-)n?@68*_<#TMUnv& z-uHI+d1Hvo`g#5+>+Ao$?W8sn7)vS5E9h$i0I;~$#P?{Pwm9==N_+UQ4OPx`*4lfV zN(2B{UWrrhwVsq&b%d`rNtd$al!)jA0QeI`lu|a" + ![Breadth-first traversal steps for graphs](graph_traversal.assets/graph_bfs_step1.png) + +=== "<2>" + ![graph_bfs_step2](graph_traversal.assets/graph_bfs_step2.png) + +=== "<3>" + ![graph_bfs_step3](graph_traversal.assets/graph_bfs_step3.png) + +=== "<4>" + ![graph_bfs_step4](graph_traversal.assets/graph_bfs_step4.png) + +=== "<5>" + ![graph_bfs_step5](graph_traversal.assets/graph_bfs_step5.png) + +=== "<6>" + ![graph_bfs_step6](graph_traversal.assets/graph_bfs_step6.png) + +=== "<7>" + ![graph_bfs_step7](graph_traversal.assets/graph_bfs_step7.png) + +=== "<8>" + ![graph_bfs_step8](graph_traversal.assets/graph_bfs_step8.png) + +=== "<9>" + ![graph_bfs_step9](graph_traversal.assets/graph_bfs_step9.png) + +=== "<10>" + ![graph_bfs_step10](graph_traversal.assets/graph_bfs_step10.png) + +=== "<11>" + ![graph_bfs_step11](graph_traversal.assets/graph_bfs_step11.png) + +!!! question "Is the sequence of breadth-first traversals unique?" + + No, it isn't. Breadth-first traversal only requires traversing in "near to far" order, **while the traversal order of multiple vertices of the same distance is allowed to be arbitrarily disrupted**. Take the figure above as an example, the access order of vertices $1$, $3$ can be swapped, and the access order of vertices $2$, $4$, $6$ can be swapped arbitrarily. + +### Complexity Analysis + +**Time complexity:** All vertices are queued in and out once, using $O(|V|)$ time; during traversal of the adjacency vertices, since it is an undirected graph, all the edges are accessed $2$ times, using $O(2|E|)$ time; and overall using $O(|V| + |E|)$ time. + +**Space complexity:** The maximum number of vertices in a list `res` , hash table `visited` , and queue `que` is $|V|$ , using $O(|V|)$ space. + +## Depth-First Traversal + +**Depth-first traversal is a type of traversal that prioritizes going to the end and turning back when there is no way out**. As shown in the figure below, starting from the top-left vertex, you visit one of the current vertex's adjacency vertices until you return at the end of your walk, then continue to the end and return, and so on until all vertex traversal is complete. + +![depth-first traversal of graph](graph_traversal.assets/graph_dfs.png) + +### Algorithm Implementation + +This algorithmic paradigm of "go to the end and back" is usually implemented based on recursion. Similar to breadth-first traversal, depth-first traversal requires a hash table `visited` to keep track of the vertices that have already been visited, in order to avoid revisiting vertices. + +```src +[file]{graph_dfs}-[class]{}-[func]{graph_dfs} +``` + +The algorithm flow for depth-first traversal is shown in the figure below. + +- The **straight dashed line represents a downward recursion**, indicating that a new recursion method is opened to access the new vertex. +- The **curved dashed line represents upward backtracking**, indicating that this recursion method has returned, back to the point where it was opened. + +For a better solution, it is recommended to combine the graphs with the code to simulate the entire DFS process in your head (or draw it down with a pen), including when each recursion method opens and when it returns. + +=== "<1>" + ![Depth-first traversal steps for graphs](graph_traversal.assets/graph_dfs_step1.png) + +=== "<2>" + ![graph_dfs_step2](graph_traversal.assets/graph_dfs_step2.png) + +=== "<3>" + ![graph_dfs_step3](graph_traversal.assets/graph_dfs_step3.png) + +=== "<4>" + ![graph_dfs_step4](graph_traversal.assets/graph_dfs_step4.png) + +=== "<5>" + ![graph_dfs_step5](graph_traversal.assets/graph_dfs_step5.png) + +=== "<6>" + ![graph_dfs_step6](graph_traversal.assets/graph_dfs_step6.png) + +=== "<7>" + ![graph_dfs_step7](graph_traversal.assets/graph_dfs_step7.png) + +=== "<8>" + ![graph_dfs_step8](graph_traversal.assets/graph_dfs_step8.png) + +=== "<9>" + ![graph_dfs_step9](graph_traversal.assets/graph_dfs_step9.png) + +=== "<10>" + ![graph_dfs_step10](graph_traversal.assets/graph_dfs_step10.png) + +=== "<11>" + ![graph_dfs_step11](graph_traversal.assets/graph_dfs_step11.png) + +!!! question "Is the sequence of depth-first traversal unique?" + + Similar to breadth-first traversal, the order of the depth-first traversal sequence is not unique. Given a vertex, it is possible to explore in either direction first, i.e., the order of the adjacency vertices can be arbitrarily disrupted, and it is all depth-first traversal. + + Taking tree traversal as an example, "root $\rightarrow$ left $\rightarrow$ right" and "left $\rightarrow$ root $\rightarrow$ right", "left $\rightarrow$ right $\rightarrow$ root" corresponds to preorder, middle-order, and postorder traversals, respectively, and they demonstrate three different traversal priorities, however, all three are depth-first traversals. + +### Complexity Analysis + +**Time complexity:** All vertices are visited $1$ times, using $O(|V|)$ time; all edges are visited $2$ times, using $O(2|E|)$ time; and overall $O(|V| + |E|)$ time. + +**Space complexity:** lists `res` , hash tables `visited` The maximum number of vertices is $|V|$ , and the maximum depth of recursion is $|V|$ , so $O(|V|)$ space is used. diff --git a/docs-en/chapter_graph/index.md b/docs-en/chapter_graph/index.md new file mode 100644 index 0000000000..0dd7ffb9f4 --- /dev/null +++ b/docs-en/chapter_graph/index.md @@ -0,0 +1,13 @@ +# Graph + +

    + +!!! abstract + + In the journey of life, we are like each node, connected by countless invisible edges. + + Every time we meet and leave each other, we leave a unique mark on this huge network graph. diff --git a/docs-en/chapter_graph/summary.md b/docs-en/chapter_graph/summary.md new file mode 100644 index 0000000000..035b3c9133 --- /dev/null +++ b/docs-en/chapter_graph/summary.md @@ -0,0 +1,30 @@ +# Summary + +### Highlights + +- A graph consists of vertices and edges and can be represented as a set consisting of a set of vertices and a set of edges. +- Compared to linear relationships (linked lists) and divide and conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex. +- The edges of a directed graph are directional, any vertex in the connected graph is reachable, and each edge of a weighted graph contains weight variables. +- An adjacency matrix uses a matrix to represent a graph, where each row (column) represents a vertex and the matrix elements represent edges, using $1$ or $0$ to indicate that there is an edge or no edge between two vertices. Adjacency matrix is very efficient in addition, deletion, and checking operations, but it takes up more space. +- An adjacency table uses multiple linked lists to represent a graph, with the $i$th linked list corresponding to vertex $i$, which stores all the neighboring vertices of that vertex. Adjacency lists are more space-efficient compared to adjacency matrices, but are less time-efficient due to the need to traverse the linked lists to find edges. +- When the linked list in the adjacency list is too long, it can be converted to a red-black tree or hash table to improve query efficiency. +- From the point of view of algorithmic thinking, the adjacency matrix embodies "space for time", and the adjacency list embodies "time for space". +- Graphs can be used to model all kinds of real-world systems, such as social networks and subway lines. +- Trees are a special case of graphs, and tree traversal is a special case of graph traversal. +- Breadth-first traversal of a graph is a search that expands from near to far, level by level, usually implemented with the help of a queue. +- Depth-first traversal of a graph is a search that prioritizes going to the bottom and backtracking when there is no way out, often based on recursion. + +### Q & A + +!!! question "Is a path defined as a sequence of vertices or a sequence of edges?" + + Wikipedia has inconsistent definitions in different languages: the English version says "a path is a sequence of edges", while the Chinese version says "a path is a sequence of vertices". Here is the original English version: In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. + In this paper, a path is considered to be a sequence of edges rather than a sequence of vertices. This is because there may be multiple edges connected between two vertices, when each edge corresponds to a path. + +!!! Question "Are there points in a disconnected graph that cannot be traversed?" + + In a disconnected graph, at least one vertex is unreachable from some vertex. Traversing a disconnected graph requires setting up multiple starting points to traverse to all connected components of the graph. + +!!! question "In the adjacency list, is there a requirement for the vertex order of "all vertices connected to this vertex"?" + + It can be any order. 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b/docs-en/chapter_greedy/fractional_knapsack_problem.md @@ -0,0 +1,50 @@ +# Fractional Knapsack Problem + +!!! question + + Given $n$ items, the weight of the $i$th item is $wgt[i-1]$, the value is $val[i-1]$ , and a backpack with capacity $cap$. Each item can be selected only once, **but a portion of the item can be selected, and the value is calculated in proportion to the weight of the selection**. Please return the maximum value of the items in the backpack without exceeding the capacity of the backpack. + +![Example data for the fractional knapsack problem](fractional_knapsack_problem.assets/fractional_knapsack_example.png) + +Fractional backpacks are very similar to 0-1 backpacks in general, in that the state contains the current item $i$ and the capacity $c$ , and the goal is to find the maximum value without exceeding the capacity of the backpack. + +The difference is that this question allows to select only a part of the item. As shown in the figure below, **we can slice the item arbitrarily and calculate the value of the item in proportion to its weight**. + +1. For an item $i$ , its value per unit weight is $val[i-1] / wgt[i-1]$ , abbreviated as unit value. +2. Assuming that a portion of the item is put in $i$ and the weight is $w$, the value added to the backpack is $w \times val[i-1] / wgt[i-1]$ . + +![Value of item per unit weight](fractional_knapsack_problem.assets/fractional_knapsack_unit_value.png) + +### Greedy Strategy Determination + +Maximizing the total value of the items in the backpack, **essentially means maximizing the value of the items per unit weight**. This leads to the greedy strategy shown in the figure below. + +1. Sort items from highest to lowest unit value. +2. Iterate over all items, **greedily choosing the item with the highest unit value each round**. +3. If the remaining backpack capacity is insufficient, simply use a portion of the current item to fill the backpack. + +![Greedy strategy for fractional knapsacks](fractional_knapsack_problem.assets/fractional_knapsack_greedy_strategy.png) + +### Code Implementation + +We create an item class `Item` in order to sort items by unit value. The loop makes a greedy selection, jumps out when the backpack is full and returns the solution. + +```src +[file]{fractional_knapsack}-[class]{}-[func]{fractional_knapsack} +``` + +In the worst case, the entire list of items needs to be traversed, **hence the time complexity is $O(n)$** , where $n$ is the number of items. + +Since a list of `Item` objects is initialized, **the space complexity is $O(n)$**. + +### Proof Of Correctness + +The counterfactual method is used. Assume that item $x$ is the item with the highest unit value, and use an algorithm to find the maximum value as `res` , but the solution does not include item $x$ . + +Now take any item of unit weight from the backpack and replace it with an item of unit weight $x$ . Since item $x$ has the highest unit value, the total value after replacement must be greater than `res` . **This contradicts the fact that `res` is the optimal solution and shows that the optimal solution must include the item $x$** . + +For the other items in this solution, we can also construct the above contradiction. In sum, **the item with a greater unit value is always the better choice**, which shows that the greedy strategy is effective. + +As shown in the figure below, the fractional knapsack problem can be transformed into "find the maximum enclosing area for a finite interval of the horizontal axis" by considering the weight of the item and the unit value of the item as the horizontal and vertical axes, respectively, of a 2D graph. This analogy helps us understand the effectiveness of the greedy strategy from a geometric perspective. + +![Geometric representation of the fractional knapsack problem](fractional_knapsack_problem.assets/fractional_knapsack_area_chart.png) diff --git a/docs-en/chapter_greedy/greedy_algorithm.assets/coin_change_greedy_strategy.png b/docs-en/chapter_greedy/greedy_algorithm.assets/coin_change_greedy_strategy.png new file mode 100644 index 0000000000000000000000000000000000000000..133b30f3ef66f6b31ead4268b95d585a9bc5e538 GIT binary patch literal 39434 zcmc$_Ra{li7dO05(A^!y4q`N~JL_t76Kv3$?AsvErNeD={z@bYT={R(Ubo282 zKlks=bMaj5Ju~aO)~uK{pQ&(7bp;$uN=yI%aFi5fwE+O-=@&eT4te@q<#D|M0MM1D zs;=DQkMB|8{qGkB*L3R#rX)FN=$dS5{W; z@9&3)hp(=#4h{|mH2K4R?46#TUR+!Z4Gj$s4c zt?xKmTwFXnJgljynV+9;Uqloa7x(w~$LDV)B_-9>)y=P5N=r*iNJwOlpQom#4h#%9 z|322%);2LQDJv`U_4RdgbITdb2?`1t*|}+NZ};-@S~)~&XlTgG%Oeno=;)Zs>M9!> z8$CU}+S=N@yu1$|J{TGrDk&-DU#(+P1c~ zdU|?>hK6QkWi>W7ofn->%mynOkxyu94f($e4G|JAEk$;rvi&d!O6i3te_ 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Suppose a factor $2$ is divided from $n$ , then their product is $2(n-2)$ . We compare this product with $n$: + +$$ +\begin{aligned} +2(n-2) & \geq n \newline +2n - n - 4 & \geq 0 \newline +n & \geq 4 +\end{aligned} +$$ + +As shown in the figure below, when $n \geq 4$, the product will be larger after cutting out a $2$, **This means that any integer greater than or equal to $4$ should be cut**. + +**Greedy Strategy I**: If the slicing scheme contains a factor of $\geq 4$, then it should be continued to be sliced. Only the factors $1$, $2$, and $3$ should appear in the final slicing scheme. + +![Cutting leads to larger product](max_product_cutting_problem.assets/max_product_cutting_greedy_infer1.png) + +Next think about which factor is optimal. Among the three factors $1$, $2$, and $3$, it is clear that $1$ is the worst because $1 \times (n-1) < n$ holds, i.e., slicing out $1$ will instead result in a smaller product. + +As shown in the figure below, when $n = 6$, there are $3 \times 3 > 2 \times 2 \times 2$ . **This means that slicing out $3$ is better than slicing out $2$**. + +**Greedy Strategy II**: In the cut scheme, at most two $2$ should exist. This is because three $2$ can always be replaced by two $3$ , resulting in a larger product. + +![Optimal Cutting Factor](max_product_cutting_problem.assets/max_product_cutting_greedy_infer2.png) + +Summarizing the above, the following greedy strategy can be introduced. + +1. Input an integer $n$ , from which the factors $3$ are continuously sliced until the remainders are $0$, $1$, $2$. +2. When the remainder is $0$, it means that $n$ is a multiple of $3$, so nothing is done. +3. When the remainder is $2$, do not continue the division and keep it. +4. When the remainder is $1$, the last $3$ should be replaced by $2$ since $2 \times 2 > 1 \times 3$. + +### Code Implementation + +As shown in the figure below, instead of slicing the integers by loops, we can use downward division to get the number of $3$ $a$ and modulo operation to get the remainder $b$, which we have: + +$$ +n = 3 a + b +$$ + +Note that for the boundary case of $n \leq 3$, a $1$ must be split and the product is $1 \times (n - 1)$ . + +```src +[file]{max_product_cutting}-[class]{}-[func]{max_product_cutting} +``` + +![Calculation of Maximum Cutting Product](max_product_cutting_problem.assets/max_product_cutting_greedy_calculation.png) + +**The time complexity depends on the programming language's implementation of the power operation**. In Python, for example, there are three commonly used power calculation functions. + +- Both the operator `**` and the function `pow()` have a time complexity of $O(\log a)$. +- The function `math.pow()` internally calls the C library's `pow()` function, which performs floating-point power-taking with a time complexity of $O(1)$. + +The variables $a$ and $b$ use extra space of constant size, **so the space complexity is $O(1)$** . + +### Proof Of Correctness + +Using the counterfactual, analyze only the case of $n \geq 3$. + +1. **All factors $\leq 3$** : Assuming that there is a factor $\geq 4$ of $x$ in the optimal cutoff scheme, it must be possible to continue dividing it into $2(x-2)$ and thus obtain a larger product. This contradicts the assumption. +2. The **-slicing scheme does not contain $1$** : Assuming that a factor $1$ exists in the optimal slicing scheme, it must be possible to merge it into another factor to obtain a larger product. This contradicts the assumption. +3. **The cut scheme contains at most two $2$** : Assuming that the optimal cut scheme contains three $2$, then it must be possible to replace it with two $3$ with a larger product. This contradicts the assumption. diff --git a/docs-en/chapter_greedy/summary.md b/docs-en/chapter_greedy/summary.md new file mode 100644 index 0000000000..94f4fb5759 --- /dev/null +++ b/docs-en/chapter_greedy/summary.md @@ -0,0 +1,12 @@ +# Summary + +- Greedy algorithms are commonly used to solve optimization problems based on the principle of making locally optimal decisions at each decision stage in the expectation of obtaining a globally optimal solution. +- The greedy algorithm iteratively makes one greedy choice after another, transforming the problem into a smaller subproblem each round until the problem is solved. +- The greedy algorithm is not only simple to implement, but also highly efficient in solving problems. Compared to dynamic programming, greedy algorithms usually have lower time complexity. +- In the change exchange problem, the greedy algorithm is guaranteed to find an optimal solution for some coin combinations; for others it is not, and the greedy algorithm may find a very poor solution. +- Problems suitable for solution with greedy algorithms have two main properties: the greedy choice property and the optimal substructure. The greedy selection property represents the effectiveness of the greedy strategy. +- For some complex problems, proof of the nature of greedy choice is not simple. It is relatively easier to falsify, for example, the change exchange problem. +- Solving greedy problems is divided into three main steps: problem analysis, greedy strategy determination, and correctness proof. Among them, greedy strategy determination is the core step, and correctness proof is often the hard part. +- The fractional knapsack problem is based on the 0-1 knapsack and allows to select a fraction of the items, so it can be solved using a greedy algorithm. The correctness of the greedy strategy can be proved using the inverse method. +- The maximum capacity problem can be solved using the exhaustive enumeration method with a time complexity of $O(n^2)$ . The time complexity can be optimized to $O(n)$ by designing a greedy strategy that moves the short board inward each round. +- In the Maximum Tangent Product Problem, we successively reason about two greedy strategies: $\geq 4$ integers should all continue to be tangent, and the optimal tangent factor is $3$ . 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zZO|!0vM>|d7f$!yz~dMOaziqqQUju3G?@k@4@3^Mr}U6Ti2`fopqPj*T(B@55*xxw zjWrd+C^ko^o_j7tigFS|T8>ca5w=GmRNN4xK!V5dGZnn5e;>`H13C9E%`w@A_yR$I z6VfQBr}Xr2b8X2Ph)gi#E#%T;pmUw75(Gk=AJ}p!e5(AzfAi+8AUKxMgyiM+Fqz?_ z0WPL*(_M%DXgdDQsKwwD7@+&qXY}#Gavfu9Z zOB=E%2+u7Z#gmXll_9)8M3yM=_pahFydS8gdwrnP!$Yy=Zw0S*6`p}Q{b|LhKm Table   Common hash algorithms

    + +| | MD5 | SHA-1 | SHA-2 | SHA-3 | +| -------------- | -------------------------------------------------- | -------------------------- | --------------------------------------------------------------- | -------------------------------------- | +| Launched | 1992 | 1995 | 2002 | 2008 | +| output length | 128 bits | 160 bits | 256 / 512 bits | 224/256/384/512 bits | +| hash collision | more | more | less | less | less | less | +| Security Level | Low, Successfully Attacked | Low, Successfully Attacked | High | High | +| Applications | Deprecated, still used for data integrity checking | Deprecated | Cryptocurrency transaction validation, digital signatures, etc. | Can be used as a replacement for SHA-2 | + +## Hash Values For Data Structures + +As we know, the `key` of a hash table can be a data type such as integer, decimal, or string. Programming languages usually provide built-in hash algorithms for these data types, which are used to compute bucket indexes in hash tables. In Python, for example, we can call the `hash()` function to compute hash values for various data types. + +- The hash values of integers and boolean quantities are themselves. +- The calculation of hash values for floating point numbers and strings is more complex, so if you are interested, please learn on your own. +- A hash of a tuple is hashed for each of its elements, and then those hashes are combined to get a single hash. +- An object's hash is generated based on its memory address. Generating hashes based on content can be achieved by overriding the object's hash method. + +!!! tip + + Note that the definitions and methods of the built-in hash function for calculating hash values differ from one programming language to another. + +=== "Python" + + ```python title="built_in_hash.py" + num = 3 + hash_num = hash(num) + # The hash value of integer 3 is 3 + + bol = True + hash_bol = hash(bol) + # The hash value of the boolean True is 1 + + dec = 3.14159 + hash_dec = hash(dec) + # The hash value of the decimal 3.14159 is 326484311674566659 + + str = "Hello Algo" + hash_str = hash(str) + # String Hello The algorithm's hash value is 4617003410720528961 + + tup = (12836, "small ha") + hash_tup = hash(tup) + # The hash value of the tuple (12836, 'smallha') is 1029005403108185979 + + obj = ListNode(0) + hash_obj = hash(obj) + # Node object hash value 274267521 + ``` + +=== "C++" + + ```cpp title="built_in_hash.cpp" + int num = 3; + size_t hashNum = hash()(num); + // The integer 3 hash is 3 + + bool bol = true; + size_t hashBol = hash()(bol); + // Boolean 1 has a hash value of 1 + + double dec = 3.14159; + size_t hashDec = hash()(dec); + // The hash value of the decimal 3.14159 is 4614256650576692846 + + string str = "Hello Algo"; + size_t hashStr = hash()(str); + // The hash value of the Hello Algo is 15466937326284535026. + + // In C++, the built-in std:hash() provides hash calculations for basic data types only. + // Hash calculations for arrays and objects are self-implemented. + ``` + +=== "Java" + + ```java title="built_in_hash.java" + int num = 3; + int hashNum = Integer.hashCode(num); + // The integer 3 hash is 3 + + boolean bol = true; + int hashBol = Boolean.hashCode(bol); + // The hash value of the boolean true is 1231 + + double dec = 3.14159; + int hashDec = Double.hashCode(dec); + // The hash value for the decimal 3.14159 is -1340954729 + + String str = "Hello Algo"; + int hashStr = str.hashCode(); + // The string Hello Algo has a hash value of -727081396. + + Object[] arr = { 12836, "littleha" }; + int hashTup = Arrays.hashCode(arr); + // The array [12836, small hash] has a hash value of 1151158. + + ListNode obj = new ListNode(0); + int hashObj = obj.hashCode(); + // The hash value of the node object utils.ListNode@7dc5e7b4 is 2110121908 + ``` + +=== "C#" + + ```csharp title="built_in_hash.cs" + int num = 3; + int hashNum = num.GetHashCode(); + // 整数 3 的哈希值为 3; + + bool bol = true; + int hashBol = bol.GetHashCode(); + // the hash value of the boolean true is 1. + + double dec = 3.14159; + int hashDec = dec.GetHashCode(); + // The hash value of the decimal 3.14159 is -1340954729. + + string str = "Hello Algo"; + int hashStr = str.GetHashCode(); + // The string Hello Algo has a hash value of -586107568. + + object[] arr = { 12836, "littleha" }; + int hashTup = arr.GetHashCode(); + // The array [12836, small hash] has a hash value of 42931033. + + ListNode obj = new(0); + int hashObj = obj.GetHashCode(); + // The hash value of node object 0 is 39053774; // The hash value of node object 0 is 39053774. + ``` + +=== "Go" + + ```go title="built_in_hash.go" + // Go does not provide a built-in hash code function + ``` + +=== "Swift" + + ```swift title="built_in_hash.swift" + let num = 3 + let hashNum = num.hashValue + // integer 3 hash value 9047044699613009734 + + let bol = true + let hashBol = bol.hashValue + // The hash value of the boolean true is -4431640247352757451 + + let dec = 3.14159 + let hashDec = dec.hashValue + // The hash value of the decimal 3.14159 is -2465384235396674631 + + let str = "Hello Algo" + let hashStr = str.hashValue + // The algorithm for the string Hello has a hash value of -7850626797806988787. + + let arr = [AnyHashable(12836), AnyHashable("small ha")] + let hashTup = arr.hashValue + // The array [AnyHashable(12836), AnyHashable("small hash")] has a hash of -2308633508154532996 + + let obj = ListNode(x: 0) + let hashObj = obj.hashValue + // The hash of the node object utils.ListNode is -2434780518035996159 + ``` + +=== "JS" + + ```javascript title="built_in_hash.js" + // JavaScript does not provide a built-in hash code function + ``` + +=== "TS" + + ```typescript title="built_in_hash.ts" + // TypeScript does not provide a built-in hash code function. + ``` + +=== "Dart" + + ```dart title="built_in_hash.dart" + int num = 3; + int hashNum = num.hashCode; + // The hash of integer 3 is 34803. + + bool bol = true; + int hashBol = bol.hashCode; + // The hash value of the boolean true is 1231 + + double dec = 3.14159; + int hashDec = dec.hashCode; + // The hash value of the decimal 3.14159 is 2570631074981783 + + String str = "Hello Algo"; + int hashStr = str.hashCode; + // The algorithm for the string Hello has a hash value of 468167534. + + List arr = [12836, "Little Hah"];; + int hashArr = arr.hashCode; + // The array [12836, small hash] has a hash value of 976512528. + + ListNode obj = new ListNode(0); + int hashObj = obj.hashCode; + // The hash value of the node object Instance of 'ListNode' is 1033450432. + ``` + +=== "Rust" + + ```rust title="built_in_hash.rs" + use std::collections::hash_map::DefaultHasher; + use std::hash::{Hash, Hasher}; + + let num = 3; + let mut num_hasher = DefaultHasher::new(); + num.hash(&mut num_hasher); + let hash_num = num_hasher.finish(); + // integer 3 hash 568126464209439262 + + let bol = true; + let mut bol_hasher = DefaultHasher::new(); + bol.hash(&mut bol_hasher); + let hash_bol = bol_hasher.finish(); + // The hash value of the boolean true is 4952851536318644461 + + let dec: f32 = 3.14159; + let mut dec_hasher = DefaultHasher::new(); + dec.to_bits().hash(&mut dec_hasher); + let hash_dec = dec_hasher.finish(); + println!("The hash of decimal {} is {}", dec, hash_dec); + // The hash value of the decimal 3.14159 is 2566941990314602357 + + let str = "Hello Algo"; + let mut str_hasher = DefaultHasher::new(); + str.hash(&mut str_hasher); + let hash_str = str_hasher.finish(); + // The hash value of the Hello Algo is 16092673739211250988. + + let arr = (&12836, &"small-ha"); + let mut tup_hasher = DefaultHasher::new(); + arr.hash(&mut tup_hasher); + let hash_tup = tup_hasher.finish(); + // The hash value of the tuple (12836, "smallha") is 1885128010422702749 + + let node = ListNode::new(42); + let mut hasher = DefaultHasher::new(); + node.borrow().val.hash(&mut hasher); + let hash = hasher.finish(); + // The node object RefCell { value: ListNode { val: 42, next: None } } has a hash value of 15387811073369036852 + ``` + +=== "C" + + ```c title="built_in_hash.c" + // C does not provide a built-in hash code function + ``` + +=== "Zig" + + ```zig title="built_in_hash.zig" + + ``` + +In many programming languages, **only immutable objects can be used as the `key`** of a hash table. If we use a list (dynamic array) as a `key`, when the contents of the list change, its hash value changes as well, and we can't look up the original `value` in the hash table. + +Although the member variables of a custom object (such as a linked list node) are mutable, it is hashable. **This is because the hash value of an object is usually generated based on the memory address**. Even if the content of the object changes, its memory address remains the same and the hash value remains the same. + +If you're careful, you may notice that the output hash value is different when you run the program in different consoles. **This is because the Python interpreter adds a random salt value to the string hash function at each startup**. 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Specifically, if we input a key `key` into a hash table, we can get the corresponding value `value` within $O(1)$ time. + +As shown in the figure below, given $n$ students, each student has two pieces of data, "name" and "student number". If we want to realize the query function of "input a student number, return the corresponding name", we can use the hash table shown in the figure below to realize it. + +![Abstract representation of hash table](hash_map.assets/hash_table_lookup.png) + +In addition to hash tables, arrays and linked lists can also realize the query function, and their efficiency comparison is shown in the table below. + +- **Add element**: simply add the element to the end of the array (linked list), using $O(1)$ time. +- **Query element**: since arrays (linked lists) are disordered, you need to iterate through all the elements in them, using $O(n)$ time. +- **Delete Element**: you need to query for the element and then delete it from the array (linked list), using $O(n)$ time. + +

    Table   Element Query Efficiency Comparison

    + +| | arrays | linked lists | hash tables | +| -------------- | ------ | -------------------- | ----------- | +| Find Elements | $O(n)$ | $O(n)$ | $O(1)$ | +| Add Element | $O(1)$ | $O(1)$ $O(1)$ $O(1)$ | +| Delete Element | $O(n)$ | $O(n)$ | $O(1)$ | + +Observe that the time complexity of all additions, deletions, checks, **and modifications in a hash table by is $O(1)$** , which is very efficient. + +## Common Operations On Hash Tables + +Common operations on hash tables include: initialization, query operations, adding key-value pairs and deleting key-value pairs. + +=== "Python" + + ```python title="hash_map.py" + # Initialize the hash table + hmap: dict = {} + + # Add operations + # Add key-value pairs (key, value) to the hash table + hmap[12836] = "little ha" + hmap[15937] = "Xiaoluo" + hmap[16750] = "small count" + hmap[13276] = "Little Law" + hmap[10583] = "Duckling" + + # Query operations + # Enter the key into the hash table to get the value value + name: str = hmap[15937] + + # Delete operations + # Remove key-value pairs from hash table (key, value) + hmap.pop(10583) + ``` + +=== "C++" + + ```cpp title="hash_map.cpp" + /* Initialize hash table */ + unordered_map map; + + /* Add operation */ + // Add the key-value pair (key, value) to the hash table. + map[12836] = "Little Hah"; + map[15937] = "Xiaoluo". + map[16750] = "Small count"; + map[13276] = "Minor Law". + map[10583] = "Duckling"; + + /* Query operations */ + // Enter the key into the hash table, and get the value + string name = map[15937]; + + /* Delete operation */ + // Remove the key-value pair (key, value) from the hash table. + map.erase(10583); + ``` + +=== "Java" + + ```java title="hash_map.java" + /* Initialize hash table */ + Map map = new HashMap<>(); + + /* Add operation */ + // Add the key-value pair (key, value) to the hash table. + map.put(12836, "smallha"); + map.put(15937, "Xiaoluo");; + map.put(16750, "Small count"); + map.put(13276, "Little Law"); + map.put(10583, "Duckling"). + + /* Query operations */ + // Enter the key into the hash table, and get the value + String name = map.get(15937); + + /* Delete operation */ + // Remove the key-value pair (key, value) from the hash table. + map.remove(10583); + ``` + +=== "C#" + + ```csharp title="hash_map.cs" + /* Initialize hash table */ + Dictionary map = new() { + /* Add operation */ + // Add the key-value pair (key, value) to the hash table. + { 12836, "little ha" }. + { 15937, "Koro" }. + { 16750, "small calculations" }. + { 13276, "Minor law" }. + { 10583, "Duckling" } + }; + + /* Query operations */ + // Enter the key into the hash table, and get the value + string name = map[15937]; + + /* Delete operation */ + // Remove the key-value pair (key, value) from the hash table. + map.Remove(10583); + ``` + +=== "Go" + + ```go title="hash_map_test.go" + /* Initialize hash table */ + hmap := make(map[int]string) + + /* Add operation */ + // Add the key-value pair (key, value) to the hash table. + hmap[12836] = "little ha" + hmap[15937] = "Xiaoluo" + hmap[16750] = "small count" + hmap[13276] = "Little Law" + hmap[10583] = "Duckling" + + /* Query operations */ + // Enter the key into the hash table, and get the value + name := hmap[15937] + + /* Delete operation */ + // Remove the key-value pair (key, value) from the hash table. + delete(hmap, 10583) + ``` + +=== "Swift" + + ```swift title="hash_map.swift" + /* Initialize hash table */ + var map: [Int: String] = [:] + + /* Add operation */ + // Add the key-value pair (key, value) to the hash table. + map[12836] = "Xiaoha" + map[15937] = "Xiaoluo" + map[16750] = "small count" + map[13276] = "Little Law" + map[10583] = "Duckling" + + /* Query operations */ + // Enter the key into the hash table, and get the value + let name = map[15937]! + + /* Delete operation */ + // Remove the key-value pair (key, value) from the hash table. + map.removeValue(forKey: 10583) + ``` + +=== "JS" + + ```javascript title="hash_map.js" + /* Initialize hash table */ + const map = new Map(); + /* Add operation */ + // Add the key-value pair (key, value) to the hash table. + map.set(12836, 'small-ha'); + map.set(15937, 'Xiaoluo'); + map.set(16750, 'Small count'); + map.set(13276, 'Small Law'); + map.set(10583, 'Duckling'); + + /* Query operations */ + // Enter the key into the hash table, and get the value + let name = map.get(15937); + + /* Delete operation */ + // Remove the key-value pair (key, value) from the hash table. + map.delete(10583); + ``` + +=== "TS" + + ```typescript title="hash_map.ts" + /* Initialize hash table */ + const map = new Map(); + /* Add operation */ + // Add the key-value pair (key, value) to the hash table. + map.set(12836, 'small-ha'); + map.set(15937, 'Xiaoluo'); + map.set(16750, 'Small count'); + map.set(13276, 'Small Law'); + map.set(10583, 'Duckling'); + console.info('\nAfter the addition is complete, the hash table is \nKey -> Value'); + console.info(map); + + /* Query operations */ + // Enter the key into the hash table, and get the value + let name = map.get(15937); + console.info('\n input student number 15937, query name ' + name); + + /* Delete operation */ + // Remove the key-value pair (key, value) from the hash table. + map.delete(10583); + console.info('After \n deletes 10583, hash table is \nKey -> Value'); + console.info(map); + ``` + +=== "Dart" + + ```dart title="hash_map.dart" + /* Initialize hash table */ + Map map = {}; + + /* Add operation */ + // Add the key-value pair (key, value) to the hash table. + map[12836] = "Little Hah"; + map[15937] = "Xiaoluo". + map[16750] = "Small count"; + map[13276] = "Minor Law". + map[10583] = "Duckling"; + + /* Query operations */ + // Enter the key into the hash table, and get the value + String name = map[15937]; + + /* Delete operation */ + // Remove the key-value pair (key, value) from the hash table. + map.remove(10583); + ``` + +=== "Rust" + + ```rust title="hash_map.rs" + use std::collections::HashMap; + + /* Initialize hash table */ + let mut map: HashMap = HashMap::new(); + + /* Add operation */ + // Add the key-value pair (key, value) to the hash table. + map.insert(12836, "Xiaoha".to_string()); + map.insert(15937, "Xiaoluo".to_string());; + map.insert(16750, "small count".to_string()); + map.insert(13279, "Small Law".to_string()); + map.insert(10583, "duckling".to_string()); + + /* Query operations */ + // Enter the key into the hash table to get the value value. + let _name: Option<&String> = map.get(&15937); + + /* Delete operation */ + // Remove the key-value pair (key, value) from the hash table. + let _removed_value: Option = map.remove(&10583); + ``` + +=== "C" + + ```c title="hash_map.c" + // C does not provide a built-in hash table + ``` + +=== "Zig" + + ```zig title="hash_map.zig" + + ``` + +Hash tables have three common traversals: traversing key-value pairs, traversing keys, and traversing values. + +=== "Python" + + ```python title="hash_map.py" + # Iterate over the hash table + # Iterate over key-value pairs key->value + for key, value in hmap.items(): + print(key, "->", value) + # Iterate over the key alone key + for key in hmap.keys(): + print(key) + # Individual traversal value value + for value in hmap.values(): + print(value) + ``` + +=== "C++" + + ```cpp title="hash_map.cpp" + /* Iterate over the hash table */ + // Iterate over key-value pairs key->value + for (auto kv: map) { + cout << kv.first << " -> " << kv.second << endl; + } + // Iterate over key->value using iteration. + for (auto iter = map.begin(); iter != map.end(); iter++) { + cout << iter->first << "->" << iter->second << endl; + } + ``` + +=== "Java" + + ```java title="hash_map.java" + /* Iterate over the hash table */ + // Iterate over key-value pairs key->value + for (Map.Entry kv: map.entrySet()) { + System.out.println(kv.getKey() + " -> " + kv.getValue()); + } + // Iterate over keys individually key + for (int key: map.keySet()) { + System.out.println(key); + } + // Iterate over values individually value + for (String val: map.values()) { + System.out.println(val); + } + ``` + +=== "C#" + + ```csharp title="hash_map.cs" + /* Iterate over the hash table */ + // Iterate over key-value pairs Key->Value + foreach (var kv in map) { + Console.WriteLine(kv.Key + " -> " + kv.Value); + } + // Iterate over keys individually key + foreach (int key in map.Keys) { + Console.WriteLine(key); + } + // Iterate over values individually value + foreach (string val in map.Values) { + Console.WriteLine(val); + } + ``` + +=== "Go" + + ```go title="hash_map_test.go" + /* Iterate over the hash table */ + // Iterate over key-value pairs key->value + for key, value := range hmap { + fmt.Println(key, "->", value) + } + // Iterate over keys individually key + for key := range hmap { + fmt.Println(key) + } + // Iterate over values individually value + for _, value := range hmap { + fmt.Println(value) + } + ``` + +=== "Swift" + + ```swift title="hash_map.swift" + /* Iterate over the hash table */ + // Iterate over key-value pairs Key->Value + for (key, value) in map { + print("\(key) -> \(value)") + } + // Iterate through the keys individually Key + for key in map.keys { + print(key) + } + // Iterate through the values individually Value + for value in map.values { + print(value) + } + ``` + +=== "JS" + + ```javascript title="hash_map.js" + /* Iterate over the hash table */ + console.info('\n traversing Key->Value pair Key->Value');; + for (const [k, v] of map.entries()) { + console.info(k + ' -> ' + v); + } + console.info('\n traversing Key Key alone');; + for (const k of map.keys()) { + console.info(k); + } + console.info('\n traversing the values individually Value');; + for (const v of map.values()) { + console.info(v); + } + ``` + +=== "TS" + + ```typescript title="hash_map.ts" + /* Iterate over the hash table */ + console.info('\n traversing Key->Value pair Key->Value');; + for (const [k, v] of map.entries()) { + console.info(k + ' -> ' + v); + } + console.info('\n traversing Key Key alone');; + for (const k of map.keys()) { + console.info(k); + } + console.info('\n traversing the values individually Value');; + for (const v of map.values()) { + console.info(v); + } + ``` + +=== "Dart" + + ```dart title="hash_map.dart" + /* Iterate over the hash table */ + // Iterate over key-value pairs Key->Value + map.forEach((key, value) { + print('$key -> $value'); + }); + + // Iterate through the keys individually Key + map.keys.forEach((key) { + print(key); + }); + + // Iterate through the values individually Value + map.values.forEach((value) { + print(value); + }); + ``` + +=== "Rust" + + ```rust title="hash_map.rs" + /* Iterate over the hash table */ + // Iterate over key-value pairs Key->Value + for (key, value) in &map { + println!("{key} -> {value}"); + } + + // Iterate through the keys individually Key + for key in map.keys() { + println!("{key}"); + } + + // Iterate through the values individually Value + for value in map.values() { + println!("{value}"); + } + ``` + +=== "C" + + ```c title="hash_map.c" + // C does not provide a built-in hash table + ``` + +=== "Zig" + + ```zig title="hash_map.zig" + + ``` + +## Simple Implementation Of Hash Tables + +Let's first consider the simplest case, **implementing a hash table using only an array**. In a hash table, we call each empty space in the array a "bucket bucket", and each bucket can store a key-value pair. Therefore, the query operation is to find the bucket corresponding to `key` and get `value` in the bucket. + +So, how to locate the corresponding bucket based on `key`? This is achieved by the hash function. The function of hash function is to map a larger input space to a smaller output space. In a hash table, the input space is all the `keys` and the output space is all the buckets (array indexes). In other words, by inputting a `key` , **we can use the hash function to get the storage location of the key-value pair corresponding to that `key` in the array**. + +Input a `key` , the hash function is computed in the following two steps. + +1. The hash value is calculated by some hash algorithm `hash()`. +2. The hash value is modulo the number of buckets (array length) `capacity` to get the index of the array `index` corresponding to the `key`. + +```shell +index = hash(key) % capacity +``` + +We can then use `index` to access the corresponding bucket in the hash table to get `value` . + +Given the length of the array `capacity = 100`, the hash algorithm `hash(key) = key`, it is easy to obtain the hash function as `key % 100`. The figure below shows how the hash function works, using `key` school number and `value` name as examples. + +![How hash functions work](hash_map.assets/hash_function.png) + +The following code implements a simple hash table. In it, we encapsulate `key` and `value` into a class `Pair` to represent key-value pairs. + +```src +[file]{array_hash_map}-[class]{array_hash_map}-[func]{} +``` + +## Hash Collision And Capacity Expansion + +Essentially, the hash function maps the input space consisting of all the `keys` to the output space consisting of all the indexes of the array, and the input space is often much larger than the output space. Therefore, **theoretically there must be "multiple inputs correspond to the same output" case**. + +For the hash function in the above example, when the last two digits of the input `key` are the same, the output of the hash function is also the same. For example, when querying the two students with student numbers 12836 and 20336, we get: + +```shell +12836 % 100 = 36 +20336 % 100 = 36 +``` + +As shown in the figure below, two school numbers point to the same name, which is obviously not correct. We call this case of multiple inputs corresponding to the same output "hash collision". + +![Example of hash collision](hash_map.assets/hash_collision.png) + +It is easy to think that the larger the hash table capacity $n$ is, the lower the probability that multiple `key` are assigned to the same bucket, and the fewer hash collisions. Therefore, **we can reduce hash collision by expanding the hash table**. + +As shown in the figure below, the key-value pairs `(136, A)` and `(236, D)` were in conflict before expansion, and the conflict disappears after expansion. + +![hash table expansion](hash_map.assets/hash_table_reshash.png) + +Similar to array expansion, hash table expansion requires migrating all key-value pairs from the original hash table to the new hash table, which is very time-consuming. And because the hash table capacity `capacity` change, we need to recalculate all the key-value pairs through the hash function to store the location, which further increases the computational overhead of the expansion process. For this reason, programming languages usually reserve a large enough hash table capacity to prevent frequent expansion. + +"Load factor" is an important concept of hash table, which is defined as the number of elements divided by the number of buckets, used to measure the severity of hash collision, **is also often used as a trigger for the expansion of the hash table capacity**. For example, in Java, when the load factor exceeds $0.75$, the system will expand the hash table capacity to $2$ times its original size. diff --git a/docs-en/chapter_hashing/index.md b/docs-en/chapter_hashing/index.md new file mode 100644 index 0000000000..7bdced46f3 --- /dev/null +++ b/docs-en/chapter_hashing/index.md @@ -0,0 +1,13 @@ +# Hash Table + +
    + +![hash table](../assets/covers/chapter_hashing.jpg){ width="600" } + +
    + +!!! abstract + + In the computer world, hash tables are like an intelligent librarian. + + He knows how to calculate the claim number so he can find the target book quickly. diff --git a/docs-en/chapter_hashing/summary.md b/docs-en/chapter_hashing/summary.md new file mode 100644 index 0000000000..fb621c0cea --- /dev/null +++ b/docs-en/chapter_hashing/summary.md @@ -0,0 +1,47 @@ +# Summary + +### Highlights + +- Input `key` , hash table can look up `value` in $O(1)$ time, which is very efficient. +- Common hash table operations include querying, adding key-value pairs, deleting key-value pairs, and traversing the hash table. +- The hash function maps `key` to an array index to access the corresponding bucket and get `value`. +- Two different `key` may be after the hash function to get the same array index, resulting in query results, this phenomenon is known as hash collision. +- The larger the hash table capacity, the lower the probability of hash collision. Therefore, hash collision can be mitigated by expanding the hash table. Similar to array expansion, hash table expansion operation has a high overhead. +- The load factor is defined as the number of elements in the hash table divided by the number of buckets, reflecting the severity of hash collision, and is often used as a condition to trigger the expansion of the hash table. +- Separate chaining addresses store all conflicting elements in the same chained list by converting individual elements into a linked list. However, a linked list that is too long reduces query efficiency, which can be further improved by converting the linked list into a red-black tree. +- Open addressing handles hash collisions by probing multiple times. Linear probing uses a fixed step size, the disadvantage is that it can not delete elements, and is prone to aggregation. Multiple hashing uses multiple hash functions for probing, compared to linear probing is less likely to produce aggregation, but multiple hash functions increase the amount of computation. +- Different programming languages take different implementations of hash tables. For example, Java's `HashMap` uses separate chaining of addresses, while Python's `Dict` uses open addressing. +- In hash table, we want the hash algorithm to be deterministic, efficient and uniformly distributed. In cryptography, the hashing algorithm should also be collision resistant and avalanche effect. +- Hashing algorithms usually use large prime numbers as moduli to maximize the even distribution of hash values and reduce hash collisions. +- Common hash algorithms include MD5, SHA-1, SHA-2, and SHA3, etc. MD5 is commonly used to verify file integrity, and SHA-2 is commonly used in security applications and protocols. +- Programming languages usually provide built-in hashing algorithms for data types to compute bucket indexes in hash tables. Typically, only immutable objects are hashable. + +### Q & A + +!!! question "Why is the time complexity of hash tables not $O(n)$ ?" + + When hash collision is more severe, the time complexity of the hash table degrades to $O(n)$ . When the hash function is well designed, the capacity is set reasonably, and hash collision is more even, the time complexity is $O(1)$ . When we use a programming language's built-in hash table, we usually consider the time complexity to be $O(1)$ . + +!!! question "Why not use the hash function $f(x) = x$? That way there is no conflict" + + Under the $f(x) = x$ hash function, each element corresponds to a unique bucket index, which is equivalent to an array. However, the input space is usually much larger than the output space (the length of the array), so the final step of a hash function is often to model the length of the array. In other words, the goal of hash tables is to map a larger state space to a smaller one and provide $O(1)$ query efficiency. + +!!! Question "The underlying implementations of hash tables are arrays, linked lists, and binary trees, but why can they be more efficient than them?" + + First, hash tables become more time efficient, but less space efficient. A significant portion of the hash table's memory is unused, the + + Second, it just becomes more time efficient in specific usage scenarios. If a function can be implemented using arrays or linked lists with the same time complexity, it is usually faster than a hash table. This is because hash functions require overhead to compute and have a larger constant term for time complexity. + + Finally, the time complexity of hash tables may deteriorate. For example, in separate chaining, we take to perform a lookup operation in a linked list or a red-black tree and still run the risk of degrading to $O(n)$ time. + +!!! Question "Does multiple hashing have the flaw of not being able to delete elements directly? For marking a deleted space, can this space be used again?" + + Multiple hashes are a type of open addressing, and all open addressing methods have the drawback of not being able to delete elements directly; they need to be deleted by marking them. Marked as deleted space can be used again. When a new element is inserted into the hash table, and the location marked as deleted is found through the hash function, the location can be used by the new element. This keeps the probe sequence of the hash table unchanged and ensures the space utilization of the hash table. + +!!! question "Why does hash collision occur when looking up elements in linear probing?" + + Find the corresponding bucket and key-value pairs through the hash function, found that the `key` does not match, which means that there is a hash collision. Therefore, the linear probing method will be based on a pre-set step down the search, until the correct key-value pairs found or can not be found to jump out until. + +!!! question "Why does hash table expansion mitigate hash collisions?" + + The last step of the hash function is often the length of the array $n$ to take the remainder, so that the output value falls into the array index range; after the expansion, the length of the array $n$ changed, and the index corresponding to the `key` may also change. 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z+L#A?%{g)JRD#7NCQ_YuV0CkcYED2yrDpD!RM048N^n%HIwj}2faI%;v%!vm?`=M3 zi&U=P>2i2Q>O0$ZhG)R|c3m&Otb^)-vb2VwwFRs|M<*Ph=E*;BN^i39j^WOKhnMG34nLPirCgm>{heCj{uIL)+6cQ<6{{ Table   Efficiency of heap operations

    , greater> minHeap; + // Initialize the max heap. + priority_queue, less> maxHeap; + + /* Element into heap */ + maxHeap.push(1); + maxHeap.push(3); + maxHeap.push(2); + maxHeap.push(5); + maxHeap.push(4); + + /* Get the top element of the heap */ + int peek = maxHeap.top(); // 5 + + /* Top-of-heap element out of heap */ + // The elements out of the heap will form a sequence from largest to smallest + maxHeap.pop(); // 5 + maxHeap.pop(); // 4 + maxHeap.pop(); // 3 + maxHeap.pop(); // 2 + maxHeap.pop(); // 1 + + /* Get heap size */ + int size = maxHeap.size(); + + /* Determine if the heap is empty */ + bool isEmpty = maxHeap.empty(); + + /* Input list and build heap */ + vector input{1, 3, 2, 5, 4}; + priority_queue, greater> minHeap(input.begin(), input.end()); + ``` + +=== "Java" + + ```java title="heap.java" + /* Initialize heap */ + // Initialize the min heap. + Queue minHeap = new PriorityQueue<>(); + // Initialize the max heap (just modify the Comparator with a lambda expression). + Queue maxHeap = new PriorityQueue<>((a, b) -> b - a); + + /* Element into heap */ + maxHeap.offer(1); + maxHeap.offer(3); + maxHeap.offer(2); + maxHeap.offer(5); + maxHeap.offer(4); + + /* Get the top element of the heap */ + int peek = maxHeap.peek(); // 5 + + /* Top-of-heap element out of heap */ + // The elements out of the heap will form a sequence from largest to smallest + peek = maxHeap.poll(); // 5 + peek = maxHeap.poll(); // 4 + peek = maxHeap.poll(); // 3 + peek = maxHeap.poll(); // 2 + peek = maxHeap.poll(); // 1 + + /* Get heap size */ + int size = maxHeap.size(); + + /* Determine if the heap is empty */ + boolean isEmpty = maxHeap.isEmpty(); + + /* Input list and build heap */ + minHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4)); + ``` + +=== "C#" + + ```csharp title="heap.cs" + /* Initialize heap */ + // Initialize the min heap. + PriorityQueue minHeap = new(); + // Initialize the max heap (just modify the Comparator with a lambda expression). + PriorityQueue maxHeap = new(Comparer.Create((x, y) => y - x)); + + /* Element into heap */ + maxHeap.Enqueue(1, 1); + maxHeap.Enqueue(3, 3); + maxHeap.Enqueue(2, 2); + maxHeap.Enqueue(5, 5); + maxHeap.Enqueue(4, 4); + + /* Get the top element of the heap */ + int peek = maxHeap.Peek();//5 + + /* Top-of-heap element out of heap */ + // The elements out of the heap will form a sequence from largest to smallest + peek = maxHeap.Dequeue(); // 5 + peek = maxHeap.Dequeue(); // 4 + peek = maxHeap.Dequeue(); // 3 + peek = maxHeap.Dequeue(); // 2 + peek = maxHeap.Dequeue(); // 1 + + /* Get heap size */ + int size = maxHeap.Count; + + /* Determine if the heap is empty */ + bool isEmpty = maxHeap.Count == 0; + + /* Input list and build heap */ + minHeap = new PriorityQueue(new List<(int, int)> { (1, 1), (3, 3), (2, 2), (5, 5), (4, 4), }); + ``` + +=== "Go" + + ```go title="heap.go" + // Integer max heaps can be built in Go by implementing heap. + // Implementing heap.Interface requires also implementing sort. + type intHeap []any + + // Push heap.Interface's method to push elements into the heap. + func (h *intHeap) Push(x any) { + // Push and Pop use the pointer receiver as an argument. + // because they not only adjust the contents of the slice, but also modify the length of the slice. + *h = append(*h, x.(int)) + } + + // Pop heap.Interface's method to pop the top-of-heap element. + func (h *intHeap) Pop() any { + // The heap elements are stored at the end. + last := (*h)[len(*h)-1] + *h = (*h)[:len(*h)-1] + return last + } + + // Methods of the Len sort. + func (h *intHeap) Len() int { + return len(*h) + } + + // Methods of Less sort. + func (h *intHeap) Less(i, j int) bool { + // If implementing a min heap, it needs to be adjusted to less than sign + return (*h)[i].(int) > (*h)[j].(int) + } + + // Methods of the Swap sort. + func (h *intHeap) Swap(i, j int) { + (*h)[i], (*h)[j] = (*h)[j], (*h)[i] + } + + // Top gets the top element of the heap. + func (h *intHeap) Top() any { + return (*h)[0] + } + + /* Driver Code */ + func TestHeap(t *testing.T) { + /* Initialize heap */ + // Initialize the max heap. + maxHeap := &intHeap{} + heap.Init(maxHeap) + /* Element into heap */ + // Call the heap. + heap.Push(maxHeap, 1) + heap.Push(maxHeap, 3) + heap.Push(maxHeap, 2) + heap.Push(maxHeap, 4) + heap.Push(maxHeap, 5) + + /* Get the top element of the heap */ + top := maxHeap.Top() + fmt.Printf("The top element of the heap is %d\n", top) + + /* Top-of-heap element out of heap */ + // Call the heap. + heap.Pop(maxHeap) // 5 + heap.Pop(maxHeap) // 4 + heap.Pop(maxHeap) // 3 + heap.Pop(maxHeap) // 2 + heap.Pop(maxHeap) // 1 + + /* Get heap size */ + size := len(*maxHeap) + fmt.Printf("The number of heap elements is %d\n", size) + + /* Determine if the heap is empty */ + isEmpty := len(*maxHeap) == 0 + fmt.Printf("Is heap empty %t\n", isEmpty) + } + ``` + +=== "Swift" + + ```swift title="heap.swift" + // Swift does not provide a built-in Heap class. + ``` + +=== "JS" + + ```javascript title="heap.js" + // JavaScript does not provide a built-in Heap class. + ``` + +=== "TS" + + ```typescript title="heap.ts" + // TypeScript does not provide a built-in Heap class. + ``` + +=== "Dart" + + ```dart title="heap.dart" + // Dart does not provide a built-in Heap class + ``` + +=== "Rust" + + ```rust title="heap.rs" + use std::collections::BinaryHeap; + use std::cmp::Reverse; + + /* Initialize heap */ + // Initialize the min heap. + let mut min_heap = BinaryHeap::>::new(); + // Initialize the max heap. + let mut max_heap = BinaryHeap::new(); + + /* Element into heap */ + max_heap.push(1); + max_heap.push(3); + max_heap.push(2); + max_heap.push(5); + max_heap.push(4); + + /* Get the top element of the heap */ + let peek = max_heap.peek().unwrap(); // 5 + + /* Top-of-heap element out of heap */ + // The elements out of the heap will form a sequence from largest to smallest + let peek = max_heap.pop().unwrap(); // 5 + let peek = max_heap.pop().unwrap(); // 4 + let peek = max_heap.pop().unwrap(); // 3 + let peek = max_heap.pop().unwrap(); // 2 + let peek = max_heap.pop().unwrap(); // 1 + + /* Get heap size */ + let size = max_heap.len(); + + /* Determine if the heap is empty */ + let is_empty = max_heap.is_empty(); + + /* Input list and build heap */ + let min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]); + ``` + +=== "C" + + ```c title="heap.c" + // C does not provide a built-in Heap class + ``` + +=== "Zig" + + ```zig title="heap.zig" + + ``` + +## Heap Implementation + +The following implementation is a max heap. To convert it to a min heap, simply invert all size logic judgments (e.g., replace $\geq$ with $\leq$). Interested readers can implement it by themselves. + +### Heap Storage And Representation + +We learned in the binary tree chapter that complete binary trees are well suited to be represented by arrays. Since a heap is exactly a complete binary tree, **we will use an array to store the heap**. + +When an array is used to represent a binary tree, the elements represent the node values and the indexes represent the position of the nodes in the binary tree. **The node pointers are realized by the index mapping formula**. + +As shown in the figure below, given an index $i$, its left-child node index is $2i + 1$, its right-child node index is $2i + 2$, and its parent node index is $(i - 1) / 2$ (rounded down). When the index is out of bounds, it means empty node or node does not exist. + +![Representation and storage of heap](heap.assets/representation_of_heap.png) + +We can encapsulate the index mapping formula into a function for easy subsequent use. + +```src +[file]{my_heap}-[class]{max_heap}-[func]{parent} +``` + +### Accessing The Top Element Of The Heap + +The top element of the heap is the root node of the binary tree, which is the first element of the list. + +```src +[file]{my_heap}-[class]{max_heap}-[func]{peek} +``` + +### Element Into The Heap + +Given the element `val` , we first add it to the bottom of the heap. After the addition, the conditions under which the heap was founded may have been broken since val may be larger than any other element in the heap. Therefore, **needs to repair each node on the path from the insertion node to the root node**, an operation known as 'heapify heapify'. + +**Consider performing heapify from bottom to top starting from the entry node**. As shown in the figure below, we compare the values of the insertion node with its parent node and swap them if the insertion node is larger. This operation is then continued by repairing each node in the heap from bottom to top until it ends when the root node is crossed or when a node is encountered that does not need to be swapped. + +=== "<1>" + ![Element into heap step](heap.assets/heap_push_step1.png) + +=== "<2>" + ![heap_push_step2](heap.assets/heap_push_step2.png) + +=== "<3>" + ![heap_push_step3](heap.assets/heap_push_step3.png) + +=== "<4>" + ![heap_push_step4](heap.assets/heap_push_step4.png) + +=== "<5>" + ![heap_push_step5](heap.assets/heap_push_step5.png) + +=== "<6>" + ![heap_push_step6](heap.assets/heap_push_step6.png) + +=== "<7>" + ![heap_push_step7](heap.assets/heap_push_step7.png) + +=== "<8>" + ![heap_push_step8](heap.assets/heap_push_step8.png) + +=== "<9>" + ![heap_push_step9](heap.assets/heap_push_step9.png) + +Let the total number of nodes be $n$ , then the height of the tree node is $O(\log n)$ . It can be seen that the maximum number of rounds of the heapify operation is $O(\log n)$ , **and the time complexity of the element into heap operation is $O(\log n)$** . + +```src +[file]{my_heap}-[class]{max_heap}-[func]{sift_up} +``` + +### The Top Element Of The Heap Is Out Of The Heap + +The top element of the heap is the root node of the binary tree, i.e. the first element of the list. If we directly remove the first element from the list, the indexes of all nodes in the binary tree will change, which will make subsequent repair using heapify difficult. In order to minimize the changes in the indexes of the elements, we use the following procedure. + +1. Swap the top element of the heap with the bottom element of the heap (i.e., swap the root node with the rightmost leaf node). +2. When the swap is complete, the bottom of the heap is removed from the list (note that since it has been swapped, it is the original top element of the heap that is actually removed). +3. Starting from the root node, **performs heapify from top to bottom**. + +As shown in the figure below, **"top-to-bottom heapify" operates in the opposite direction of "bottom-to-top heapify"**we compare the value of the root node with the values of its two child nodes, and exchange the biggest We compare the value of the root node with the values of its two children, and exchange the largest child node with the root node. We then loop through this operation until we cross a leaf node or encounter a node that does not need to be exchanged. + +=== "<1>" + ![Heap top element exit step](heap.assets/heap_pop_step1.png) + +=== "<2>" + ![heap_pop_step2](heap.assets/heap_pop_step2.png) + +=== "<3>" + ![heap_pop_step3](heap.assets/heap_pop_step3.png) + +=== "<4>" + ![heap_pop_step4](heap.assets/heap_pop_step4.png) + +=== "<5>" + ![heap_pop_step5](heap.assets/heap_pop_step5.png) + +=== "<6>" + ![heap_pop_step6](heap.assets/heap_pop_step6.png) + +=== "<7>" + ![heap_pop_step7](heap.assets/heap_pop_step7.png) + +=== "<8>" + ![heap_pop_step8](heap.assets/heap_pop_step8.png) + +=== "<9>" + ![heap_pop_step9](heap.assets/heap_pop_step9.png) + +=== "<10>" + ![heap_pop_step10](heap.assets/heap_pop_step10.png) + +Similar to the element-in operation, the top-of-heap element-out operation has a time complexity of $O(\log n)$ . + +```src +[file]{my_heap}-[class]{max_heap}-[func]{sift_down} +``` + +## Common Applications Of Heaps + +- **Priority Queue**: heaps are often used as the preferred data structure to implement priority queues, with time complexity of $O(\log n)$ for both in-queue and out-queue operations and $O(n)$ for queue-building operations, which are very efficient. +- **Heap sort**: given a set of data, we can build a heap with them and then keep performing element-out operations to get ordered data. However, we usually implement heap sorting using a more elegant approach, as detailed in the subsequent chapter on heap sorting. +- **Getting the largest $k$ elements**: this is a classic algorithmic problem and also a typical application, e.g., selecting the top 10 hot news as a microblogging hot search, selecting the top 10 selling products, etc. diff --git a/docs-en/chapter_heap/index.md b/docs-en/chapter_heap/index.md new file mode 100644 index 0000000000..94b84e5bb0 --- /dev/null +++ b/docs-en/chapter_heap/index.md @@ -0,0 +1,13 @@ +# Heap + +
    + +![heap](../assets/covers/chapter_heap.jpg){ width="600" } + +
    + +!!! abstract + + The heaps are like the peaks of a mountain; they are layered and undulating and have different shapes. + + Every mountain has its height and the highest peaks are always the first to come into view. diff --git a/docs-en/chapter_heap/summary.md b/docs-en/chapter_heap/summary.md new file mode 100644 index 0000000000..606af8b40d --- /dev/null +++ b/docs-en/chapter_heap/summary.md @@ -0,0 +1,17 @@ +# Summary + +### Highlights + +- A heap is a complete binary tree that can be categorized into a max heap and a min heap based on the establishment conditions. The top element of the heap of a big (small) top heap is the largest (smallest). +- A priority queue is defined as a queue with an outgoing priority, usually implemented using a heap. +- The common operations of heap and their corresponding time complexity include: elements into the heap $O(\log n)$, elements out of the top of the heap $O(\log n)$ and access to the top of the heap $O(1)$, and so on. +- Complete binary trees are well suited for representation in arrays, so we usually use arrays to store heaps. +- The heapify operation is used to maintain the properties of the heap and is used in both in-heap and out-heap operations. +- The time complexity of inputting $n$ elements and building a heap can be optimized to $O(n)$ , which is very efficient. +- Top-K is a classical algorithmic problem that can be solved efficiently using a heap data structure with a time complexity of $O(n \log k)$ . + +### Q & A + +!!! Question "Is the "heap" of data structure the same concept as the "heap" of memory management?" + + The two are not the same concept, they just happen to both be called heaps. The heap in a computer system's memory is part of a dynamic memory allocation that a program can use to store data at runtime. Programs can request a certain amount of heap memory for storing complex structures such as objects and arrays. When this data is no longer needed, the program needs to free this memory to prevent memory leaks. 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z=<$Wrddi(aKtY8A1$Xt4+m3ev*_VB+l!xt~KV8TdDp$t3e(gI?YCiT*XpQWXIrRiZ zXor;zzjfg(Si8}j49GM~B;x@@Vwjv^@p9t0%Ww9uKY98iI*k1O(o>A@u)IX$tJmzs zSx8QEJH0s(04_?QN&9JNCgI{;b>?`7{8?l$21|t_%DwZCrVa@5|M2@etPoVG zIuQ2ssCE9Kz@y06 + +!!! abstract + + A young girl dances, intertwining with the data, a melody of algorithms fluttering from her skirts. + + She invites you to dance, so follow her lead and step into the world of algorithms full of logic and beauty. diff --git a/docs-en/chapter_introduction/summary.md b/docs-en/chapter_introduction/summary.md new file mode 100644 index 0000000000..ac56c258e5 --- /dev/null +++ b/docs-en/chapter_introduction/summary.md @@ -0,0 +1,9 @@ +# Summary + +- Algorithms are ubiquitous in our daily lives and are not a distant knowledge. In fact, we have learned many algorithms without even realizing it to solve life's problems, big and small. +- The principle of looking up a dictionary is consistent with the binary search algorithm. The binary search algorithm embodies the important algorithmic idea of divide and conquer. +- The process of sorting poker is very similar to the insertion sort algorithm. The insertion sort algorithm is suitable for sorting small data sets. +- The steps in currency change are essentially greedy algorithms, taking the best option that currently seems to be available at each step. +- An algorithm is a set of instructions or operational steps to solve a specific problem in a limited amount of time, while a data structure is the way data is organized and stored in a computer. +- Data structures and algorithms are closely intertwined. 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a/docs-en/chapter_introduction/what_is_dsa.md b/docs-en/chapter_introduction/what_is_dsa.md new file mode 100644 index 0000000000..784af7023e --- /dev/null +++ b/docs-en/chapter_introduction/what_is_dsa.md @@ -0,0 +1,53 @@ +# What is Algorithms + +## Algorithm Definition + +An "algorithm" is a set of instructions or operational steps to solve a particular problem in a limited time, which has the following characteristics. + +- Problems are well defined and contain clear definitions of inputs and outputs. +- Feasible and able to be accomplished with limited steps, time and memory space. +- Each step has a defined meaning and the output is always the same for the same inputs and operating conditions. + +## Data Structure Definitions + +"A data structure is a way of organizing and storing data in a computer and has the following design goals. + +- Space occupation is minimized to save computer memory. +- Data manipulation is as fast as possible, covering data access, additions, deletions, updates, etc. +- Provide concise data representation and logical information in order to make the algorithm work efficiently. + +**Data structure design is a process full of tradeoffs**. If you want to improve in one area, you often need to compromise in another. Two examples are given below. + +- Compared to arrays, linked lists are more convenient for data addition and deletion operations, but at the expense of data access speed. +- Graphs provide richer logical information compared to linked lists, but take up more memory space. + +## Relationship Between Data Structures And Algorithms + +As shown in the figure below, data structure and algorithm are highly correlated and tightly integrated in the following three ways. + +- Data structures are the cornerstone of algorithms. Data structures provide algorithms with structured storage of data, and methods for manipulating the data. +- Algorithms are the arena where data structures come into play. Data structures by themselves only store information about the data, and it is in combination with an algorithm that a particular problem can be solved. +- Algorithms can often be implemented based on different data structures, but the efficiency of execution can vary greatly, and choosing the right data structure is key. + +![Relationship between data structure and algorithm](what_is_dsa.assets/relationship_between_data_structure_and_algorithm.png) + +Data structures and algorithms are like the building blocks shown in the figure below. A set of building blocks, in addition to containing many parts, comes with detailed assembly instructions. By following the instructions step by step, we can assemble a beautiful model of the blocks. + +![assembling blocks](what_is_dsa.assets/assembling_blocks.jpg) + +The detailed correspondence between the two is shown in the table below. + +

    Table   Analogizing Data Structures and Algorithms to Building Blocks

    + +| Data Structures and Algorithms | Building Blocks | +| ------------------------------ | ----------------------------------------------------------------------------- | +| Input data | Unassembled blocks | +| data structure | building block organization, including shape, size, and connections | +| algorithms | a series of operational steps to put the blocks together into the target form | +| Output Data | Building Block Models | + +It is worth stating that data structures and algorithms are independent of programming languages. Because of this, this book is able to provide implementations in multiple programming languages. + +!!! tip "agreed upon acronym" + + In practice, we usually refer to "data structures and algorithms" as "algorithms". 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+1,45 @@ +# The Book + +The aim of this project is to create an open source, free, novice-friendly introductory tutorial on data structures and algorithms. + +- Animated graphs are used throughout the book to structure the knowledge of data structures and algorithms in a way that is clear and easy to understand with a smooth learning curve. +- The source code of the algorithms can be run with a single click, supporting Java, C++, Python, Go, JS, TS, C#, Swift, Rust, Dart, Zig and other languages. +- Readers are encouraged to help each other and make progress in the chapter discussion forums, and questions and comments can usually be answered within two days. + +## Target Readers + +If you are a beginner to algorithms, have never touched an algorithm before, or already have some experience brushing up on data structures and algorithms, and have a vague understanding of data structures and algorithms, repeatedly jumping sideways between what you can and can't do, then this book is just for you! + +If you have already accumulated a certain amount of questions and are familiar with most of the question types, then this book can help you review and organize the algorithm knowledge system, and the repository source code can be used as a "brushing tool library" or "algorithm dictionary". + +If you are an algorithm expert, we look forward to receiving your valuable suggestions or [participate in the creation together](https://www.hello-algo.com/chapter_appendix/contribution/). + +!!! success "precondition" + + You will need to have at least a basic knowledge of programming in any language and be able to read and write simple code. + +## Content Structure + +The main contents of the book are shown in the figure below. + +- **Complexity Analysis**: dimensions and methods of evaluation of data structures and algorithms. Methods of deriving time complexity, space complexity, common types, examples, etc. +- **Data Structures**: basic data types, classification methods of data structures. Definition, advantages and disadvantages, common operations, common types, typical applications, implementation methods of data structures such as arrays, linked lists, stacks, queues, hash tables, trees, heaps, graphs, etc. +- **Algorithms**: definitions, advantages and disadvantages, efficiency, application scenarios, solution steps, sample topics of search, sorting algorithms, divide and conquer, backtracking algorithms, dynamic programming, greedy algorithms, and more. + +![Hello Algo content structure](about_the_book.assets/hello_algo_mindmap.jpg) + +## Acknowledgements + +During the creation of this book, I received help from many people, including but not limited to: + +- Thank you to my mentor at the company, Dr. Shih Lee, for encouraging me to "get moving" during one of our conversations, which strengthened my resolve to write this book. +- I would like to thank my girlfriend Bubbles for being the first reader of this book, and for making many valuable suggestions from the perspective of an algorithm whiz, making this book more suitable for newbies. +- Thanks to Tengbao, Qibao, and Feibao for coming up with a creative name for this book that evokes fond memories of writing the first line of code "Hello World!". +- Thanks to Sutong for designing the beautiful cover and logo for this book and patiently revising it many times under my OCD. +- Thanks to @squidfunk for writing layout suggestions and for developing the open source documentation theme [Material-for-MkDocs](https://github.com/squidfunk/mkdocs-material/tree/master). + +During the writing process, I read many textbooks and articles on data structures and algorithms. These works provide excellent models for this book and ensure the accuracy and quality of its contents. I would like to thank all my teachers and predecessors for their outstanding contributions! + +This book promotes a hands-on approach to learning, and in this respect is heavily inspired by ["Hands-On Learning for Depth"](https://github.com/d2l-ai/d2l-zh). I highly recommend this excellent book to all readers. + +A heartfelt thank you to my parents, it is your constant support and encouragement that gives me the opportunity to do this fun-filled thing. diff --git a/docs-en/chapter_preface/index.md b/docs-en/chapter_preface/index.md new file mode 100644 index 0000000000..9a6b66aa6e --- /dev/null +++ b/docs-en/chapter_preface/index.md @@ -0,0 +1,13 @@ +# Preface + +

    + +![Preface](../assets/covers/chapter_preface.jpg){ width="600" } + +
    + +!!! abstract + + Algorithms are like a beautiful symphony, with each line of code flowing like a rhythm. + + May this book ring softly in your head, leaving a unique and profound melody. diff --git a/docs-en/chapter_preface/suggestions.assets/code_md_to_repo.png b/docs-en/chapter_preface/suggestions.assets/code_md_to_repo.png new file mode 100644 index 0000000000000000000000000000000000000000..0b33c528f774819454ec215065d325cecdaba0c3 GIT binary patch literal 60126 zcmb4q^;?xqwD#=1>5>M?O-M;g=O!eiQBb;5Qb9oKkw#iTx+El|Lll7xC?zGG(kWfi ze7@f~|G@cW>bhoKYd!0pHEU+>CsJEenGlZ#4*)=@s-mb10Or3YdeP=@Z{v=_V$*UotNi<-J>?ms)h|8&|XdB)a5M$>Ss%YxwP3+uk-F>}7;%!`C={$Yh@uO2hMYpTFyQa4G ziJ85jxh3@FFTaTF*RNm0!^6kM#(H~uD=RAlYIl4JN7K{OV`5_B22q4HaF za@q^OpyJBf@1x&~YSG!H4L-pUc^^I`XXVAD6=t?%Xg<*nibQn~FXbxkZ<>3^lK99W?aSCuEhh4UF1)2Ak)Y zh1Z7h$~>&pM`veeFKsUMj`w^S_)^wcW)nFi>(c3wG^*j<`!cz)5Y?ERSFUHI|7Uh* za(=h5b41wgyGMGv{xhSSKc7|A)%qv?eEIQnvif(?kD|P`yuI~<^W}>Dzx%p=eabJp 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zddmvXybL@i;58WVvF722H?`c46os@NG{?`FkVY~L!a6U3G?7m}*P0lJQO7i|-@1s` z2$&Tl7#B$U-4Mc3@^?@Z5KlW8ntBBqPL+9y0eamR!xXyx?dyur=d)HG^i(tq2(W;i z{B`#0Mu4|)I@pI>h!@)m4Ia8`#9vu(sPa$o3NZ+F$*|SL^D+P`qE)ZfGRDd7_vTZ+ ztyX@-PQ}2N!$fL|d&ds&VnMW@kP7->loB~-Et;Y_l~^#nQ%)Pe4P<2ew1vn0hJ{sLn0XZ?cLbCb LG~{b#VekGITmG|- literal 0 HcmV?d00001 diff --git a/docs-en/chapter_preface/suggestions.md b/docs-en/chapter_preface/suggestions.md new file mode 100644 index 0000000000..d19ca9bc25 --- /dev/null +++ b/docs-en/chapter_preface/suggestions.md @@ -0,0 +1,224 @@ +# How To Read + +!!! tip + + For the best reading experience, it is recommended that you read through this section. + +## Conventions Of Style + +- Those labeled `*` after the title are optional chapters with relatively difficult content. If you have limited time, it is advisable to skip them. +- Proper nouns and words and phrases with specific meanings are marked with `"double quotes"` to avoid ambiguity. +- Important proper nouns and their English translations are marked with `" "` in parentheses, e.g. `"array array"` . It is recommended to memorize them for reading the literature. +- **Bolded text** Indicates key content or summary statements, which deserve special attention. +- When it comes to terms that are inconsistent between programming languages, this book follows Python, for example using $\text{None}$ to mean "empty". +- This book partially abandons the specification of annotations in programming languages in exchange for a more compact layout of the content. There are three main types of annotations: title annotations, content annotations, and multi-line annotations. + +=== "Python" + + ```python title="" + """Header comments for labeling functions, classes, test samples, etc."""" + + # Content comments for detailed code solutions + + """ + multi-line + marginal notes + """ + ``` + +=== "C++" + + ```cpp title="" + /* Header comments for labeling functions, classes, test samples, etc. */ + + // Content comments for detailed code solutions. + + /** + * multi-line + * marginal notes + */ + ``` + +=== "Java" + + ```java title="" + /* Header comments for labeling functions, classes, test samples, etc. */ + + // Content comments for detailed code solutions. + + /** + * multi-line + * marginal notes + */ + ``` + +=== "C#" + + ```csharp title="" + /* Header comments for labeling functions, classes, test samples, etc. */ + + // Content comments for detailed code solutions. + + /** + * multi-line + * marginal notes + */ + ``` + +=== "Go" + + ```go title="" + /* Header comments for labeling functions, classes, test samples, etc. */ + + // Content comments for detailed code solutions. + + /** + * multi-line + * marginal notes + */ + ``` + +=== "Swift" + + ```swift title="" + /* Header comments for labeling functions, classes, test samples, etc. */ + + // Content comments for detailed code solutions. + + /** + * multi-line + * marginal notes + */ + ``` + +=== "JS" + + ```javascript title="" + /* Header comments for labeling functions, classes, test samples, etc. */ + + // Content comments for detailed code solutions. + + /** + * multi-line + * marginal notes + */ + ``` + +=== "TS" + + ```typescript title="" + /* Header comments for labeling functions, classes, test samples, etc. */ + + // Content comments for detailed code solutions. + + /** + * multi-line + * marginal notes + */ + ``` + +=== "Dart" + + ```dart title="" + /* Header comments for labeling functions, classes, test samples, etc. */ + + // Content comments for detailed code solutions. + + /** + * multi-line + * marginal notes + */ + ``` + +=== "Rust" + + ```rust title="" + /* Header comments for labeling functions, classes, test samples, etc. */ + + // Content comments for detailed code solutions. + + /** + * multi-line + * marginal notes + */ + ``` + +=== "C" + + ```c title="" + /* Header comments for labeling functions, classes, test samples, etc. */ + + // Content comments for detailed code solutions. + + /** + * multi-line + * marginal notes + */ + ``` + +=== "Zig" + + ```zig title="" + // Header comments for labeling functions, classes, test samples, etc. + + // Content comments for detailed code solutions. + + // Multi-line + // Annotation + ``` + +## Learn Efficiently In Animated Graphic Solutions + +Compared with text, videos and pictures have a higher degree of information density and structure and are easier to understand. In this book, **key and difficult knowledge will be presented mainly in the form of animations and graphs**, while the text serves as an explanation and supplement to the animations and graphs. + +If, while reading the book, you find that a particular paragraph provides an animation or a graphic solution as shown below, **please use the figure as the primary source and the text as a supplement and synthesize the two to understand the content**. + +![Example animation](../index.assets/animation.gif) + +## Deeper Understanding In Code Practice + +The companion code for this book is hosted in the [GitHub repository](https://github.com/krahets/hello-algo). As shown in the figure below, **the source code is accompanied by test samples that can be run with a single click**. + +If time permits, **it is recommended that you refer to the code and knock it through on your own**. If you have limited time to study, please read through and run all the code at least once. + +The process of writing code is often more rewarding than reading it. **Learning by doing is really learning**. + +![Running code example](../index.assets/running_code.gif) + +The preliminaries for running the code are divided into three main steps. + +**Step 1: Install the local programming environment**. Please refer to [Appendix Tutorial](https://www.hello-algo.com/chapter_appendix/installation/) for installation, or skip this step if already installed. + +**Step 2: Clone or download the code repository**. If [Git](https://git-scm.com/downloads) is already installed, you can clone this repository with the following command. + +```shell +git clone https://github.com/krahets/hello-algo.git +``` + +Of course, you can also in the location shown in the figure below, click "Download ZIP" directly download the code zip, and then in the local solution. + +![Clone repository with download code](suggestions.assets/download_code.png) + +**Step 3: Run the source code**. As shown in the figure below, for the code block labeled with the file name at the top, we can find the corresponding source code file in the `codes` folder of the repository. The source code files can be run with a single click, which will help you save unnecessary debugging time and allow you to focus on what you are learning. + +![Code block with corresponding source file](suggestions.assets/code_md_to_repo.png) + +## Growing Together In Questioning And Discussion + +While reading this book, please don't skip over the points that you didn't learn. **Feel free to ask your questions in the comment section**. We will be happy to answer them and can usually respond within two days. + +As you can see in the figure below, each post comes with a comment section at the bottom. I hope you'll pay more attention to the comments section. On the one hand, you can learn about the problems that people encounter, so as to check the gaps and stimulate deeper thinking. On the other hand, we expect you to generously answer other partners' questions, share your insights, and help others improve. + +![Example of comment section](../index.assets/comment.gif) + +## Algorithm Learning Route + +From a general point of view, we can divide the process of learning data structures and algorithms into three stages. + +1. **Introduction to Algorithms**. We need to familiarize ourselves with the characteristics and usage of various data structures and learn about the principles, processes, uses and efficiency of different algorithms. +2. **Brush up on algorithm questions**. It is recommended to start brushing from popular topics, such as [Sword to Offer](https://leetcode.cn/studyplan/coding-interviews/) and [LeetCode Hot 100](https://leetcode.cn/studyplan/top-100- liked/), first accumulate at least 100 questions to familiarize yourself with mainstream algorithmic problems. Forgetfulness can be a challenge when first brushing up, but rest assured that this is normal. We can follow the "Ebbinghaus Forgetting Curve" to review the questions, and usually after 3-5 rounds of repetitions, we will be able to memorize them. +3. **Build the knowledge system**. In terms of learning, we can read algorithm column articles, solution frameworks and algorithm textbooks to continuously enrich the knowledge system. In terms of brushing, we can try to adopt advanced brushing strategies, such as categorizing by topic, multiple solutions, multiple solutions, etc. Related brushing tips can be found in various communities. + +As shown in the figure below, this book mainly covers "Phase 1" and is designed to help you start Phase 2 and 3 more efficiently. + +![algorithm learning route](suggestions.assets/learning_route.png) diff --git a/docs-en/chapter_preface/summary.md b/docs-en/chapter_preface/summary.md new file mode 100644 index 0000000000..2ea2c18ff6 --- /dev/null +++ b/docs-en/chapter_preface/summary.md @@ -0,0 +1,8 @@ +# Summary + +- The main audience of this book is beginners in algorithm. If you already have some basic knowledge, this book can help you systematically review your algorithm knowledge, and the source code in this book can also be used as a "brushing tool library". +- The book consists of three main sections, Complexity Analysis, Data Structures, and Algorithms, covering most of the topics in the field. +- For newcomers to algorithms, it is crucial to read an introductory book in the beginning stages to take the road less traveled. +- Animations and graphs within the book are usually used to introduce key points and difficult knowledge. These should be given more attention when reading the book. +- Practice is the best way to learn programming. It is highly recommended that you run the source code and type in the code yourself. +- Each chapter in the web version of the book has a discussion forum, so feel free to share your doubts and insights. diff --git a/docs-en/chapter_reference/index.md b/docs-en/chapter_reference/index.md new file mode 100644 index 0000000000..008092ebb4 --- /dev/null +++ b/docs-en/chapter_reference/index.md @@ -0,0 +1,19 @@ +# References + +[1] Thomas H. Cormen, et al. Introduction to Algorithms (3rd Edition). + +[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1st Edition). + +[3] Yan Weimin. Data structure (C language version). + +[4] Deng, J. F.. Data structure (C++ language edition, 3rd edition). + +[5] Mark Allen Weiss. Data structures and algorithm analysis: a description of the Java language (3rd ed.). + +[6] J. Cheng. Big talk about data structures. + +[7] Wang Q. Data structures and the beauty of algorithms. + +[8] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6th Edition). + +[9] Aston Zhang, et al. 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It utilizes the ordered nature of data to reduce the search range by half in each round until the target element is found or the search interval is empty. + +!!! question + + Given an array `nums` of length $n$ with elements in descending order, the array contains no duplicate elements. Find and return the index of the element `target` in the array. If the array does not contain the element, return $-1$. + +![binary search example data](binary_search.assets/binary_search_example.png) + +As shown in the figure below, we first initialize pointers $i = 0$ and $j = n - 1$ , pointing to the first and last elements of the array, representing the search interval $[0, n - 1]$ . Note that the middle bracket denotes the closed interval, which contains the boundary values themselves. + +Next, loop through the following two steps. + +1. Calculate the midpoint index $m = \lfloor {(i + j) / 2} \rfloor$ , where $\lfloor \: \rfloor$ denotes the downward rounding operation. +2. To determine the size relationship between `nums[m]` and `target`, there are three cases. + 1. When `nums[m] < target`, it means that `target` is in the interval $[m + 1, j]$, so execute $i = m + 1$ . + 2. When `nums[m] > target` , it means that `target` is in the interval $[i, m - 1]$ , so execute $j = m - 1$ . + 3. When `nums[m] = target`, it means that `target` is found, so it returns the index $m$. + +If the array does not contain the target element, the search interval is eventually reduced to empty. This returns $-1$ . + +=== "<1>" + ![binary search process](binary_search.assets/binary_search_step1.png) + +=== "<2>" + ![binary_search_step2](binary_search.assets/binary_search_step2.png) + +=== "<3>" + ![binary_search_step3](binary_search.assets/binary_search_step3.png) + +=== "<4>" + ![binary_search_step4](binary_search.assets/binary_search_step4.png) + +=== "<5>" + ![binary_search_step5](binary_search.assets/binary_search_step5.png) + +=== "<6>" + ![binary_search_step6](binary_search.assets/binary_search_step6.png) + +=== "<7>" + ![binary_search_step7](binary_search.assets/binary_search_step7.png) + +It is worth noting that since $i$ and $j$ are both of type `int`, **so $i + j$ may be out of the range of values of type `int`**. In order to avoid large numbers crossing the boundary, we usually use the formula $m = \lfloor {i + (j - i) / 2} \rfloor$ to calculate the midpoint. + +```src +[file]{binary_search}-[class]{}-[func]{binary_search} +``` + +**Time complexity $O(\log n)$** : In a bisection loop, the interval is reduced by half in each round and the number of loops is $\log_2 n$ . + +**Space complexity $O(1)$** : Pointers $i$ and $j$ use constant size space. + +## Interval Representation + +In addition to the double-closed intervals mentioned above, a common interval representation is the "left-closed-right-open" interval, defined as $[0, n)$, where the left edge contains itself and the right edge does not contain itself. In this representation, the interval $[i, j]$ is empty when $i = j$. + +We can implement a binary search algorithm with the same functionality based on this representation. + +```src +[file]{binary_search}-[class]{}-[func]{binary_search_lcro} +``` + +As shown in the figure below, the initialization, loop condition and interval reduction operations of the binary search algorithm are different in the two interval representations. + +Since both the left and right edges of the "double-closed interval" representation are defined as closed intervals, the narrowing of the interval by the pointers $i$ and $j$ is also symmetric. This is less prone to errors, **so the "double-closed interval" style is generally recommended**. + +![two interval definitions](binary_search.assets/binary_search_ranges.png) + +## Strengths And Limitations + +Binary search has better performance in both time and space. + +- Binary search is time efficient. The logarithmic order has a significant time complexity advantage in large data sizes. For example, when the data size $n = 2^{20}$, linear search requires $2^{20} = 1048576$ rounds of cycles, while binary search requires only $\log_2 2^{20} = 20$ rounds of cycles. +- Binary search requires no extra space. Compared to searching algorithms that require extra space (such as hash search), binary search is more space-efficient. + +However, binary search is not suitable for all cases, mainly for the following reasons. + +- Binary search only works with ordered data. If the input data is unordered, it is not worthwhile to specialize in sorting in order to use binary search. This is because the time complexity of the sorting algorithm is usually $O(n \log n)$ , which is higher than both linear and binary search. For scenarios where elements are frequently inserted into specific locations to keep the array organized, the time complexity is $O(n)$ , which is also very expensive. +- Binary search is only applicable to arrays. Binary search requires jumping (non-contiguous) access to elements, and performing jumping accesses in a linked list is inefficient, so it is not suitable for application to linked lists or data structures based on linked list implementations. +- Linear lookup performs better with small data size. 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z$z6$+keIe6Vd2noX**U~3`mQF*%DH-i!+hWDLqeusqd4vq_6D~JQdY*u^~N;57{ll z$h}Lgocr+$tyMae{!qPkTtC>68Lj&nB8t;u;=`zhCsf(-U61J6qc5GlLkztm3Bd;0 zdH&Dn><%n>gV*RXyE-6z+}Pou%T9a{&tbNL5qtdoZKU`SJ2qR!H;vOuc)u*@<5|oc z5mbmjXL)sfghF3-W{KWIH>EFxI&bNS53j1=8FqY8azA-teLgeSo~|X4S&&bHymKUnCM89xFM>2EqvbJSb@DIK=Th1#XGsZ=o)mr#)cpz$ z55dQ~qIxk&9Mg5d!MuF{|vZ?zO1hPIDm!r$L$Kew^&-DWfib~THPZ_b)zMt|tU zYDjdBT)&aMCES9qEz(+?7u7e*ArP>8I0_Z$T3FT++t^BkEQzdA(dImuS3Rq82OhK%7RuXZ&4h(qBMSYMJM+*ECfW4+_`T%%bGQ+C2F>k1KXG@cst + +| | Linear search | Binary search | Tree search | Hash search | +| --------------------------- | ------------- | ------------------ | ------------------------ | ----------------------- | +| Find Elements | $O(n)$ | $O(\log n)$ | $O(\log n)$ | $O(1)$ | +| insert element | $O(1)$ | $O(n)$ | $O(\log n)$ | $O(1)$ $O(1)$ | +| delete element | $O(n)$ | $O(n)$ | $O(\log n)$ | $O(1)$ | +| Extra Space | $O(1)$ | $O(1)$ | $O(n)$ | $O(n)$ | +| Data Preprocessing | / | Sort $O(n \log n)$ | Build Tree $O(n \log n)$ | Build Hash Table $O(n)$ | +| Whether the data is ordered | Unordered | Ordered | Ordered | Unordered | + +The choice of searching algorithm also depends on the volume of data, search performance requirements, frequency of data queries and updates, etc. + +**Linear search** + +- It is more general and does not require any data preprocessing. If we only need to query the data once, the data preprocessing of the other three methods takes longer than the linear search. +- Suitable for smaller volumes of data, in which case time complexity has less impact on efficiency. +- Suitable for scenarios where data is updated more frequently, as this method does not require any additional maintenance of the data. + +**binary search** + +- It is suitable for the case of large data volume and has a stable efficiency performance with a worst-case time complexity of $O(\log n)$ . +- The amount of data cannot be too large because storing arrays requires contiguous memory space. +- Not suitable for high-frequency data addition and deletion scenarios because of the overhead of maintaining an ordered array. + +**hash lookup** + +- It is suitable for scenarios that require high query performance with an average time complexity of $O(1)$ . +- Not suitable for scenarios that require ordered data or range lookups, as hash tables cannot maintain the orderliness of the data. +- The high dependency on hash functions and hash collision handling strategies carries a high risk of performance degradation. +- Not suitable for excessive data volumes, as hash tables require extra space to minimize conflicts and thus provide good query performance. + +**Tree Finder** + +- Suitable for massive data because tree nodes are stored decentralized in memory. +- Ideal for scenarios where you need to maintain ordered data or range lookups. +- During the continuous addition and removal of nodes, the binary search tree may become skewed and the time complexity deteriorates to $O(n)$ . +- If an AVL tree or a red-black tree is used, the operations can be stabilized with $O(\log n)$ efficiency, but the operation of maintaining the tree balance will add extra overhead. diff --git a/docs-en/chapter_searching/summary.md b/docs-en/chapter_searching/summary.md new file mode 100644 index 0000000000..af130d5f0a --- /dev/null +++ b/docs-en/chapter_searching/summary.md @@ -0,0 +1,8 @@ +# Summary + +- Binary search relies on the orderliness of the data and is achieved by gradually reducing the search interval by half through a loop. It requires ordered input data and only works with arrays or data structures implemented based on arrays. +- Brute-force search locates data by traversing data structures. Linear search is applicable to arrays and linked lists, breadth-first search and depth-first search are applicable to graphs and trees. These algorithms are generalizable and do not require data preprocessing, but have high time complexity $O(n)$. +- Hash lookups, tree lookups and binary searches are efficient search methods that can quickly locate a target element in a particular data structure. These algorithms are efficient, with a time complexity of $O(\log n)$ or even $O(1)$, but usually require the use of additional data structures. +- In practice, we need to specifically analyze factors such as data volume, search performance requirements, data query and update frequency, so as to choose a suitable search method. +- Linear search is suitable for small or frequently updated data; binary search is suitable for large, sorted data; hash search is suitable for data that requires high query efficiency and does not require a range query; tree search is suitable for large, dynamic data that requires maintenance of order and support for range queries. +- Replacing linear lookups with hash lookups is a commonly used strategy to optimize runtime, reducing the time complexity from $O(n)$ to $O(1)$ . diff --git a/docs-en/chapter_sorting/bubble_sort.assets/bubble_operation_step1.png b/docs-en/chapter_sorting/bubble_sort.assets/bubble_operation_step1.png new file mode 100644 index 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The process is like a bubble rising from the bottom to the top, hence the name bubble sort. + +As shown in the figure below, the bubbling process can be simulated by using the element swapping operation: start traversing the array from the leftmost end to the right, compare the sizes of the neighboring elements in turn, and swap them if "left element > right element". After the traversal is complete, the largest element will be moved to the right end of the array. + +=== "<1>" + ![Simulate bubbling using element swapping operations](bubble_sort.assets/bubble_operation_step1.png) + +=== "<2>" + ![bubble_operation_step2](bubble_sort.assets/bubble_operation_step2.png) + +=== "<3>" + ![bubble_operation_step3](bubble_sort.assets/bubble_operation_step3.png) + +=== "<4>" + ![bubble_operation_step4](bubble_sort.assets/bubble_operation_step4.png) + +=== "<5>" + ![bubble_operation_step5](bubble_sort.assets/bubble_operation_step5.png) + +=== "<6>" + ![bubble_operation_step6](bubble_sort.assets/bubble_operation_step6.png) + +=== "<7>" + ![bubble_operation_step7](bubble_sort.assets/bubble_operation_step7.png) + +## Algorithmic Flow + +Let the length of the array be $n$ , the steps of bubble sort are shown in the figure below. + +1. First, perform a "bubble" on $n$ elements, **swapping the largest element of the array to the correct position**. +2. Next, perform a "bubble" on the remaining $n - 1$ elements, **swapping the second largest element to the correct position**. +3. And so on, after $n - 1$ rounds of "bubbling", **the first $n - 1$ larger elements are swapped to the correct position**. +4. The only remaining element must be the smallest element and does not need to be sorted, so the array is sorted. + +![bubble sort process](bubble_sort.assets/bubble_sort_overview.png) + +```src +[file]{bubble_sort}-[class]{}-[func]{bubble_sort} +``` + +## Efficiency Optimization + +We have found that if no swap operation is performed during a round of bubbling, the array has been sorted and the result can be returned directly. Therefore, you can add a flag `flag` to monitor this situation and return the result as soon as it occurs. + +After optimization, the worst and average time complexity of bubble sort is still $O(n^2)$; however, the best time complexity $O(n)$ can be achieved when the input array is completely ordered. + +```src +[file]{bubble_sort}-[class]{}-[func]{bubble_sort_with_flag} +``` + +## Algorithm Properties + +- **Time complexity is $O(n^2)$, Adaptive sorting**: each round of "bubbling" traverses the array with lengths $n - 1$, $n - 2$, $\dots$, $2$, $1$, and the total is $(n - 1) n / 2$. 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The time complexity of these sorting algorithms cannot exceed $O(n \log n)$. Next, we will look at some non-comparison sorting algorithms, which can have time complexity up to linear order. + +"Bucket sort" is a typical application of divide and conquer strategy. It distributes data evenly among buckets by setting up some buckets with size order, each bucket corresponds to a data range; then, it performs sorting inside each bucket; finally, it merges all the data according to the order of the buckets. + +## Algorithmic Flow + +Consider an array of length $n$ with elements of floating point numbers in the range $[0, 1)$. The flow of bucket sort is shown in the figure below. + +1. Initialize $k$ buckets and assign $n$ elements to $k$ buckets. +2. Perform a separate sort for each bucket (in this article, we use the programming language's built-in sort function). +3. Merge the results in order of the buckets from smallest to largest. + +![Bucket sorting algorithm flow](bucket_sort.assets/bucket_sort_overview.png) + +```src +[file]{bucket_sort}-[class]{}-[func]{bucket_sort} +``` + +## Algorithm Properties + +Bucket sort is suitable for processing data with large volume. For example, the input data contains 1 million elements, and due to space constraints, the system memory cannot load all the data at once. At this point, you can divide the data into 1000 buckets, then sort each bucket separately, and finally merge the results. + +- **Time complexity $O(n + k)$** : Assuming that the elements are evenly distributed across the buckets, the number of elements in each bucket is $\frac{n}{k}$ . Assuming that sorting a single bucket uses $O(\frac{n}{k} \log\frac{n}{k})$ time, then sorting all buckets uses $O(n \log\frac{n}{k})$ time. **When the number of buckets $k$ is large, the time complexity tends to $O(n)$** . Merging the results requires traversing all buckets and elements, taking $O(n + k)$ time. +- **Adaptive Sorting**: in the worst case, all data is assigned to a bucket and sorting that bucket uses $O(n^2)$ time. +- **Space complexity $O(n + k)$, non-in-place sorting**: requires additional space with the help of $k$ buckets and a total of $n$ elements. +- The stability of a bucket sort depends on the stability of the sorting algorithm that sorts the elements in the bucket. + +## How To Achieve Equal Distribution + +The time complexity of bucket sort can theoretically reach $O(n)$ , **the key is to evenly distribute the elements into each bucket**, because the actual data is often not evenly distributed. For example, we want to evenly distribute all the products on Taobao into 10 buckets by price range, but the product prices are unevenly distributed, with very many below $100 and very few above $1000. If the price range is divided into 10 equal parts, the difference in the number of items in each bucket will be very large. + +To achieve an even distribution, we can start by setting an approximate dividing line and roughly dividing the data into 3 buckets. **Afterallocation, the buckets with more commodities are then continued to be divided into 3 buckets until the number of elements in all buckets is roughly equal**. + +As shown in the figure below, this method essentially creates a recursion tree, and the goal is to make the values of the leaf nodes as average as possible. Of course, it is not necessary to divide the data into 3 buckets per round, the specific division can be flexibly selected according to the data characteristics. + +![Recursion dividing bucket](bucket_sort.assets/scatter_in_buckets_recursively.png) + +If we know the probability distribution of the commodity price in advance, **then we can set the price cutoff for each bucket based on the data probability distribution**. It is worth noting that the data distribution does not necessarily need to be intentionally statistical, but can also be approximated using some probability model based on the data characteristics. + +As shown in the figure below, we assume that the price of the commodity follows a normal distribution so that the price interval can be set reasonably well so that the commodity can be distributed evenly across the buckets. + +![Sort buckets according to probability distribution](bucket_sort.assets/scatter_in_buckets_distribution.png) diff --git a/docs-en/chapter_sorting/counting_sort.assets/counting_sort_overview.png b/docs-en/chapter_sorting/counting_sort.assets/counting_sort_overview.png new file mode 100644 index 0000000000000000000000000000000000000000..6afdf0ea1667264117a544e16e9342124d439216 GIT binary patch literal 32476 zcmc$^bx<5%@Gm;Mi@Uo!L4yUi;4UFZAS`adJ=o$NG`J-pKoWuzBq1#B?!LIY`y=1` zt6R6~z5m{;dUJMac4m6|boc2`pYAggt)-!ejX{9{006eKlAI0zAV2*D52Hbz4p(2e 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steps](counting_sort.assets/counting_sort_step1.png) + +=== "<2>" + ![counting_sort_step2](counting_sort.assets/counting_sort_step2.png) + +=== "<3>" + ![counting_sort_step3](counting_sort.assets/counting_sort_step3.png) + +=== "<4>" + ![counting_sort_step4](counting_sort.assets/counting_sort_step4.png) + +=== "<5>" + ![counting_sort_step5](counting_sort.assets/counting_sort_step5.png) + +=== "<6>" + ![counting_sort_step6](counting_sort.assets/counting_sort_step6.png) + +=== "<7>" + ![counting_sort_step7](counting_sort.assets/counting_sort_step7.png) + +=== "<8>" + ![counting_sort_step8](counting_sort.assets/counting_sort_step8.png) + +The implementation code for counting sort is shown below. + +```src +[file]{counting_sort}-[class]{}-[func]{counting_sort} +``` + +## Algorithm Properties + +- **Time complexity $O(n + m)$** : Involves traversing `nums` and traversing `counter` , both using linear time. In general $n \gg m$ , the time complexity tends to $O(n)$ . +- **Space complexity $O(n + m)$, Non-in-place sorting**: with the help of arrays `res` and `counter` of lengths $n$ and $m$ respectively. +- **Stable Sort**: Since the order of filling `res` with elements is "right-to-left", traversing `nums` in reverse order avoids changing the relative positions of the equal elements, and thus achieves a stable sort. In fact, an orthogonal traversal of `nums` will also give the correct sorting result, but the result will be non-stable. + +## Limitations + +You might think that counting sorts are very clever, and that you can achieve an efficient sorting job just by counting quantities. However, the preconditions for using counting sort are relatively strict. + +**The counting sort only works with non-negative integers**. If you want to use it for other types of data, you need to make sure that they can be converted to non-negative integers and that the relative size relationships between the elements cannot be changed during the conversion. For example, for an array of integers containing negative numbers, you can just add a constant to all the numbers, convert all the numbers to positive numbers, and then convert them back when the sorting is complete. + +**Counting sort is suitable for cases where the data volume is large but the data range is small**. For example, in the above example $m$ cannot be too large or it will take up too much space. 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"<3>" + ![heap_sort_step3](heap_sort.assets/heap_sort_step3.png) + +=== "<4>" + ![heap_sort_step4](heap_sort.assets/heap_sort_step4.png) + +=== "<5>" + ![heap_sort_step5](heap_sort.assets/heap_sort_step5.png) + +=== "<6>" + ![heap_sort_step6](heap_sort.assets/heap_sort_step6.png) + +=== "<7>" + ![heap_sort_step7](heap_sort.assets/heap_sort_step7.png) + +=== "<8>" + ![heap_sort_step8](heap_sort.assets/heap_sort_step8.png) + +=== "<9>" + ![heap_sort_step9](heap_sort.assets/heap_sort_step9.png) + +=== "<10>" + ![heap_sort_step10](heap_sort.assets/heap_sort_step10.png) + +=== "<11>" + ![heap_sort_step11](heap_sort.assets/heap_sort_step11.png) + +=== "<12>" + ![heap_sort_step12](heap_sort.assets/heap_sort_step12.png) + +In the code implementation, we use the same top-to-bottom heap `sift_down()` function as in the heap chapter. It is worth noting that since the length of the heap decreases as the largest element is extracted, we need to add a length parameter $n$ to the `sift_down()` function to specify the current effective length of the heap. + +```src +[file]{heap_sort}-[class]{}-[func]{heap_sort} +``` + +## Algorithm Properties + +- **Time complexity $O(n \log n)$, non-adaptive sorting**: the heap building operation uses $O(n)$ time. The time complexity of extracting the largest element from the heap is $O(\log n)$ , with a total of $n - 1$ rounds. +- **Space complexity $O(1)$, in-place sorting**: several pointer variables use $O(1)$ space. Element swapping and heapify operations are performed on the original array. +- **Unstable ordering**: the relative positions of equal elements may change when exchanging the top element of the heap and the bottom element of the heap. diff --git a/docs-en/chapter_sorting/index.md b/docs-en/chapter_sorting/index.md new file mode 100644 index 0000000000..8a28b1c49b --- /dev/null +++ b/docs-en/chapter_sorting/index.md @@ -0,0 +1,13 @@ +# Sort + +

    + +!!! abstract + + Sorting is like a magic key that turns chaos into order, allowing us to understand and process data in a more efficient way. + + Whether it's a simple ascending order or a complex categorical arrangement, sorting shows us the beauty of data harmony. diff --git a/docs-en/chapter_sorting/insertion_sort.assets/insertion_operation.png b/docs-en/chapter_sorting/insertion_sort.assets/insertion_operation.png new file mode 100644 index 0000000000000000000000000000000000000000..1bcf4602253ed3a17ee4b4bced280999e7bc4745 GIT binary patch literal 28959 zcmc$_by$>5^glfJE-c+hBP9ru3P=hoQU)a{Atfa(C`iakhmPH4h~96N={EtS65fFva&KVGRDTn78e&= zT3V{As%GbAXJ%#w2M51?{TdPyQdCs*?c2A`&d&V&{I<5Xj*gC=o}Q$nq(6WDG&D5C z#Kc5JMg98q3q6gltE=1J-*1|3N=r-gs{1Sl-ZlHbG(z*lN!?+U0zh>qod=m@n3m) zc@ygsy1KgB-(JPH$9?(orLwY8MMb4(uuw-wM>Af-t43n0Y-ZF(~ny!-o6L||cI;pp^O+i+V%ZN$4A`>{Wx4h{|{7iR?n`KTgP*3@X%d{^B>ZE{nx zpPyfRe7sY>Q}bYRXjMo+qrbPeciN9MRGEi+cT3*v#Ngy$R7>Q*=AXsw#p0nNjg&Yo 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+1,44 @@ +# Insertion Sort + +The "insertion sort insertion sort" is a simple sorting algorithm that works very similar to the process of manually organizing a deck of cards. + +Specifically, we select a baseline element in the unsorted interval, compare the size of that element with the elements in the sorted interval to its left, one by one, and insert that element into the correct position. + +The figure below shows the process of inserting elements into an array. Let the base element be `base`, we need to move all the elements from the target index to `base` one place to the right, and then assign `base` to the target index. + +![Single insertion operation](insertion_sort.assets/insertion_operation.png) + +## Algorithmic Flow + +The overall flow of the insertion sort is shown in the figure below. + +1. In the initial state, the 1st element of the array is already sorted. +2. After selecting the 2nd element of the array as `base` , and inserting it in the correct position, **the first 2 elements of the array are sorted**. +3. After selecting the 3rd element as `base` , and inserting it in the correct position, **the first 3 elements of the array are sorted**. +4. And so on, in the last round, the last element is selected as `base` , and after inserting it in the correct position, **all elements are sorted**. + +![insertion sort process](insertion_sort.assets/insertion_sort_overview.png) + +```src +[file]{insertion_sort}-[class]{}-[func]{insertion_sort} +``` + +## Algorithm Properties + +- **Time complexity $O(n^2)$, Adaptive sorting**: in the worst case, each insertion operation needs to loop $n - 1$, $n-2$, $\dots$, $2$, $1$ respectively, and the summation yields $(n - 1) n / 2$ , so the time complexity is $O(n^2)$ . The insertion operation terminates early when encountering ordered data. When the input array is completely ordered, insertion sort reaches the optimal time complexity $O(n)$ . +- **Space complexity $O(1)$, In-place sorting**: pointers $i$ and $j$ use extra space of constant size. +- **Stable sort**: during insertion operation, we will insert the elements to the right of the equal elements without changing their order. + +## Insertion Sort Advantages + +The time complexity of insertion sort is $O(n^2)$ and the time complexity of the quick sort we are about to learn is $O(n \log n)$ . Even though the time complexity of insertion sort is higher compared to quick sort, **insertion sort is usually faster when the amount of data is small**. + +This conclusion is similar to the conclusion about the applicability of linear and binary search. Algorithms such as quick sorting algorithm $O(n \log n)$ are partition-based sorting algorithms, which tend to contain more unitary computational operations. While the values of $n^2$ and $n \log n$ are closer and complexity does not play a dominant role when the amount of data is small; the number of unit operations in each round plays a decisive factor. + +In fact, the built-in sorting functions of many programming languages (e.g., Java) use insertion sort, along the lines of a partition-based sorting algorithm, such as quick sort, for long arrays, or direct insertion sort for short arrays. + +Although the time complexity of bubble sort, selection sort and insertion sort is $O(n^2)$ , in practice, **insertion sort is used significantly more often than bubble sort and selection sort for the following main reasons**. + +- The bubble sort is implemented based on element swapping and involves 3 unit operations with the help of a temporary variable; the insertion sort is implemented based on element assignment and requires only 1 unit operation. 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/dev/null +++ b/docs-en/chapter_sorting/quick_sort.md @@ -0,0 +1,98 @@ +# Quick Sort + +"Quick sort" is a sorting algorithm based on the divide and conquer strategy, which is efficient and widely used. + +The core operation of quick sort is the "sentinel division", the goal is: selection of an element in the array as the "base number", all the elements smaller than the base number of elements moved to its left, and greater than the base number of elements moved to its right. Specifically, the process of sentinel division is shown in the figure below. + +1. Select the leftmost element of the array as the reference number, and initialize two pointers `i` and `j` to point to the two ends of the array respectively. +2. Set up a loop that uses `i` (`j`) in each round to find the first element that is larger (smaller) than the baseline number, respectively, and then swaps the two elements. +3. Loop through step `2.` until `i` and `j` meet and stop, and finally swap the datums to the dividing line of the two subarrays. + +=== "<1>" + ![Sentinel division step](quick_sort.assets/pivot_division_step1.png) + +=== "<2>" + ![pivot_division_step2](quick_sort.assets/pivot_division_step2.png) + +=== "<3>" + ![pivot_division_step3](quick_sort.assets/pivot_division_step3.png) + +=== "<4>" + ![pivot_division_step4](quick_sort.assets/pivot_division_step4.png) + +=== "<5>" + ![pivot_division_step5](quick_sort.assets/pivot_division_step5.png) + +=== "<6>" + ![pivot_division_step6](quick_sort.assets/pivot_division_step6.png) + +=== "<7>" + ![pivot_division_step7](quick_sort.assets/pivot_division_step7.png) + +=== "<8>" + ![pivot_division_step8](quick_sort.assets/pivot_division_step8.png) + +=== "<9>" + ![pivot_division_step9](quick_sort.assets/pivot_division_step9.png) + +After the sentinel division is done, the original array is divided into three parts: the left subarray, the base number, and the right subarray, and it satisfies "any element of the left subarray $\leq$ base number $\leq$ any element of the right subarray". Therefore, we only need to sort these two arrays next. + +!!! note "A divide and conquer strategy for quick sort" + + The essence of sentinel division is to simplify the problem of sorting a longer array into a problem of sorting two shorter arrays. + +```src +[file]{quick_sort}-[class]{quick_sort}-[func]{partition} +``` + +## Algorithmic Flow + +The overall process of quick sort is shown in the figure below. + +1. First, perform a "sentinel division" on the original array to get the unsorted left and right subarrays. +2. Then, the "sentinel division" is performed recursionally on the left and right subarrays. +3. The recursion continues until the length of the subarray reaches 1, thus completing the sorting of the entire array. + +![Quick sort process](quick_sort.assets/quick_sort_overview.png) + +```src +[file]{quick_sort}-[class]{quick_sort}-[func]{quick_sort} +``` + +## Algorithm Properties + +- **Time complexity $O(n \log n)$, Adaptive Sorting**: In the average case, the number of recursion levels for sentinel division is $\log n$ , the total number of loops in each level is $n$, and the overall time complexity used is $O(n \log n)$. In the worst case, each round of sentinel division operation divides the array of length $n$ into two subarrays of length $0$ and $n - 1$, at which point the number of recursion levels reaches $n$, the number of loops in each level is $n$, and the overall time used is $O(n^2)$. +- **Space complexity $O(n)$, in-place sorting**: reaches the worst recursion depth $n$ with the input array completely inverted, using $O(n)$ stack frame space. The sorting operation is performed on the original array without the help of additional arrays. +- **Unstable ordering**: in the last step of the sentinel division, the base numbers may be swapped to the right of equal elements. + +## Why Is Fast Platoon Fast? + +As you can see from the name, quick sort should have a certain advantage in terms of efficiency. Although the average time complexity of quick sort is the same as that of merge sort and heap sort, quick sort is usually more efficient for the following reasons. + +- **Low probability of worst-case**: although the worst-case time complexity of quick sort is $O(n^2)$, which is not as stable as merge sort, in the vast majority of cases, quick sort can run at a time complexity of $O(n \log n)$. +- **Efficient use of cache**: When performing a sentinel division operation, the system can load the entire subarray into the cache, thus accessing the elements more efficiently. Algorithms such as heap sort lack this feature as they require jumping to access elements. +- Low constant factor of **complexity**: Among the above three algorithms, quick sorting has the lowest total number of comparison, assignment, and swap operations. This is similar to the reason why Insertion Sort is faster than Bubble Sort. + +## Benchmark Optimization + +**Quick sort may be less time efficient with certain inputs**. As an extreme example, suppose the input array is completely inverted, and since we choose the leftmost element as the base number, then after the completion of the sentinel division, the base number is exchanged to the rightmost end of the array, resulting in a left subarray length of $n - 1$ and a right subarray length of $0$ . As this recursion continues, the length of the right subarray after each round of sentinel division is $0$, the divide and conquer strategy fails, and the quick sort degenerates into a "bubble sort". + +To try to avoid this situation, **we can optimize the selection strategy of the benchmark number in sentinel division**. For example, we can randomly pick an element as the benchmark number. However, if we are unlucky and pick an undesirable benchmark number every time, the efficiency is still unsatisfactory. + +Note that programming languages often generate "pseudo-random numbers". If we construct a specific test case for a sequence of pseudo-random numbers, the efficiency of the quick sort may still be degraded. + +For further improvement, we can select three candidate elements in the array (usually the first, last, and midpoint elements of the array), **and use the median of these three candidate elements as the base number**. In this way, the probability that the base number is "neither too small nor too large" will be greatly increased. Of course, we can also select more candidate elements to further improve the robustness of the algorithm. With this approach, the probability of time complexity degradation to $O(n^2)$ is greatly reduced. + +```src +[file]{quick_sort}-[class]{quick_sort_median}-[func]{partition} +``` + +## Tail Recursion Optimization + +**Under certain inputs, the quick sort may take up more space**. Taking a completely inverted input array as an example, since the length of the right child array after each round of sentinel division is $0$, the height of the recursion tree will reach $n - 1$, which will take up $O(n)$ size of stack frame space. + +To prevent the accumulation of stack frame space, we can compare the lengths of the two subarrays after each round of sentinel sorting is completed, **recursion only on the shorter subarray**. 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suitable for cases where the amount of data $n$ is large but the data range $m$ is small. Suppose we need to sort $n = 10^6$ school numbers, which is a $8$ bit number, which means the data range $m = 10^8$ is very large, and using counting sort would require allocating a lot of memory space, which can be avoided by radix sort. + +The core idea of "radix sort" is the same as counting sort, which is also realized by counting the number of digits. Based on this, radix sort utilizes the progressive relationship between each digit and sorts each digit in turn to get the final sort result. + +## Algorithmic Flow + +Taking the school number data as an example, assuming that the lowest digit of the number is the $1$th digit and the highest digit is the $8$th digit, the flow of radix sort is shown in the figure below. + +1. Initialization bits $k = 1$ . +2. Performs "counting sort" on the $k$ digit of the student number. When finished, the data is sorted from smallest to largest by the $k$ digit. +3. Increase $k$ by $1$ and return to step `2.` Continue iteration until it ends when all bits are sorted. + +![Radix sort algorithm flow](radix_sort.assets/radix_sort_overview.png) + +Let's dissect the code implementation. For a $d$ progressive number $x$, to get its first $k$ bit $x_k$, you can use the following formula: + +$$ +x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d +$$ + +where $\lfloor a \rfloor$ denotes rounding down the floating point number $a$ and $\bmod \: d$ denotes rounding up $d$. For school number data, $d = 10$ and $k \in [1, 8]$ . + +In addition, we need to make a small change to the counting sort code so that it can sort the numbers according to the $k$th bit of the number. + +```src +[file]{radix_sort}-[class]{}-[func]{radix_sort} +``` + +!!! Question "Why do you start sorting from the lowest position?" + + In consecutive sorting rounds, the later sorting round overwrites the result of the previous sorting round. For example, if the result of the first sorting round $a < b$ and the result of the second sorting round $a > b$, then the result of the second round will replace the result of the first round. Since the higher digits of a number have higher priority than the lower digits, we should sort the lower digits first before sorting the higher digits. + +## Algorithm Properties + +Compared to counting sort, radix sort is suitable for larger ranges of values, **provided that the data can be represented in a fixed bit format and the number of bits is not too large**. For example, floating point numbers are not suitable for radix sort because the number of bits $k$ is too large, which may result in a time complexity $O(nk) \gg O(n^2)$ . + +- **Time complexity $O(nk)$**: Given a data size of $n$, a data $d$ progression, and a maximum number of bits $k$, it takes $O(n + d)$ time to perform a counting sort on a particular bit, and $O((n + d)k)$ time to sort all the $k$ bits. Typically, both $d$ and $k$ are relatively small and the time complexity tends to be $O(n)$. +- **Space complexity $O(n + d)$, Non-in-place sort**: like counting sort, radix sort requires the help of arrays `res` and `counter` of length $n$ and $d$. +- **Stable sort**: same as counting sort. diff --git a/docs-en/chapter_sorting/selection_sort.assets/selection_sort_instability.png b/docs-en/chapter_sorting/selection_sort.assets/selection_sort_instability.png new file mode 100644 index 0000000000000000000000000000000000000000..32560b438106f37887f3a5de9f5adab48b6a80ec GIT binary patch literal 18862 zcmce-cUV+Uw=TGMcLPn%IW>rY5(Gp*Vv7V(P_iIdKysEOX_IqCKu`oxf|4c2CWAyJ z2cZc9A|N>n)Be8mJ!fXlotbm*bD#NxX78$1RjXF5wW{iUyCc;fDw7g15di=|s&Y?3 z69DjVkFa4R9QW@ySHJ@IGOn(wt$2BPxxBm_R3EsqvT|{8aejV|{_x8&bUrdNa$#X% zcX#*T;NbN1^yuj5_wV29>+4cdQY|el9UUD#Jw0dVXP8;c_V)I>{+yGOlhV@C&CN|L z7JL8x{npmjiHV6%pFYXU%P%c0DJv@Q;8XE84zt76bGBYz97#Qg9@2{(?GchsI)zximY%DJ?fBpKkv$J!2 ze7v5X-u(Rh*w|P|NJvdh&Cj1d*VfivzI+)S9c^G>ke!`9G&B?*9&T=K-rnB+=+UFH zvNCCDX?J&bBO{}|y*)=qM=L9scbC%<2vTAzylviEcE=TUL|(b;kS;Fr^HZMPF6HnnIr9}rp9{BD_TDJdyQZciFo zJN>h}`DbNrcHwBpUH-?;Z>_iKcT(eGF)`crVv(awG%jQwQ$-xmBArzek!HeAXItv|hg z)BEP|eel6+@9rNbJzw``e$AwJwFcK$R*aSZHsZWE+V5M$cF%t|FUZ-_WBIjzyzqM= zs=03a=+Be#tzEx|wcn3MrVpB?Cu$}}OGk#XdOIsdzcx%YoOG1#AD=C(9B&?ME^jZ_ zPmLwFH^*Zdrnk;MO&t2P?qm$_xAbmfW?Qu5A{>;);W*@T$VFgRbuMYdTOyJdC3LfUhY1Q*OD}&$q{#rtgB!Qzqxwa7R+yZy&*dvHY_*5{u7>%eKRlzYIm3`CcN*5YC8NTH<7i&cfbrJ zQM}8m4AjKKu0bD$y&MZc}Sl3J*J8celcFoX46X{6<18`5? z`zxaQXkz5_Zkv_DSt>X@q2@UlkSDrNj=q0C$0quw>BOk962_BR47;f34m@`OL@UYB zLHDVZ(fL=fFKEmNWUn{D<|HR}H#cp8Y>ylbRuV4g>11aPH6~3igJ#=K?k0T@Y zR(mt}GXRlLCc*{_15R@%+Yu68%yEGC9~jEv12P;MCpUu?Jp~SEML|SDr36?l4C#=V zmZo746zKI5+Syv;xkan6wVd0liZ5t)qWvF`ae*BLKdC zq>LubHYp)ghp!<~(7p}u{ZBWMDgZ7o5Sd=eK&*U15J?Da?G*?RQ2;=TXaLqjhZ|G? zhC~_QMytj(vZq7?Q=AY`q4@tT5NNV7Gh{by%F?|PUlcfGKdVRjj_jo-u?>0Z%X+F0 z>inOEqwMXAy%?mB0>325o2q`Xf0groM5VP9AGak$*R$%r6V3E!m)ko@S@Qu%lQ^mg zC~{TNpea=Z@YEIlHy5P@e9-(<%;9+BMefCgA4l5IZ__ZXKfWhaL(ZuaZ?$O9Dq`{W zsIBy(*2kQCJ{IIR1@T)~6JvF4G-%5n_6QcS-~33S0bIS4#f*9E#Pfnnoab|}r{Po5 zvM|x`nb$j{oJ84NnEM=~9No!ZPqct}=g78XP~mWvgb`02*EuSjSOoZDV5HSi?3z1o z#ehCM`x7jOFT_4Fz|6?->%Bp-s?l}m9dKWspWHMs$0IxjW*rj*Y>BPL1_4ic{NxMT z2CGBWCwxNF!(hC)u{w5v)LbB|-jN19;og1|Lq(Q$a?z(PEO+ibM78Nh0v1Y~X%HF~U`u zgyn}W?6?E^6}#dTG}>b~IQ2aOc6^=aL8RQlhaaQuIQ_@^8qoYaJTll#IJBGiP9_f47 zlCUGE$bcbWie&-v&A4x{q30Ot5p+1ib)MWYpFdP>C}N7e<->$-yx8p|hEIx3lwGk; zg7eW~=%y;ULH>*PKJk!D-s4fGzpHIAKBzX?;>;7V^i|@rYIpfz0yVXC2Q1<@`H${y z)7Rpbs`f0B<;Bj*i3|7sS)F(C^Hr=4p`S~C_T|>TsNjbV1-*lY-S z?+cC0Qv^`qh=jZqCUeH1+c@dU6}B5b89}!>q5aPv$B#zl!~)M%fRmW2LmllgHo3#s zXOg3}x4`~NTiwl1dM(%8-iZD-dbT@=5K5ABpd$uVbslEcz6%G!Txaz_N{sg0%H9|8 zTpqB0sc1)^%(~_(X^%uvJo-OP0#=To zh`9PcoZo*|sqXnoJHFnpl_Ov!G)+H?t(he-+4$aP%VfsTwG&RBISaKLcFfK~aZ5_! 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step](selection_sort.assets/selection_sort_step1.png) + +=== "<2>" + ![selection_sort_step2](selection_sort.assets/selection_sort_step2.png) + +=== "<3>" + ![selection_sort_step3](selection_sort.assets/selection_sort_step3.png) + +=== "<4>" + ![selection_sort_step4](selection_sort.assets/selection_sort_step4.png) + +=== "<5>" + ![selection_sort_step5](selection_sort.assets/selection_sort_step5.png) + +=== "<6>" + ![selection_sort_step6](selection_sort.assets/selection_sort_step6.png) + +=== "<7>" + ![selection_sort_step7](selection_sort.assets/selection_sort_step7.png) + +=== "<8>" + ![selection_sort_step8](selection_sort.assets/selection_sort_step8.png) + +=== "<9>" + ![selection_sort_step9](selection_sort.assets/selection_sort_step9.png) + +=== "<10>" + ![selection_sort_step10](selection_sort.assets/selection_sort_step10.png) + +=== "<11>" + ![selection_sort_step11](selection_sort.assets/selection_sort_step11.png) + +In the code, we use $k$ to record the smallest element in the unsorted interval. + +```src +[file]{selection_sort}-[class]{}-[func]{selection_sort} +``` + +## Algorithm Properties + +- **Time complexity is $O(n^2)$, non-adaptive sorting**: the outer loop consists of a total of $n - 1$ rounds, the length of the unsorted interval in the first round is $n$ , and the length of the unsorted interval in the last round is $2$ , i.e., each round of the outer loop contains $n$, $n - 1$, $\dots$, $3$, $2$ rounds of inner loops, and the summation is $\frac{(n - 1)(n + 2)}{2}$ . +- **Space complexity $O(1)$, In-place sorting**: pointers $i$ and $j$ use extra space of constant size. +- **Unstable ordering**: as shown in the figure below, it is possible for an element `nums[i]` to be swapped to the right of an element equal to it, resulting in a change in the relative order of the two. + +![Example of selection sort non-instability](selection_sort.assets/selection_sort_instability.png) diff --git 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Non-comparative sorting does not use comparison operators, and the time complexity can be up to $O(n)$, but its generality is relatively poor. + +## Ideal Sorting Algorithm + +**runs fast, in situ, stable, positively adaptive, and generalizes well**. Obviously, no sorting algorithm that combines all of these properties has been found so far. Therefore, when selecting a sorting algorithm, the decision needs to be based on the specific data characteristics and problem requirements. + +Next, we will learn various sorting algorithms together and analyze the advantages and disadvantages of each sorting algorithm based on the evaluation dimensions mentioned above. diff --git a/docs-en/chapter_sorting/summary.assets/sorting_algorithms_comparison.png b/docs-en/chapter_sorting/summary.assets/sorting_algorithms_comparison.png new file mode 100644 index 0000000000000000000000000000000000000000..1fdb02ee521bdd5ea2fa5fa4ae9978f293f0ab42 GIT binary patch literal 60946 zcmd42^gdqf^yCo$Qq$LFP;`2P8 z@4feR|AG5jKg{fP_Bw07SDwAkOstlMA{IJ1IsgDz%1UxN004db3Z6zqdi=O8d^y4z}s@}AkKeO5SzLT1oIx;e{zJ2NG6J+n` zy19F;tEcKmK3wzfLCI2BhF zTRgMqKy>&AhBY=eT3T5KguSY(tIN*L&MnH_JKI}cUS9fko1C2N>FGJQe-#)S=;P;; zQ(oun;xYGm4hDmjl$6xh*U!(-XBT8!*;oyZELBug--d^q=?t6QC{bT)a-n@y4OOH#AODiZ}+FpuDjA?FewzIdZsq4tet8{tp 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Assuming that we "search from left to right" first, then if we can't find an element smaller than the base number, **we will jump out of the loop at `i == j`, and then `nums[j] == nums[i] > nums[left]`**. That is, at this point, the last step of the swap operation will swap an element larger than the baseline number to the leftmost end of the array, causing the sentinel division to fail. + + For example, given an array `[0, 0, 0, 0, 0, 1]`, if you first "find from left to right", the sentinel divides the array into `[1, 0, 0, 0, 0]`, which is incorrect. + + Thinking deeper, if we choose `nums[right]` as the base number, then it's the other way around: we have to "find from left to right" first. + +!!! question "Regarding tail recursion optimization, why is it that choosing a shorter array ensures that the recursion depth does not exceed $\log n$ ?" + + The recursion depth is the number of recursion methods that are currently not returned. Each round of sentinel division we divide the original array into two subarrays. After tail recursion optimization, the maximum length of the downward recursion subarray is half the length of the original array. Assuming the worst case, which is always half the length, the final depth of recursion is $\log n$ . + + Recalling the original quick sort, it is possible to recursion successively through arrays of large lengths, in the worst case $n$, $n - 1$, $\dots$, $2$, $1$, with a recursion depth of $n$. The tail recursion optimization avoids this. + +!!! question "Is the time complexity of quick sort $O(n^2)$ when all elements in the array are equal? How should I handle this degenerate case?" + + Yes. This situation can be considered by dividing the array into three parts by sentinel division: less than, equal to, and greater than the base number. Only the less than and greater than parts are recursed downwards. In this method, an array with all equal elements is input and only one round of sentinel division completes the sorting. + +!!! question "Why is the worst time complexity of bucket sort $O(n^2)$ ?" + + In the worst case, all elements are sorted into the same bucket. 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zM@L8F;^K00avB;MIyyR%lat@PdGqz_*Xru(;^N|rjEuInw*20_`1tt3!ot+l)UB + +| method name | description | time complexity | +| ----------- | ------------------------------------- | --------------- | +| pushFirst() | Add element to top of queue | $O(1)$ | +| pushLast() | Add element to end of queue | $O(1)$ | +| popFirst() | Delete the first element of the queue | $O(1)$ | +| popLast() | Delete end-of-queue element | $O(1)$ | +| peekFirst() | access the first element of the queue | $O(1)$ | +| peekLast() | access the end-of-queue element | $O(1)$ | + +Similarly, we can directly use the double-ended queue class already implemented in the programming language. + +=== "Python" + + ```python title="deque.py" + # Initialize the double-ended queue + deque: deque[int] = collections.deque() + + # Element in the queue + deque.append(2) # add to the end of the queue + deque.append(5) + deque.append(4) + deque.appendleft(3) # add to head of queue + deque.appendleft(1) + + # Access to elements + front: int = deque[0] # head element of the queue + rear: int = deque[-1] # end of queue element + + # Element out of line + pop_front: int = deque.popleft() # head element out of queue + pop_rear: int = deque.pop() # end-of-queue element out of queue + + # Get the length of the double-ended queue + size: int = len(deque) + + # Determine if the double-ended queue is empty + is_empty: bool = len(deque) == 0 + ``` + +=== "C++" + + ```cpp title="deque.cpp" + /* Initialize double-ended queue */ + deque deque; + + /* Element in queue */ + deque.push_back(2); // add to the end of the queue + deque.push_back(5); + deque.push_back(4); + deque.push_front(3); // add to the top of the queue + deque.push_front(1); + + /* Access to elements */ + int front = deque.front(); // head element of the queue + int back = deque.back(); // end of queue element + + /* Element out of line */ + deque.pop_front(); // the top element of the queue is out of the queue + deque.pop_back(); // the last element of the queue is out of the queue + + /* Get the length of the double-ended queue */ + int size = deque.size(); + + /* Determine if the double-ended queue is empty */ + bool empty = deque.empty(); + ``` + +=== "Java" + + ```java title="deque.java" + /* Initialize double-ended queue */ + Deque deque = new LinkedList<>(); + + /* Element in queue */ + deque.offerLast(2); // add to the end of the queue + deque.offerLast(5); + deque.offerLast(4); + deque.offerFirst(3); // add to the top of the queue + deque.offerFirst(1); + + /* Access to elements */ + int peekFirst = deque.peekFirst(); // first element of the queue + int peekLast = deque.peekLast(); // the last element of the queue + + /* Element out of line */ + int popFirst = deque.pollFirst(); // first element out of queue + int popLast = deque.pollLast(); // end of queue out of queue + + /* Get the length of the double-ended queue */ + int size = deque.size(); + + /* Determine if the double-ended queue is empty */ + boolean isEmpty = deque.isEmpty(); + ``` + +=== "C#" + + ```csharp title="deque.cs" + /* Initialize double-ended queue */ + // In C#, think of a linked list, a LinkedList, as a double-ended queue. + LinkedList deque = new(); + + /* Element in queue */ + deque.AddLast(2); // add to the end of the queue + deque.AddLast(5); + deque.AddLast(4); + deque.AddFirst(3); // add to the top of the queue + deque.AddFirst(1); + + /* Access to elements */ + int peekFirst = deque.First.Value; // first element of the queue + int peekLast = deque.Last.Value; // the last element of the queue + + /* Element out of line */ + deque.RemoveFirst(); // the first element of the queue is out of the queue + deque.RemoveLast(); // the end of the queue is out of the queue + + /* Get the length of the double-ended queue */ + int size = deque.Count; + + /* Determine if the double-ended queue is empty */ + bool isEmpty = deque.Count == 0; + ``` + +=== "Go" + + ```go title="deque_test.go" + /* Initialize double-ended queue */ + // Use lists as double-ended queues in Go. + deque := list.New() + + /* Element in queue */ + deque.PushBack(2) // add to the end of the queue + deque.PushBack(5) + deque.PushBack(4) + deque.PushFront(3) // add to the top of the queue + deque.PushFront(1) + + /* Access to elements */ + front := deque.Front() // head element of the queue + rear := deque.Back() // the end element of the queue + + /* Element out of line */ + deque.Remove(front) // head element out of queue + deque.Remove(rear) // the end of the queue is out of the queue + + /* Get the length of the double-ended queue */ + size := deque.Len() + + /* Determine if the double-ended queue is empty */ + isEmpty := deque.Len() == 0 + ``` + +=== "Swift" + + ```swift title="deque.swift" + /* Initialize double-ended queue */ + // Swift doesn't have a built-in double-ended queue class, so you can use Array as a double-ended queue. + var deque: [Int] = [] + + /* Element in queue */ + deque.append(2) // add to the end of the queue + deque.append(5) + deque.append(4) + deque.insert(3, at: 0) // add to top of queue + deque.insert(1, at: 0) + + /* Access to elements */ + let peekFirst = deque.first! // The first element of the queue. + let peekLast = deque.last! // The last element of the queue. + + /* Element out of line */ + // The complexity of popFirst is O(n) when simulating with Array. + let popFirst = deque.removeFirst() // the first element of the queue is out of the queue + let popLast = deque.removeLast() // the last element of the queue is out of the queue + + /* Get the length of the double-ended queue */ + let size = deque.count + + /* Determine if the double-ended queue is empty */ + let isEmpty = deque.isEmpty + ``` + +=== "JS" + + ```javascript title="deque.js" + /* Initialize double-ended queue */ + // JavaScript doesn't have a built-in double-ended queue, you can only use Array as a double-ended queue. + const deque = []; + + /* Element in queue */ + deque.push(2); + deque.push(5); + deque.push(4); + // Note that the unshift() method has a time complexity of O(n) because it is an array. + deque.unshift(3); + deque.unshift(1); + console.log("Double-ended queue dequeue = ", deque); + + /* Access to elements */ + const peekFirst = deque[0]; + console.log("Team head element peekFirst = " + peekFirst); + const peekLast = deque[deque.length - 1]; + console.log("Last element of queue peekLast = " + peekLast); + + /* Element out of line */ + // Note that the shift() method has a time complexity of O(n) because it's an array. + const popFront = deque.shift(); + console.log("First element out of the queue popFront = " + popFront + ", first out of the queue after deque = " + deque); + const popBack = deque.pop(); + console.log("End-of-queue out element popBack = " + popBack + ", end-of-queue out after deque = " + deque); + + /* Get the length of the double-ended queue */ + const size = deque.length; + console.log("Double-ended queue length size = " + size); + + /* Determine if the double-ended queue is empty */ + const isEmpty = size === 0; + console.log("Whether double-ended queue is empty = " + isEmpty); + ``` + +=== "TS" + + ```typescript title="deque.ts" + /* Initialize double-ended queue */ + // TypeScript doesn't have a built-in double-ended queue; you can only use Array as a double-ended queue. + const deque: number[] = []; + + /* Element in queue */ + deque.push(2); + deque.push(5); + deque.push(4); + // Note that the unshift() method has a time complexity of O(n) because it is an array. + deque.unshift(3); + deque.unshift(1); + console.log("Double-ended queue dequeue = ", deque); + + /* Access to elements */ + const peekFirst: number = deque[0]; + console.log("Team head element peekFirst = " + peekFirst); + const peekLast: number = deque[deque.length - 1]; + console.log("Last element of queue peekLast = " + peekLast); + + /* Element out of line */ + // Note that the shift() method has a time complexity of O(n) because it's an array. + const popFront: number = deque.shift() as number; + console.log("First element out of the queue popFront = " + popFront + ", first out of the queue after deque = " + deque); + const popBack: number = deque.pop() as number; + console.log("End-of-queue out element popBack = " + popBack + ", end-of-queue out after deque = " + deque); + + /* Get the length of the double-ended queue */ + const size: number = deque.length; + console.log("length of double-ended queue size = " + size); + + /* Determine if the double-ended queue is empty */ + const isEmpty: boolean = size === 0; + console.log("Whether double-ended queue is empty = " + isEmpty); + ``` + +=== "Dart" + + ```dart title="deque.dart" + /* Initialize double-ended queue */ + // In Dart, Queues are defined as double-ended queues. + Queue deque = Queue(); + + /* Element in queue */ + deque.addLast(2); // add to the end of the queue + deque.addLast(5); + deque.addLast(4); + deque.addFirst(3); // add to the top of the queue + deque.addFirst(1); + + /* Access to elements */ + int peekFirst = deque.first; // first element of the queue + int peekLast = deque.last; // the last element of the queue + + /* Element out of line */ + int popFirst = deque.removeFirst(); // first element out of queue + int popLast = deque.removeLast(); // the last element out of the queue + + /* Get the length of the double-ended queue */ + int size = deque.length; + + /* Determine if the double-ended queue is empty */ + bool isEmpty = deque.isEmpty;W + ``` + +=== "Rust" + + ```rust title="deque.rs" + /* Initialize double-ended queue */ + let mut deque: VecDeque = VecDeque::new(); + + /* Element in queue */ + deque.push_back(2); // add to the end of the queue + deque.push_back(5); + deque.push_back(4); + deque.push_front(3); // add to the top of the queue + deque.push_front(1); + + /* Access to elements */ + if let Some(front) = deque.front() { // the first element of the queue + } + if let Some(rear) = deque.back() { // end-of-queue element + } + + /* Element out of line */ + if let Some(pop_front) = deque.pop_front() { // the top element of the queue is out of the queue + } + if let Some(pop_rear) = deque.pop_back() { // end-of-queue element out of queue + } + + /* Get the length of the double-ended queue */ + let size = deque.len(); + + /* Determine if the double-ended queue is empty */ + let is_empty = deque.is_empty(); + ``` + +=== "C" + + ```c title="deque.c" + // C does not provide a built-in double-ended queue + ``` + +=== "Zig" + + ```zig title="deque.zig" + + ``` + +## Double-Ended Queue Implementation + +The implementation of a double-ended queue is similar to a queue, and you can choose either a linked list or an array as the underlying data structure. + +### Implementation Based On Bi-Directional Linked Lists + +Recall from the previous section that we used an ordinary unidirectional linked list to implement the queue because it makes it easy to delete the head node (corresponding to an out-of-queue operation) and add a new node after the tail node (corresponding to an in-queue operation). + +For a double-ended queue, both the head and the tail can perform in-queue and out-queue operations. In other words, a double-ended queue needs to realize another symmetric direction. For this reason, we use a "bidirectional linked list" as the underlying data structure of a double-ended queue. + +As shown in the figure below, we consider the head and tail nodes of a double-ended queue as the head and tail of the double-ended queue, and realize the function of adding and deleting nodes at both ends at the same time. + +=== "LinkedListDeque" + ![Implementing in-queue and out-queue operations for double-ended queues based on linked lists](deque.assets/linkedlist_deque.png) + +=== "pushLast()" + ![linkedlist_deque_push_last](deque.assets/linkedlist_deque_push_last.png) + +=== "pushFirst()" + ![linkedlist_deque_push_first](deque.assets/linkedlist_deque_push_first.png) + +=== "popLast()" + ![linkedlist_deque_pop_last](deque.assets/linkedlist_deque_pop_last.png) + +=== "popFirst()" + ![linkedlist_deque_pop_first](deque.assets/linkedlist_deque_pop_first.png) + +The implementation code is shown below. + +```src +[file]{linkedlist_deque}-[class]{linked_list_deque}-[func]{} +``` + +### An Array-Based Implementation + +As shown in the figure below, similar to implementing queues based on arrays, we can also implement double-ended queues using ring arrays. + +=== "ArrayDeque" + ![Implementing in-queue and out-queue operations for double-ended queues based on arrays](deque.assets/array_deque.png) + +=== "pushLast()" + ![array_deque_push_last](deque.assets/array_deque_push_last.png) + +=== "pushFirst()" + ![array_deque_push_first](deque.assets/array_deque_push_first.png) + +=== "popLast()" + ![array_deque_pop_last](deque.assets/array_deque_pop_last.png) + +=== "popFirst()" + ![array_deque_pop_first](deque.assets/array_deque_pop_first.png) + +On the basis of the queue implementation, only need to add the "queue head into the queue" and "queue tail out of the queue" method. + +```src +[file]{array_deque}-[class]{array_deque}-[func]{} +``` + +## Double-Ended Queue Application + +A double-ended queue combines the logic of both stacks and queues, **so it can realize all the application scenarios of both, while providing a higher degree of freedom**. + +As we know, the "undo" function of software is usually realized by using a stack: the system `push` each change operation to the stack, and then `pop` to realize the undo. However, software usually limits the number of undo steps (e.g. only $50$ steps are allowed to be saved) due to the limitation of system resources. When the length of the stack exceeds $50$, the software needs to perform a delete operation at the bottom of the stack (i.e., the head of the queue). **However, the stack is unable to fulfill this function, and then a double-ended queue needs to be used instead of the stack**. Note that the core logic of "undo" still follows the stack's first-in, first-out principle, but the double-ended queue is able to implement some additional logic more flexibly. diff --git a/docs-en/chapter_stack_and_queue/index.md b/docs-en/chapter_stack_and_queue/index.md new file mode 100644 index 0000000000..7731ec39fd --- /dev/null +++ b/docs-en/chapter_stack_and_queue/index.md @@ -0,0 +1,13 @@ +# Stack And Queue + +
    + +![stack and queue](../assets/covers/chapter_stack_and_queue.jpg){ width="600" } + +
    + +!!! abstract + + Stacks are like stacking cats and queues are like cats queuing. + + The two represent the logical relationships of first-in-last-out and first-in-first-out, respectively. diff --git a/docs-en/chapter_stack_and_queue/queue.assets/array_queue.png b/docs-en/chapter_stack_and_queue/queue.assets/array_queue.png new file mode 100644 index 0000000000000000000000000000000000000000..4e5dfd095cb216374e50c1d38979051fe3184dbe GIT binary patch literal 25642 zcmcG#byQSQ_W*kC%rJBbNT-xasif3^f`F7tNh?UVNJ(5&S`kE$lmQixE|F#c5l~XP z2LT5r9#*5jXZ_uX;!*>%p@bBWN^zClUGLIwb!R9Cxx8vr80Bcg`{ zNf`Fu30o6@ZCy=0m6MZ`+1c4A)xn#ao5z3u?(gp(9UUDW9!5k&EG{liPfzde?(XdD z{QmuWZEa0fR<@y`fg$+&_V)J5O){C-@R5-bCnqN_clTeve!YA54mXY)8yj0*UdG{Y zot>QrhX>8g%`aZO`1b8vUteEsZ7m*;ud1q2Q&ZE|*KcWQ!D6wQ9T}#krc+Z>t*xy} zN=gO!`AJDh3Gwl1X=y1bDMLd;a&mGT8ygD?3pO@34h{|j0|TX{rPMTwJ`my4u^@TU1n(o11H4VNqUQ{^7%inC9Q@?d=K*3XzeKj~_o)QBjGC zihB0!*~G*|SXh{rmR9TZmXVRs{QP`HMFkAQw{PG6{P}ZiZ0y$7*5u@5cXzj!m)Gd% 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As the name suggests, a queue simulates the phenomenon of queuing, where newcomers keep joining the tail end of the queue and those at the head of the queue leave one by one. + +As shown in the figure below, we will call the head of the queue "head", the tail is called "tail", the operation of adding elements to the tail is called "queue", the operation of deleting elements from the head of the queue is called "queue". called "out of the queue". + +![First-in-first-out rule for queues](queue.assets/queue_operations.png) + +## Queue Common Operations + +The common operations of queues are shown in the table below. Note that method names may vary from one programming language to another. We use the same method names as stack here. + +

    Table   Efficiency of Queue Operations

    + +| method name | description | time complexity | +| ----------- | ------------------------------------------------------------- | --------------- | +| push() | Element in queue, i.e. adding element to the end of the queue | $O(1)$ | +| pop() | queue head element out | $O(1)$ | +| peek() | access the first element of the queue | $O(1)$ | + +We can just use the queue class that is readily available in the programming language. + +=== "Python" + + ```python title="queue.py" + # Initialize the queue + # In Python, we generally think of the double-ended queue class, deque, as a queue. + # Although queue.Queue() is a pure queue class, it's not very usable, so it's not recommended. + que: deque[int] = collections.deque() + + # Elements in the queue + que.append(1) + que.append(3) + que.append(2) + que.append(5) + que.append(4) + + # Access to the first element of the queue + front: int = que[0]; + + # Element out of line + pop: int = que.popleft() + + # Get the length of the queue + size: int = len(que) + + # Determine if the queue is empty + is_empty: bool = len(que) == 0 + ``` + +=== "C++" + + ```cpp title="queue.cpp" + /* Initialize Queue */ + queue queue; + + /* Element in queue */ + queue.push(1); + queue.push(3); + queue.push(2); + queue.push(5); + queue.push(4); + + /* Access to the first element of the queue */ + int front = queue.front(); + + /* Element out of line */ + queue.pop(); + + /* Get the length of the queue */ + int size = queue.size(); + + /* Determine if the queue is empty */ + bool empty = queue.empty(); + ``` + +=== "Java" + + ```java title="queue.java" + /* Initialize Queue */ + Queue queue = new LinkedList<>(); + + /* Element in queue */ + queue.offer(1); + queue.offer(3); + queue.offer(2); + queue.offer(5); + queue.offer(4); + + /* Access to the first element of the queue */ + int peek = queue.peek(); + + /* Element out of line */ + int pop = queue.poll(); + + /* Get the length of the queue */ + int size = queue.size(); + + /* Determine if the queue is empty */ + boolean isEmpty = queue.isEmpty(); + ``` + +=== "C#" + + ```csharp title="queue.cs" + /* Initialize Queue */ + Queue queue = new(); + + /* Element in queue */ + queue.Enqueue(1); + queue.Enqueue(3); + queue.Enqueue(2); + queue.Enqueue(5); + queue.Enqueue(4); + + /* Access to the first element of the queue */ + int peek = queue.Peek(); + + /* Element out of line */ + int pop = queue.Dequeue(); + + /* Get the length of the queue */ + int size = queue.Count; + + /* Determine if the queue is empty */ + bool isEmpty = queue.Count == 0; + ``` + +=== "Go" + + ```go title="queue_test.go" + /* Initialize Queue */ + // In Go, use lists as queues. + queue := list.New() + + /* Element in queue */ + queue.PushBack(1) + queue.PushBack(3) + queue.PushBack(2) + queue.PushBack(5) + queue.PushBack(4) + + /* Access to the first element of the queue */ + peek := queue.Front() + + /* Element out of line */ + pop := queue.Front() + queue.Remove(pop) + + /* Get the length of the queue */ + size := queue.Len() + + /* Determine if the queue is empty */ + isEmpty := queue.Len() == 0 + ``` + +=== "Swift" + + ```swift title="queue.swift" + /* Initialize Queue */ + // Swift doesn't have a built-in queue class, so you can use Array as a queue. + var queue: [Int] = [] + + /* Element in queue */ + queue.append(1) + queue.append(3) + queue.append(2) + queue.append(5) + queue.append(4) + + /* Access to the first element of the queue */ + let peek = queue.first! + + /* Element out of line */ + // The complexity of removeFirst is O(n) because it's an array. + let pool = queue.removeFirst() + + /* Get the length of the queue */ + let size = queue.count + + /* Determine if the queue is empty */ + let isEmpty = queue.isEmpty + ``` + +=== "JS" + + ```javascript title="queue.js" + /* Initialize Queue */ + // JavaScript doesn't have a built-in queue, you can use Array as a queue. + const queue = []; + + /* Element in queue */ + queue.push(1); + queue.push(3); + queue.push(2); + queue.push(5); + queue.push(4); + + /* Access to the first element of the queue */ + const peek = queue[0]; + + /* Element out of line */ + // The underlying level is an array, so the time complexity of the shift() method is O(n) + const pop = queue.shift(); + + /* Get the length of the queue */ + const size = queue.length; + + /* Determine if the queue is empty */ + const empty = queue.length === 0; + ``` + +=== "TS" + + ```typescript title="queue.ts" + /* Initialize Queue */ + // TypeScript doesn't have a built-in queue; you can use Array as a queue. + const queue: number[] = []; + + /* Element in queue */ + queue.push(1); + queue.push(3); + queue.push(2); + queue.push(5); + queue.push(4); + + /* Access to the first element of the queue */ + const peek = queue[0]; + + /* Element out of line */ + // The underlying level is an array, so the time complexity of the shift() method is O(n) + const pop = queue.shift(); + + /* Get the length of the queue */ + const size = queue.length; + + /* Determine if the queue is empty */ + const empty = queue.length === 0; + ``` + +=== "Dart" + + ```dart title="queue.dart" + /* Initialize Queue */ + // In Dart, the queue class Qeque is a double-ended queue and can also be used as a queue + Queue queue = Queue(); + + /* Element in queue */ + queue.add(1); + queue.add(3); + queue.add(2); + queue.add(5); + queue.add(4); + + /* Access to the first element of the queue */ + int peek = queue.first; + + /* Element out of line */ + int pop = queue.removeFirst(); + + /* Get the length of the queue */ + int size = queue.length; + + /* Determine if the queue is empty */ + bool isEmpty = queue.isEmpty; + ``` + +=== "Rust" + + ```rust title="queue.rs" + /* Initialize double-ended queue */ + // Use a double-ended queue as a regular queue in Rust + let mut deque: VecDeque = VecDeque::new(); + + /* Element in queue */ + deque.push_back(1); + deque.push_back(3); + deque.push_back(2); + deque.push_back(5); + deque.push_back(4); + + /* Access to the first element of the queue */ + if let Some(front) = deque.front() { + } + + /* Element out of line */ + if let Some(pop) = deque.pop_front() { + } + + /* Get the length of the queue */ + let size = deque.len(); + + /* Determine if the queue is empty */ + let is_empty = deque.is_empty(); + ``` + +=== "C" + + ```c title="queue.c" + // C does not provide a built-in queue + ``` + +=== "Zig" + + ```zig title="queue.zig" + + ``` + +## Queue Implementation + +To implement a queue, we need a data structure that can add elements at one end and remove elements at the other. Therefore, both linked lists and arrays can be used to implement queues. + +### A Linked List Based Implementation + +As shown in the figure below, we can consider the "head node" and "tail node" of the linked list as the "head" and "tail" respectively. "The tail node can only add nodes and the head node can only delete nodes. + +=== "LinkedListQueue" + ![Implementing queue entry and exit operations based on linked lists](queue.assets/linkedlist_queue.png) + +=== "push()" + ![linkedlist_queue_push](queue.assets/linkedlist_queue_push.png) + +=== "pop()" + ![linkedlist_queue_pop](queue.assets/linkedlist_queue_pop.png) + +Here is the code to implement a queue with a linked list. + +```src +[file]{linkedlist_queue}-[class]{linked_list_queue}-[func]{} +``` + +### An Array-Based Implementation + +Since the time complexity of deleting the first element of an array is $O(n)$ , this can lead to less efficient out-of-queue operations. However, we can avoid this problem by using the following clever approach. + +We can use a variable `front` to point to the index of the element at the head of the queue, and maintain a variable `size` to record the length of the queue. Define `rear = front + size` and this formula calculates `rear` to point to the next position after the end element of the queue. + +Based on this design, **the valid interval of the elements contained in the array is `[front, rear - 1]`**, and the various operations are implemented as shown in the figure below. + +- Queue entry operation: assign the input element at the `rear` index and increase `size` by 1. +- Operation: Simply increase `front` by 1 and decrease `size` by 1. + +It can be seen that both the queue-in and queue-out operations need to be performed only once, and both have a time complexity of $O(1)$ . + +=== "ArrayQueue" + ![Implement queue in/queue out operations based on arrays](queue.assets/array_queue.png) + +=== "push()" + ![array_queue_push](queue.assets/array_queue_push.png) + +=== "pop()" + ![array_queue_pop](queue.assets/array_queue_pop.png) + +You may notice a problem: `front` and `rear` are both moving to the right as they keep moving in and out of the array, **and when they reach the end of the array, they can't move any further**. To solve this problem, we can think of the array as a "circular array" with the first and last arrays connected. + +For circular arrays, we need to have `front` or `rear` return to the head of the array when it crosses the end of the array to continue traversing. This periodicity can be achieved by the "remainder operation", as shown in the following code. + +```src +[file]{array_queue}-[class]{array_queue}-[func]{} +``` + +The above implementation of the queue still has the limitation that its length is immutable. However, this problem is not difficult to solve, we can replace the array with a dynamic array, thus introducing an expansion mechanism. Interested students can try to implement their own. + +The conclusions of the comparison of the two implementations are consistent with the stack and will not be repeated here. + +## Typical Applications Of Queues + +- **Taobao Orders**. After a shopper places an order, the order will be added to the queue, and the system will then process the orders in the queue sequentially according to the order. During the double eleven period, a huge number of orders will be generated in a short period of time, and high concurrency becomes a problem that engineers need to focus on. +- **All kinds of to-do lists**. Any scenario that requires "first-come-first-served" functionality, such as a task queue for a printer, a queue for a restaurant, and so on. 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We replace the plates with elements of various types (e.g., integers, characters, objects, etc.) to get the stack data structure. + +As shown in the figure below, we call the top of the stack the "top of the stack" and the bottom the "bottom of the stack". The operation of adding an element to the top of the stack is called "Into the stack", and the operation of removing an element from the top of the stack is called "Out of the stack". + +![First-in-first-out rule for stacks](stack.assets/stack_operations.png) + +## Common Operations On Stacks + +The common operations of the stack are shown in the table below, and the specific method names need to be determined according to the programming language used. Here, we take the common names `push()`, `pop()`, `peek()` as examples. + +

    Table   Efficiency of stack operations

    + +| Methods | Description | Time Complexity | +| ------- | -------------------------------------- | --------------- | +| push() | element to stack (add to top of stack) | $O(1)$ | +| pop() | top element of stack out | $O(1)$ | +| peek() | access top stack element | $O(1)$ | + +Usually, we can just use the built-in stack class of a programming language. However, some languages may not provide a stack class, so we can use the language's "arrays" or "linked lists" as stacks, and ignore stack-independent operations in the program logic. + +=== "Python" + + ```python title="stack.py" + # Initialize the stack + # Python doesn't have a built-in stack class, so you can use List as a stack. + stack: list[int] = [] + + # Element on the stack + stack.append(1) + stack.append(3) + stack.append(2) + stack.append(5) + stack.append(4) + + # Access the top element of the stack + peek: int = stack[-1] + + # Element out of the stack + pop: int = stack.pop() + + # Get the length of the stack + size: int = len(stack) + + # Determine if it's empty + is_empty: bool = len(stack) == 0 + ``` + +=== "C++" + + ```cpp title="stack.cpp" + /* Initialize the stack */ + stack stack; + + /* Element on stack */ + stack.push(1); + stack.push(3); + stack.push(2); + stack.push(5); + stack.push(4); + + /* Accesses the top element of the stack */ + int top = stack.top(); + + /* Element out of stack */ + stack.pop(); // no return value + + /* Get the length of the stack */ + int size = stack.size(); + + /* Determine if it is empty */ + bool empty = stack.empty(); + ``` + +=== "Java" + + ```java title="stack.java" + /* Initialize the stack */ + Stack stack = new Stack<>(); + + /* Element on stack */ + stack.push(1); + stack.push(3); + stack.push(2); + stack.push(5); + stack.push(4); + + /* Accesses the top element of the stack */ + int peek = stack.peek(); + + /* Element out of stack */ + int pop = stack.pop(); + + /* Get the length of the stack */ + int size = stack.size(); + + /* Determine if it is empty */ + boolean isEmpty = stack.isEmpty(); + ``` + +=== "C#" + + ```csharp title="stack.cs" + /* Initialize the stack */ + Stack stack = new(); + + /* Element on stack */ + stack.Push(1); + stack.Push(3); + stack.Push(2); + stack.Push(5); + stack.Push(4); + + /* Accesses the top element of the stack */ + int peek = stack.Peek(); + + /* Element out of stack */ + int pop = stack.Pop(); + + /* Get the length of the stack */ + int size = stack.Count; + + /* Determine if it is empty */ + bool isEmpty = stack.Count == 0; + ``` + +=== "Go" + + ```go title="stack_test.go" + /* Initialize the stack */ + // In Go, it is recommended to use Slice as a stack. + var stack []int + + /* Element on stack */ + stack = append(stack, 1) + stack = append(stack, 3) + stack = append(stack, 2) + stack = append(stack, 5) + stack = append(stack, 4) + + /* Accesses the top element of the stack */ + peek := stack[len(stack)-1] + + /* Element out of stack */ + pop := stack[len(stack)-1] + stack = stack[:len(stack)-1] + + /* Get the length of the stack */ + size := len(stack) + + /* Determine if it is empty */ + isEmpty := len(stack) == 0 + ``` + +=== "Swift" + + ```swift title="stack.swift" + /* Initialize the stack */ + // Swift doesn't have a built-in stack class, so you can use Array as a stack. + var stack: [Int] = [] + + /* Element on stack */ + stack.append(1) + stack.append(3) + stack.append(2) + stack.append(5) + stack.append(4) + + /* Accesses the top element of the stack */ + let peek = stack.last! + + /* Element out of stack */ + let pop = stack.removeLast() + + /* Get the length of the stack */ + let size = stack.count + + /* Determine if it is empty */ + let isEmpty = stack.isEmpty + ``` + +=== "JS" + + ```javascript title="stack.js" + /* Initialize the stack */ + // JavaScript doesn't have a built-in stack class, you can use Array as a stack. + const stack = []; + + /* Element on stack */ + stack.push(1); + stack.push(3); + stack.push(2); + stack.push(5); + stack.push(4); + + /* Accesses the top element of the stack */ + const peek = stack[stack.length-1]; + + /* Element out of stack */ + const pop = stack.pop(); + + /* Get the length of the stack */ + const size = stack.length; + + /* Determine if it is empty */ + const is_empty = stack.length === 0; + ``` + +=== "TS" + + ```typescript title="stack.ts" + /* Initialize the stack */ + // TypeScript doesn't have a built-in stack class, so you can use Array as a stack. + const stack: number[] = []; + + /* Element on stack */ + stack.push(1); + stack.push(3); + stack.push(2); + stack.push(5); + stack.push(4); + + /* Accesses the top element of the stack */ + const peek = stack[stack.length - 1]; + + /* Element out of stack */ + const pop = stack.pop(); + + /* Get the length of the stack */ + const size = stack.length; + + /* Determine if it is empty */ + const is_empty = stack.length === 0; + ``` + +=== "Dart" + + ```dart title="stack.dart" + /* Initialize the stack */ + // Dart doesn't have a built-in stack class, so you can use List as a stack. + List stack = []; + + /* Element on stack */ + stack.add(1); + stack.add(3); + stack.add(2); + stack.add(5); + stack.add(4); + + /* Accesses the top element of the stack */ + int peek = stack.last; + + /* Element out of stack */ + int pop = stack.removeLast(); + + /* Get the length of the stack */ + int size = stack.length; + + /* Determine if it is empty */ + bool isEmpty = stack.isEmpty; + ``` + +=== "Rust" + + ```rust title="stack.rs" + /* Initialize the stack */ + // Use the Vec as a stack. + let mut stack: Vec = Vec::new(); + + /* Element on stack */ + stack.push(1); + stack.push(3); + stack.push(2); + stack.push(5); + stack.push(4); + + /* Accesses the top element of the stack */ + let top = stack[stack.len() - 1]; + + /* Element out of stack */ + let pop = stack.pop().unwrap(); + + /* Get the length of the stack */ + let size = stack.len(); + + /* Determine if it is empty */ + let is_empty = stack.is_empty(); + ``` + +=== "C" + + ```c title="stack.c" + // C does not provide a built-in stack. + ``` + +=== "Zig" + + ```zig title="stack.zig" + + ``` + +## Stack Implementation + +In order to get a deeper understanding of how stacks work, let's try to implement a stack class ourselves. + +A stack follows the first-in-first-out principle, so we can only add or remove elements at the top of the stack. However, arrays and linked lists can add and remove elements anywhere, **so the stack can be thought of as a restricted array or linked list**. In other words, we can "mask" some of the extraneous operations of arrays or linked lists so that their external logic conforms to the properties of a stack. + +### A Linked List Based Implementation + +When using a linked list to implement a stack, we can consider the head node of the linked list as the top of the stack and the tail node as the bottom of the stack. + +As shown in the figure below, for the stack operation, we only need to insert the element into the head of the linked list node insertion method is called "head insertion method". For stack out operation, we just need to remove the head node from the linked list. + +=== "LinkedListStack" + ![Implementing stack-in-stack-out operations based on linked lists](stack.assets/linkedlist_stack.png) + +=== "push()" + ![linkedlist_stack_push](stack.assets/linkedlist_stack_push.png) + +=== "pop()" + ![linkedlist_stack_pop](stack.assets/linkedlist_stack_pop.png) + +The following is sample code for implementing a stack based on a linked list. + +```src +[file]{linkedlist_stack}-[class]{linked_list_stack}-[func]{} +``` + +### An Array-Based Implementation + +When using an array to implement a stack, we can use the tail of the array as the top of the stack. As shown in the figure below, the stack-in and stack-out operations correspond to adding and removing elements to the tail of the array, respectively, both with a time complexity of $O(1)$ . + +=== "ArrayStack" + ![Implementing stack in/out operations based on arrays](stack.assets/array_stack.png) + +=== "push()" + ![array_stack_push](stack.assets/array_stack_push.png) + +=== "pop()" + ![array_stack_pop](stack.assets/array_stack_pop.png) + +Since the elements of the stack may be constantly increasing, we can use dynamic arrays so that we do not have to deal with the array expansion problem. Here is the sample code. + +```src +[file]{array_stack}-[class]{array_stack}-[func]{} +``` + +## Comparison Of Two Implementations + +**Support Operations** + +Both implementations support the operations in the stack definition. The array implementation additionally supports random access, but this is beyond the scope of the stack definition, so it is generally not used. + +**Time Efficiency** + +In array-based implementations, both the stack-in and stack-out operations are performed in pre-allocated contiguous memory with good cache locality, and thus are more efficient. However, if the stack entry exceeds the capacity of the array, the expansion mechanism is triggered, causing the time complexity of that stack entry operation to become $O(n)$ . + +In the linked list implementation, the expansion of linked lists is very flexible and does not suffer from the efficiency degradation when expanding arrays as described above. However, the stack entry operation requires initialization of the node object and modification of the pointer, and is therefore relatively inefficient. However, if the stacked element is itself a node object, then the initialization step can be omitted, thus improving efficiency. + +In summary, when the elements of the in-stack and out-stack operations are basic data types, such as `int` or `double` , we can conclude the following. + +- Stacks implemented based on arrays are less efficient when expanding is triggered, but are more efficient on average because expanding is a low-frequency operation. +- Stacks implemented based on linked lists provide a more consistent performance in terms of efficiency. + +**Space efficiency** + +When initializing a list, the system assigns an "initial capacity" to the list, which may exceed the actual demand. Moreover, the expansion mechanism is usually based on a specific multiplier (e.g., 2x), and the expanded capacity may also exceed the actual demand. As a result, **stacks implemented based on arrays may cause a certain amount of wasted space**. + +However, since linked lists nodes require additional storage of pointers, **therefore linked nodes take up a relatively large amount of space**. + +In summary, we cannot simply determine which implementation is more memory-efficient; we need to analyze the situation on a case-by-case basis. + +## Typical Applications Of Stacks + +- **Backward and Forward in Browsers, Undo and Undoverse in Software**. Whenever we open a new web page, the browser executes the previous page into the stack so that we can go back to the previous page with the backward operation. The backward operation actually executes out of the stack. If we want to support both backward and forward, then we need two stacks to implement it together. +- **Program memory management**. Each time a function is called, the system adds a stack frame to the top of the stack to record the context information of the function. In a recursion function, the downward recursion phase performs constant in-stack operations, while the upward backward phase performs out-stack operations. diff --git a/docs-en/chapter_stack_and_queue/summary.md b/docs-en/chapter_stack_and_queue/summary.md new file mode 100644 index 0000000000..263c66da99 --- /dev/null +++ b/docs-en/chapter_stack_and_queue/summary.md @@ -0,0 +1,31 @@ +# Summary + +### Highlights + +- A stack is a data structure that follows the first-in-first-out principle and can be implemented as an array or a linked list. +- From a time efficiency perspective, array implementations of stacks have a high average efficiency, but the time complexity of a single stack entry operation deteriorates to $O(n)$ during the expansion process. In contrast, the linked list-based implementation of the stack has a more stable efficiency performance. +- In terms of space efficiency, an array implementation of a stack may result in a degree of space wastage. However, it should be noted that linked list nodes take up more memory space than array elements. +- A queue is a data structure that follows the first-in-first-out principle and can be implemented equally well with arrays or linked lists. The conclusions for queues are similar to the aforementioned conclusions for stacks in terms of the comparison of time efficiency and space efficiency. +- A double-ended queue is a type of queue with a higher degree of freedom that allows adding and removing operations of elements at both ends. + +### Q & A + +!!! question "Is the browser's forward and backward implemented as a two-way linked list?" + + The browser's forward and backward functions are essentially a manifestation of the "stack". When the user visits a new page, it is added to the top of the stack; when the user clicks the back button, the page is ejected from the top of the stack. Using a double-ended queue makes it easy to implement some additional operations, which are mentioned in the section on double-ended queues. + +!!! question "Is it necessary to free the memory of the outgoing node after going out of the stack?" + + If the popup node is still needed later, it does not need to be freed. If you don't need it later, languages like `Java` and `Python` have automatic garbage collection mechanisms, so you don't need to free the memory manually; in `C` and `C++` you need to free the memory manually. + +!!! Question "A double-ended queue looks like two stacks spliced together, what is its purpose?" + + A double-ended queue is like a combination of a stack and a queue, or two stacks put together. It shows the logic of the stack + queue, so it can realize all the applications of the stack and queue, and more flexible. + +!!! question "How exactly is undo (undo) and reverse undo (redo) implemented?" + + Two stacks are used, stack `A` for undo and stack `B` for anti-undo. + + 1. Whenever the user performs an operation, press this operation onto the stack `A` and empty the stack `B`. + 2. When the user executes "Undo", the most recent operation is popped from stack `A` and pressed into stack `B`. + 3. 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``` + +=== "C#" + + ```csharp title="" + /* Array representation of a binary tree */ + // With the int? nullable type, you can use null to mark the null position. + int?[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 }; + ``` + +=== "Go" + + ```go title="" + /* Array representation of a binary tree */ + // With any type of slice, you can use nil to mark empty spaces. + tree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15} + ``` + +=== "Swift" + + ```swift title="" + /* Array representation of a binary tree */ + // With the Int? nullable type, you can use nil to mark the null position. + let tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15] + ``` + +=== "JS" + + ```javascript title="" + /* Array representation of a binary tree */ + // Use null for empty space + let tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15]; + ``` + +=== "TS" + + ```typescript title="" + /* Array representation of a binary tree */ + // Use null for empty space + let tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15]; + ``` + +=== "Dart" + + ```dart title="" + /* Array representation of a binary tree */ + // With the int? nullable type, you can use null to mark the null position. + List tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15]; + ``` + +=== "Rust" + + ```rust title="" + /* Array representation of a binary tree */ + // Use None to mark the empty space + let tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)]; + ``` + +=== "C" + + ```c title="" + /* Array representation of a binary tree */ + // Use the int maximum value to mark empty spaces, thus requiring that the node value not be INT_MAX + int tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15}; + ``` + +=== "Zig" + + ```zig title="" + + ``` + +![Array representation of any type of binary tree](array_representation_of_tree.assets/array_representation_with_empty.png) + +It is worth stating that **complete binary trees are well suited to be represented using arrays**. Recall that the definition of a complete binary tree is that $\text{None}$ only occurs at the bottom and to the right, **so all $\text{None}$ must occur at the end of the level-order traversal sequence**. + +This means that when using an array to represent a complete binary tree, it is convenient to omit storing all $\text{None}$. An example is given in the figure below. + +![Array representation of complete binary tree](array_representation_of_tree.assets/array_representation_complete_binary_tree.png) + +The following code implements a binary tree based on an array representation and includes the following operations. + +- Given a node, get its value, left (right) child node, and parent node. +- Gets the sequence of preorder traversals, middle order traversals, postorder traversals, and level-order traversals. + +```src +[file]{array_binary_tree}-[class]{array_binary_tree}-[func]{} +``` + +## Strengths And Limitations + +The array representation of a binary tree has the following main advantages. + +- Arrays are stored in contiguous memory space, are cache friendly, and are faster to access and traverse. +- No need to store pointers, which is more space efficient. +- Allow random access to nodes. + +However, there are some limitations to the array representation. + +- Array storage requires contiguous memory 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right: None + })) + } + } + ``` + +=== "C" + + ```c title="" + /* AVL tree node structure */ + TreeNode struct TreeNode { + int val; + int height; + struct TreeNode *left; + struct TreeNode *right; + } TreeNode; + + /* Constructors */ + TreeNode *newTreeNode(int val) { + TreeNode *node; + + node = (TreeNode *)malloc(sizeof(TreeNode)); + node->val = val; + node->height = 0; + node->left = NULL; + node->right = NULL; + return node; + } + ``` + +=== "Zig" + + ```zig title="" + + ``` + +The "node height" is the distance from the node to its farthest leaf node, i.e. the number of "edges" it passes through. In particular, leaf nodes have a height of 0 and empty nodes have a height of -1. We will create two utility functions to get and update the height of the node. + +```src +[file]{avl_tree}-[class]{a_v_l_tree}-[func]{update_height} +``` + +### Node Balance Factor + +The balance factor of a node is defined as the height of the left subtree minus the height of the right subtree of the node, and the balance factor of an empty tree node is zero. We also encapsulate the function to get the balance factor of a node into a function for future use. + +```src +[file]{avl_tree}-[class]{a_v_l_tree}-[func]{balance_factor} +``` + +!!! note + + Let the balance factor be $f$ , then the balance factor of any node of an avl tree node satisfies $-1 \le f \le 1$ . + +## AVL Tree Rotation + +AVL trees are characterized by the "rotate" operation, which can rebalance the balanced binary tree nodes without affecting the middle-order traversal sequence of the binary tree. In other words, **rotation operation can maintain the nature of the "binary search tree", but also can make the tree back to "balanced binary tree"**. + +Nodes with absolute value of balance factor $> 1$ are called "out-of-balance nodes". Depending on the imbalance of a node, there are four types of rotation operations: right rotation, left rotation, right rotation followed by left rotation, and left rotation followed by right rotation. These rotations are described in detail below. + +### Right-Handed + +As shown in the figure below, the balance factor is below the node. From the bottom to the top, the first out-of-balance node in the binary tree is "node 3". We focus on the subtree rooted at this out-of-balance node, denote this node as `node` and its left-child node as `child`, and perform a "right-hand rotation". After the right-hand rotation, the subtree is balanced and still maintains the characteristics of a binary search tree. + +=== "<1>" + ![Right rotation step](avl_tree.assets/avltree_right_rotate_step1.png) + +=== "<2>" + ![avltree_right_rotate_step2](avl_tree.assets/avltree_right_rotate_step2.png) + +=== "<3>" + ![avltree_right_rotate_step3](avl_tree.assets/avltree_right_rotate_step3.png) + +=== "<4>" + ![avltree_right_rotate_step4](avl_tree.assets/avltree_right_rotate_step4.png) + +As shown in the figure below, when the node `child` has a right-child node (notated as `grandChild`), you need to add a step to the right-child node: make `grandChild` the left-child node of `node`. + +![Right rotation with grandChild](avl_tree.assets/avltree_right_rotate_with_grandchild.png) + +"Rotate to the right" is a figurative statement, in fact, you need to modify the node pointer to achieve, the code is shown below. + +```src +[file]{avl_tree}-[class]{a_v_l_tree}-[func]{right_rotate} +``` + +### Levitra + +Accordingly, if we consider the "mirror image" of the above imbalanced binary tree, we need to perform the "left-handed" operation shown in the figure below. + +![left-rotate operation](avl_tree.assets/avltree_left_rotate.png) + +Similarly, as shown in the figure below, when the node `child` has a left-child node (denoted as `grandChild`), you need to add a step in the left-hand rotation: make `grandChild` the right-child node of `node`. + +![Left rotation with grandChild](avl_tree.assets/avltree_left_rotate_with_grandchild.png) + +It can be observed that **the right-toration and left-rotation operations are logically mirror-symmetric, and the two imbalance cases they solve respectively are also symmetric**. Based on the symmetry, we simply replace all `left` with `right` and all `right` with `left` in the implementation code for right-handedness to obtain the implementation code for left-handedness. + +```src +[file]{avl_tree}-[class]{a_v_l_tree}-[func]{left_rotate} +``` + +### Left Then Right + +For the out-of-balance node 3 in the figure below, neither left-rotation nor right-rotation can be used to rebalance the tree node. In this case, you need to perform a "left spin" on `child` and then a "right spin" on `node`. + +![Rotate left then right](avl_tree.assets/avltree_left_right_rotate.png) + +### First Right, Then Left + +As shown in the figure below, for the mirror case of the above imbalanced binary tree, we need to perform a "right rotation" on `child` and then a "left rotation" on `node`. + +![Rotate right then left](avl_tree.assets/avltree_right_left_rotate.png) + +### Selection Of Rotation + +The four imbalance cases shown in the figure below correspond to each of the above cases, and require the use of right, left, right then left, and left then right rotation operations, respectively. + +![The four rotation cases of an avl tree](avl_tree.assets/avltree_rotation_cases.png) + +As shown in the table below, we determine which of the figure above the imbalanced node belongs to by determining the balance factor of the imbalanced node as well as the plus or minus sign of the balance factor of the child node on the higher side. + +

    Table   Selection conditions for the four rotation cases

    + +| Balance factor of imbalanced node | Balance factor of child node | Rotational method to be used | +| --------------------------------- | ---------------------------- | ---------------------------- | +| $> 1$ (i.e., left-biased tree) | $\geq 0$ | right-handed | +| $> 1$ (i.e., left-biased tree) | $<0$ | First left then right | +| $< -1$ (i.e., right biased tree) | $\leq 0$ | left-handed | +| $< -1$ (i.e., right-biased tree) | $>0$ | first right then left | + +For ease of use, we encapsulate the rotation operation into a function. **With this function, we can rotate various imbalance cases to bring the imbalanced nodes back into balance**. + +```src +[file]{avl_tree}-[class]{a_v_l_tree}-[func]{rotate} +``` + +## AVL Tree Common Operations + +### Insert Node + +The node insertion operation of an AVL tree is similar in principal to a binary search tree. The only difference is that after inserting a node in an AVL tree, a series of out-of-balance nodes may appear on the path from that node to the root node. Therefore, **we need to start from this node and perform a rotation operation from the bottom up to bring all the out-of-balance nodes back to balance**. + +```src +[file]{avl_tree}-[class]{a_v_l_tree}-[func]{insert_helper} +``` + +### Delete Node + +Similarly, based on the method of deleting nodes in a binary search tree, a rotation operation needs to be performed from the bottom to the top to bring back the balance to all the out-of-balance nodes. + +```src +[file]{avl_tree}-[class]{a_v_l_tree}-[func]{remove_helper} +``` + +### Find The Node + +The node finding operation of an avl tree is consistent with a binary search tree and will not be repeated here. + +## Typical Applications Of AVL Trees + +- Organize and store large data for high-frequency lookups and low-frequency additions and deletions. +- Used to build an indexing system in a database. +- Red-black trees are preferred over avl trees in many applications. 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The number of rounds is at most the height of the binary tree, using $O(\log n)$ time when the binary tree is balanced. + +```src +[file]{binary_search_tree}-[class]{binary_search_tree}-[func]{search} +``` + +### Insert Node + +Given an element `num` to be inserted, in order to maintain the nature of the binary search tree "left subtree < root node < right subtree", the insertion process is shown in the figure below. + +1. **Find insertion position**: similar to Find, start from the root node and loop down according to the relationship between the current node value and the size of `num` until the leaf node is crossed (traversing to $\text{None}$ ). +2. **Insert node at this position**: initialize node `num` and place the node at position $\text{None}$. + +![Insert node in binary search tree](binary_search_tree.assets/bst_insert.png) + +In the code implementation, the following two points need to be noted. + +- A binary search tree node is not allowed to have duplicate nodes, otherwise its definition would be violated. Therefore, if the node to be inserted already exists in the tree node, the insertion is not performed and it is returned directly. +- In order to insert a node, we need to save the node from the previous loop with the help of the node `pre`. This way, when traversing to $\text{None}$, we can get its parent node, and thus complete the node insertion operation. + +```src +[file]{binary_search_tree}-[class]{binary_search_tree}-[func]{insert} +``` + +As with finding nodes, inserting nodes uses $O(\log n)$ time. + +### Delete Node + +First find the target node in the binary tree and then remove it from the binary tree. + +Similar to inserting a node, we need to ensure that the "left subtree < root node < right subtree" property of the binary search tree is still satisfied after the deletion operation is completed. + +Therefore, we need to perform the corresponding node deletion operations according to the number of children of the target node, which are categorized into three cases: 0, 1 and 2. + +As shown in the figure below, when the degree of the node to be deleted is $0$, it means that the node is a leaf node and can be deleted directly. + +![Remove node in binary search tree (degree 0)](binary_search_tree.assets/bst_remove_case1.png) + +As shown in the figure below, when the degree of the node to be deleted is $1$, it is sufficient to replace the node to be deleted with its children. + +![Remove node in binary search tree (degree 1)](binary_search_tree.assets/bst_remove_case2.png) + +When the degree of the node to be deleted is $2$, we can't delete it directly, but need to replace it with a node. This node can be the smallest node of the right-child node or the largest node of the left-child node because we want to keep the nature of the binary search tree "left $<$ root $<$ right". + +Assuming that we select the smallest node of the right-child node (i.e., the next node in the middle-order traversal), the flow of the deletion operation is shown in the figure below. + +1. Find the next node in the "middle-order traversal sequence" of the node to be deleted, denoted as `tmp`. +2. Overwrite the value of `tmp` with the value of the node to be deleted and recursion tree node `tmp` . + +=== "<1>" + ![Remove node in binary search tree (degree 2)](binary_search_tree.assets/bst_remove_case3_step1.png) + +=== "<2>" + ![bst_remove_case3_step2](binary_search_tree.assets/bst_remove_case3_step2.png) + +=== "<3>" + ![bst_remove_case3_step3](binary_search_tree.assets/bst_remove_case3_step3.png) + +=== "<4>" + ![bst_remove_case3_step4](binary_search_tree.assets/bst_remove_case3_step4.png) + +The node deletion operation also uses $O(\log n)$ time, where $O(\log n)$ time is needed to find the node to be deleted, and $O(\log n)$ time is needed to get the succeeding node for mid-order traversal. + +```src +[file]{binary_search_tree}-[class]{binary_search_tree}-[func]{remove} +``` + +### Middle-Order Traversal Is Ordered + +As shown in the figure below, the middle-order traversal of a binary tree follows the traversal order of "left $\rightarrow$ root $\rightarrow$ right", and the binary search tree satisfies the size relationship of "left-child node $<$ root node $<$ right-child node". The binary search tree satisfies the "left child $<$ root $<$ right child" size relation. + +This means that when performing a mid-order traversal in a binary search tree, the next smallest node will always be traversed preferentially, leading to an important property:**The sequence of mid-order traversals of a binary search tree is ascending**. + +Using the ascending nature of the middle-order traversal, it takes only $O(n)$ time to fetch ordered data in a binary search tree without additional sorting operations, which is very efficient. + +![Sequence of intermediate order traversals of a binary search tree](binary_search_tree.assets/bst_inorder_traversal.png) + +## Efficiency Of Binary Search Trees + +Given a set of data, we consider storing it using an array or a binary search tree. Observing the table below, the time complexity of each operation of the binary search tree is of logarithmic degree, which provides a stable and efficient performance. Arrays are more efficient than binary search trees only in data applicability scenarios with high frequency additions and low frequency lookups for deletions. + +

    Table   Efficiency of arrays vs. search trees

    + +| | Unordered arrays | Binary search trees | +| -------------- | ---------------- | ------------------- | +| Find Elements | $O(n)$ | $O(\log n)$ | +| insert element | $O(1)$ | $O(\log n)$ | +| Delete Element | $O(n)$ | $O(\log n)$ | + +Ideally, the binary search tree is "balanced" so that any node can be found in a $\log n$ round-robin. + +However, if we keep inserting and deleting nodes in the binary search tree, it may cause the binary tree to degenerate into a linked list as shown in the figure below, at which point the time complexity of the various operations also degenerates to $O(n)$ . + +![Degradation of binary search tree](binary_search_tree.assets/bst_degradation.png) + +## Common Applications Of Binary Search Trees + +- Used as a multi-level index in the system for efficient find, insert, and delete operations. +- as the underlying data structure for certain searching algorithms. +- Used to store a stream of data to keep it in an organized state. diff --git 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Similar to a linked list, the basic unit of a binary tree is a node, and each node contains: a value, a reference to a left-child node, and a reference to a right-child node. + +=== "Python" + + ```python title="" + class TreeNode: + """Binary tree node classes"""" + def __init__(self, val: int): + self.val: int = val # node value + self.left: TreeNode | None = None # left-child node reference + self.right: TreeNode | None = None # right-child node reference + ``` + +=== "C++" + + ```cpp title="" + /* Binary tree node structure */ + struct TreeNode { + int val; // node value + TreeNode *left; // Pointer to left-child node. + TreeNode *right; // Pointer to right-child node. + TreeNode(int x) : val(x), left(nullptr), right(nullptr) {} + }; + ``` + +=== "Java" + + ```java title="" + /* Binary tree node class */ + class TreeNode { + int val; // node value + TreeNode left; // Reference to left-child node. + TreeNode right; // right-child node reference + TreeNode(int x) { val = x; } + } + ``` + +=== "C#" + + ```csharp title="" + /* Binary tree node class */ + class TreeNode { + int val; // node value + TreeNode? left; // Left-child node reference. + TreeNode? right; // Reference to right-child node. + TreeNode(int x) { val = x; } + } + ``` + +=== "Go" + + ```go title="" + /* Binary tree node structure */ + type TreeNode struct { + Val int + Left *TreeNode + Right *TreeNode + } + /* Constructor */ + func NewTreeNode(v int) *TreeNode { + return &TreeNode{ + Left: nil, // pointer to left-child node + Right: nil, // pointer to right-child node + Val: v, // node value + } + } + ``` + +=== "Swift" + + ```swift title="" + /* Binary tree node class */ + class TreeNode { + var val: Int // node value + var left: TreeNode? // Left-child node reference. + var right: TreeNode? // Right-child node reference. + + init(x: Int) { + val = x + } + } + ``` + +=== "JS" + + ```javascript title="" + /* Binary tree node class */ + class TreeNode { + val; // node value + left; // Pointer to the left-child node. + right; // pointer to right-child node + constructor(val, left, right) { + this.val = val === undefined ? 0 : val; + this.left = left === undefined ? null : left; + this.right = right === undefined ? null : right; + } + } + ``` + +=== "TS" + + ```typescript title="" + /* Binary tree node class */ + class TreeNode { + val: number; + left: TreeNode | null; + right: TreeNode | null; + + constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) { + this.val = val === undefined ? 0 : val; // node value + this.left = left === undefined ? null : left; // left-child node reference + this.right = right === undefined ? null : right; // right-child node reference + } + } + ``` + +=== "Dart" + + ```dart title="" + /* Binary tree node class */ + class TreeNode { + int val; // node value + TreeNode? left; // Left-child node reference. + TreeNode? right; // Reference to right-child node. + TreeNode(this.val, [this.left, this.right]); + } + ``` + +=== "Rust" + + ```rust title="" + use std::rc::Rc; + use std::cell::RefCell; + + /* Binary tree node structure */ + struct TreeNode { + val: i32, // node value + left: Option>>, // left-child node reference + right: Option>>, // right-child node reference + } + + impl TreeNode { + /* Constructor */ + fn new(val: i32) -> Rc> { + Rc::new(RefCell::new(Self { + val, + left: None, + right: None + })) + } + } + ``` + +=== "C" + + ```c title="" + /* Binary tree node structure */ + typedef struct TreeNode { + int val; // node value + int height; // height of the node + struct TreeNode *left; // Pointer to left-child node. + struct TreeNode *right; // Pointer to right-child node. + } TreeNode; + + /* Constructors */ + TreeNode *newTreeNode(int val) { + TreeNode *node; + + node = (TreeNode *)malloc(sizeof(TreeNode)); + node->val = val; + node->height = 0; + node->left = NULL; + node->right = NULL; + return node; + } + ``` + +=== "Zig" + + ```zig title="" + + ``` + +Each node has two references (pointers) to the left-child node and the right-child node, which are called the parent nodes of the two child nodes. When given a node in a binary tree, the tree formed by the left-child node and the following nodes of the node is called the "left subtree" of the node, and similarly, the "right subtree" of the node is called the "right subtree". + +**In a binary tree node, all nodes except leaf nodes contain children and non-empty subtrees**. As shown in the figure below, if "node 2" is considered as the parent node, its left-child node and right-child node are "node 4" and "node 5" respectively. The left-child node is "the tree formed by node 4 and its following nodes" and the right-child node is "the tree formed by node 5 and its following nodes". + +![parent node, child node, subtree](binary_tree.assets/binary_tree_definition.png) + +## Common Terminology For Binary Trees + +The common terminology for binary trees is shown in the figure below. + +- "root node root node": a node at the top level of a binary tree with no parent node. +- "leaf node": a node with no children, both pointers point to $\text{None}$. +- "edge": a line segment connecting two nodes, i.e. a node reference (pointer). +- The "level" of the node: increasing from top to bottom, the level of the root node is 1. +- Degree of a node degree: the number of children of a node. In a binary tree, the degree can take values in the range 0, 1, 2. +- Height of a binary tree: the number of edges from the root node to the farthest leaf node. +- Depth of a node depth: the number of edges that pass from the root node to the node. +- The "height" of a node: the number of edges passing from the farthest leaf node to the node. + +![Common terminology for binary trees](binary_tree.assets/binary_tree_terminology.png) + +!!! tip + + Note that we usually define "height" and "depth" as "number of edges traveled", but some topics or textbooks may define them as "the number of nodes traveled". In this case, add 1 to both height and depth. + +## Basic Binary Tree Operations + +### Initialize A Binary Tree + +Similar to a linked list, the nodes are initialized first and then the references (pointers) are constructed. + +=== "Python" + + ```python title="binary_tree.py" + # Initialize binary tree + # Initialize nodes + n1 = TreeNode(val=1) + n2 = TreeNode(val=2) + n3 = TreeNode(val=3) + n4 = TreeNode(val=4) + n5 = TreeNode(val=5) + # Construct references to point (i.e. pointers) + n1.left = n2 + n1.right = n3 + n2.left = n4 + n2.right = n5 + ``` + +=== "C++" + + ```cpp title="binary_tree.cpp" + /* Initialize the binary tree */ + // Initialize nodes + TreeNode* n1 = new TreeNode(1); + TreeNode* n2 = new TreeNode(2); + TreeNode* n3 = new TreeNode(3); + TreeNode* n4 = new TreeNode(4); + TreeNode* n5 = new TreeNode(5); + // Constructing reference pointers (i.e. pointers) + n1->left = n2; + n1->right = n3; + n2->left = n4; + n2->right = n5; + ``` + +=== "Java" + + ```java title="binary_tree.java" + // Initialize nodes + TreeNode n1 = new TreeNode(1); + TreeNode n2 = new TreeNode(2); + TreeNode n3 = new TreeNode(3); + TreeNode n4 = new TreeNode(4); + TreeNode n5 = new TreeNode(5); + // Constructing reference pointers (i.e. pointers) + n1.left = n2; + n1.right = n3; + n2.left = n4; + n2.right = n5; + ``` + +=== "C#" + + ```csharp title="binary_tree.cs" + /* Initialize the binary tree */ + // Initialize nodes + TreeNode n1 = new(1); + TreeNode n2 = new(2); + TreeNode n3 = new(3); + TreeNode n4 = new(4); + TreeNode n5 = new(5); + // Constructing reference pointers (i.e. pointers) + n1.left = n2; + n1.right = n3; + n2.left = n4; + n2.right = n5; + ``` + +=== "Go" + + ```go title="binary_tree.go" + /* Initialize the binary tree */ + // Initialize nodes + n1 := NewTreeNode(1) + n2 := NewTreeNode(2) + n3 := NewTreeNode(3) + n4 := NewTreeNode(4) + n5 := NewTreeNode(5) + // Constructing reference pointers (i.e. pointers) + n1.Left = n2 + n1.Right = n3 + n2.Left = n4 + n2.Right = n5 + ``` + +=== "Swift" + + ```swift title="binary_tree.swift" + // Initialize nodes + let n1 = TreeNode(x: 1) + let n2 = TreeNode(x: 2) + let n3 = TreeNode(x: 3) + let n4 = TreeNode(x: 4) + let n5 = TreeNode(x: 5) + // Constructing reference pointers (i.e. pointers) + n1.left = n2 + n1.right = n3 + n2.left = n4 + n2.right = n5 + ``` + +=== "JS" + + ```javascript title="binary_tree.js" + /* Initialize the binary tree */ + // Initialize nodes + let n1 = new TreeNode(1), + n2 = new TreeNode(2), + n3 = new TreeNode(3), + n4 = new TreeNode(4), + n5 = new TreeNode(5); + // Constructing reference pointers (i.e. pointers) + n1.left = n2; + n1.right = n3; + n2.left = n4; + n2.right = n5; + ``` + +=== "TS" + + ```typescript title="binary_tree.ts" + /* Initialize the binary tree */ + // Initialize nodes + let n1 = new TreeNode(1), + n2 = new TreeNode(2), + n3 = new TreeNode(3), + n4 = new TreeNode(4), + n5 = new TreeNode(5); + // Constructing reference pointers (i.e. pointers) + n1.left = n2; + n1.right = n3; + n2.left = n4; + n2.right = n5; + ``` + +=== "Dart" + + ```dart title="binary_tree.dart" + /* Initialize the binary tree */ + // Initialize nodes + TreeNode n1 = new TreeNode(1); + TreeNode n2 = new TreeNode(2); + TreeNode n3 = new TreeNode(3); + TreeNode n4 = new TreeNode(4); + TreeNode n5 = new TreeNode(5); + // Constructing reference pointers (i.e. pointers) + n1.left = n2; + n1.right = n3; + n2.left = n4; + n2.right = n5; + ``` + +=== "Rust" + + ```rust title="binary_tree.rs" + // Initialize nodes + let n1 = TreeNode::new(1); + let n2 = TreeNode::new(2); + let n3 = TreeNode::new(3); + let n4 = TreeNode::new(4); + let n5 = TreeNode::new(5); + // Constructing reference pointers (i.e. pointers) + n1.borrow_mut().left = Some(n2.clone()); + n1.borrow_mut().right = Some(n3); + n2.borrow_mut().left = Some(n4); + n2.borrow_mut().right = Some(n5); + ``` + +=== "C" + + ```c title="binary_tree.c" + /* Initialize the binary tree */ + // Initialize nodes + TreeNode *n1 = newTreeNode(1); + TreeNode *n2 = newTreeNode(2); + TreeNode *n3 = newTreeNode(3); + TreeNode *n4 = newTreeNode(4); + TreeNode *n5 = newTreeNode(5); + // Constructing reference pointers (i.e. pointers) + n1->left = n2; + n1->right = n3; + n2->left = n4; + n2->right = n5; + ``` + +=== "Zig" + + ```zig title="binary_tree.zig" + + ``` + +### Insertion And Deletion Of Nodes + +Similar to a linked list, insertion and deletion of nodes in a binary tree can be accomplished by modifying pointers. An example is given in the figure below. + +![Inserting and removing nodes in a binary tree](binary_tree.assets/binary_tree_add_remove.png) + +=== "Python" + + ```python title="binary_tree.py" + # Insertion and deletion of nodes + p = TreeNode(0) + # Insert node P in the middle of n1 -> n2 + n1.left = p + p.left = n2 + # Delete node P + n1.left = n2 + ``` + +=== "C++" + + ```cpp title="binary_tree.cpp" + /* Insertion and deletion of nodes */ + TreeNode* P = new TreeNode(0); + // Insert node P in the middle of n1 -> n2. + n1->left = P; + P->left = n2; + // Delete node P + n1->left = n2; + ``` + +=== "Java" + + ```java title="binary_tree.java" + TreeNode P = new TreeNode(0); + // Insert node P in the middle of n1 -> n2. + n1.left = P; + P.left = n2; + // Delete node P + n1.left = n2; + ``` + +=== "C#" + + ```csharp title="binary_tree.cs" + /* Insertion and deletion of nodes */ + TreeNode P = new(0); + // Insert node P in the middle of n1 -> n2. + n1.left = P; + P.left = n2; + // Delete node P + n1.left = n2; + ``` + +=== "Go" + + ```go title="binary_tree.go" + /* Insertion and deletion of nodes */ + // Insert node P in the middle of n1 -> n2. + p := NewTreeNode(0) + n1.Left = p + p.Left = n2 + // Delete node P + n1.Left = n2 + ``` + +=== "Swift" + + ```swift title="binary_tree.swift" + let P = TreeNode(x: 0) + // Insert node P in the middle of n1 -> n2. + n1.left = P + P.left = n2 + // Delete node P + n1.left = n2 + ``` + +=== "JS" + + ```javascript title="binary_tree.js" + /* Insertion and deletion of nodes */ + let P = new TreeNode(0); + // Insert node P in the middle of n1 -> n2. + n1.left = P; + P.left = n2; + // Delete node P + n1.left = n2; + ``` + +=== "TS" + + ```typescript title="binary_tree.ts" + /* Insertion and deletion of nodes */ + const P = new TreeNode(0); + // Insert node P in the middle of n1 -> n2. + n1.left = P; + P.left = n2; + // Delete node P + n1.left = n2; + ``` + +=== "Dart" + + ```dart title="binary_tree.dart" + /* Insertion and deletion of nodes */ + TreeNode P = new TreeNode(0); + // Insert node P in the middle of n1 -> n2. + n1.left = P; + P.left = n2; + // Delete node P + n1.left = n2; + ``` + +=== "Rust" + + ```rust title="binary_tree.rs" + let p = TreeNode::new(0); + // Insert node P in the middle of n1 -> n2. + n1.borrow_mut().left = Some(p.clone()); + p.borrow_mut().left = Some(n2.clone()); + // Delete node p + n1.borrow_mut().left = Some(n2); + ``` + +=== "C" + + ```c title="binary_tree.c" + /* Insertion and deletion of nodes */ + TreeNode *P = newTreeNode(0); + // Insert node P in the middle of n1 -> n2. + n1->left = P; + P->left = n2; + // Delete node P + n1->left = n2; + ``` + +=== "Zig" + + ```zig title="binary_tree.zig" + + ``` + +!!! note + + It is important to note that inserting a node may change the original logical structure of a binary tree, while deleting a node usually means deleting the node and all its subtrees. Therefore, in a binary tree, insertion and deletion operations are usually done in conjunction with a set of operations to achieve meaningful operations. + +## Common Binary Tree Types + +### Perfect Binary Tree + +The nodes in all levels of a "perfect binary tree" are completely filled. In a perfect binary tree, the degree of the leaf nodes is $0$, and the degree of all the other nodes is $2$; if the height of the tree is $h$, then the total number of nodes is $2^{h+1} - 1$, which shows the standard exponential relationship, reflecting the common phenomenon of cell division in nature. + +!!! tip + + Note that in the Chinese community, a perfect binary tree is often referred to as a "full binary tree". + +![perfect binary tree](binary_tree.assets/perfect_binary_tree.png) + +### Complete Binary Tree + +As shown in the figure below, only the nodes at the bottom level of a complete binary tree are not filled, and the nodes at the bottom level are filled as far to the left as possible. + +![complete binary tree](binary_tree.assets/complete_binary_tree.png) + +### Full Binary Tree + +As shown in the figure below, a "full binary tree full binary tree" has all nodes except leaf nodes with two children. + +![full binary tree](binary_tree.assets/full_binary_tree.png) + +### Balanced Binary Tree + +As shown in the figure below, the absolute value of the difference between the heights of the left-child node and the right subtree of any node in a balanced binary tree does not exceed 1. + +![balanced binary tree](binary_tree.assets/balanced_binary_tree.png) + +## Degeneration Of Binary Trees + +The figure below shows the ideal and degenerate states of a binary tree. A binary tree reaches a "perfect binary tree" when each level of nodes is filled, while a binary tree degenerates into a "linked list" when all nodes are skewed to one side. + +- A perfect binary tree is the ideal case to take full advantage of the "divide and conquer" of binary trees. +- Linked lists are the other extreme, where each operation becomes linear and the time complexity degrades to $O(n)$ . + +![Best and worst structures of binary trees](binary_tree.assets/binary_tree_best_worst_cases.png) + +As shown in the table below, the number of leaf nodes, the total number of nodes, and the height of the binary tree reaches extreme or minimal values under the best and worst structure. + +

    Table   Best and worst case scenarios for binary trees

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zN2wwUf-q_Jywe0>t6J8;xqrU)iHzae$vamF3N>VsoWfYi()-kk)hdu3(y%&a_IIG< zvyk9V-R69umk7_c!-n*b;wYDcvEVwaiZF?4*O2O;jYFk*=T?u@75F{sO7cC}M$}kQ zrHf48xO>3AZ&i`PMq0&VZ+i2d8HEaPpwslo87uVh2eCxEB+s$hzj+nMjiDdTbkIld z3Sb(ff~^##HFv+^(FjYGF^NrK=(i=n6v + +!!! abstract + + The tree is full of life, with its deep roots and branches. + + It shows us the vivid shape of the data divide and conquer. diff --git a/docs-en/chapter_tree/summary.md b/docs-en/chapter_tree/summary.md new file mode 100644 index 0000000000..1d574d6c56 --- /dev/null +++ b/docs-en/chapter_tree/summary.md @@ -0,0 +1,54 @@ +# Summary + +### Highlights + +- A binary tree is a non-linear data structure that embodies the logic of divide and conquer. Each binary tree node contains a value and two pointers to its left-child node and right-child node. +- For a node in a binary tree, its left (right) child node and the tree formed below it are called the left (right) child tree of that node. +- Terms related to binary trees include root node, leaf node, level, degree, edge, height and depth. +- Initialization, node insertion and node deletion operations for binary trees are performed in a similar way to linked list operations. +- The common types of binary trees are perfect binary tree, complete binary tree, full binary tree and balanced binary tree. A perfect binary tree is the most desirable state, while a linked list is the worst state after degradation. +- A binary tree can be represented as an array by arranging the node values and empty spaces in a level-order traversal order and implementing pointers based on the index mapping relationship between parent nodes and child nodes. +- Binary tree level-order traversal is a breadth-first search method, which reflects the "circle by circle outward" hierarchical traversal, usually through the queue to realize. +- Pre-order, middle-order, and back-order traversals are all depth-first searches, which embody the "go to the end, and then go back to continue" back traversal method, usually using recursion to achieve. +- A binary search tree is an efficient element-finding data structure with $O(\log n)$ time complexity for find, insert and delete operations. When the binary search tree degenerates into a linked list, the time complexity of each operation deteriorates to $O(n)$ . +- An AVL tree, also known as a balanced binary search tree, ensures that the tree remains balanced after constant insertion and deletion of nodes by means of a rotation operation. +- The rotation operations of AVL tree include right rotation, left rotation, right rotation then left rotation, and left rotation then right rotation. After a node is inserted or deleted, the AVL tree performs a rotation operation from the bottom to the top to rebalance the tree. + +### Q & A + +!!! question "For a binary tree with only one node, is the height of the tree and the depth of the root node both $0$?" + + Yes, because height and depth are usually defined as "the number of edges traveled". + +!!! Question "Insertion and deletion in a binary tree are generally accomplished by a set of operations, what is meant by "a set of operations" here? What does "one set of operations" mean? Can it be interpreted as the resource release of a child node of a resource?" + + Taking a binary search tree as an example, node deletion operations are handled in three cases, each of which requires multiple steps of node operations. + +!!! question "Why does DFS traverse a binary tree in first, middle and last order, and what is the use of each?" + + DFS traversal in the first, middle and last order is similar to the order of accessing an array, and it is the basic method of traversing a binary tree. Using these three traversal methods, we can get the result of traversal in a specific order. For example, in a binary search tree, since the node size satisfies `left-child node value < root node value < right-child node value`, we can obtain an ordered sequence of nodes by traversing the tree according to the priority of `left -> root -> right`. + +!!! question "The right-rotation operation deals with the relationship between the out-of-balance nodes `node`, `child`, and `grand_child`, so doesn't the connection between the parent node of `node` and the original connection of `node` need to be maintained? Wouldn't it be broken after the right-rotation operation?" + + We need to look at this from a recursion perspective. The right-rotate operation `right_rotate(root)` is passed in the root node of the subtree, and eventually `return child` returns the root node of the subtree after the rotation. The connection between the root node of the subtree and its parent node is done after the function returns, and is not part of the maintenance of the right-rotate operation. + +!!! question "In C++, functions are divided into `private` and `public`, is there any consideration for this? Why did you put the `height()` function and the `updateHeight()` function in `public` and `private` respectively?" + + It mainly depends on the scope of use of the method, if the method is only used inside the class, then it is designed as `private`. For example, it makes no sense for the user to call `updateHeight()` alone, it is just a step in the insertion and deletion operations. `height()` accesses the height of a node, similar to `vector.size()`, so it is set to `public` for ease of use. + +!!! Question "How to construct a binary search tree from a set of input data, please? Is the choice of root node important?" + + Yes, the method for building the tree is given in the `build_tree()` method in the binary search tree code. As for the selection of the root node, we usually sort the input data, then use the midpoint element as the root node, and then recursively build the left and right subtrees. Doing so maximizes the balance of the tree. + +!!! question "In Java, is the `equals()` method always used for string comparison?" + + In Java, for basic data types, `==` is used to compare whether the values of two variables are equal. For reference types, the two notations work differently. + + - `==` : Used to compare two variables to see if they point to the same object, i.e. if they are at the same location in memory. + - `equals()`: Used to compare whether the values of two objects are equal. + + So if we want to compare values, we would normally use `equals()` . 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    + +

    + + +

    + +

    Hello Algo

    + +

    Data Structures and Algorithms: Animated Illutrations and Off-the-Shelf Code

    + +

    + + GitHub Repo stars + +   + + GitHub contributors + +

    + +--- + +

    Praise

    + +!!! quote + + "An accessible and easy-to-understand introduction to data structures and algorithms, guiding readers to learn actively and hands-on. Highly recommended for beginners in algorithms." + + **— Junhui Deng, Professor of Computer Science at Tsinghua University** + +!!! quote + + "If I'd had 'Hello Algo' back when I was learning Data Structures and Algorithms, it would have been 10 times easier to learn!" + + **— Mu Li, Senior Principal Scientist at Amazon** + +--- + +

    Animated illustrations throughout the book

    + +

    Clear and easy-to-understand content with a smooth learning curve
    Computer, tablet, and cell phone all-terminal reading

    + +![algorithm_animation](index.assets/animation.gif) + +!!! quote "" + +

    "A picture is worth a thousand words."

    + +--- + +

    Code Runs in One Click

    + +

    Provides concise implementations and test samples of algorithms and data structures that are ready to run
    Support for Java, C++, Python, Go, JS, TS, C#, Swift, Zig and more

    . + +![running_code](index.assets/running_code.gif) + +!!! quote "" + +

    "Talk is cheap. Show me the code."

    + +--- + +

    Possibility of discussion and questions

    + +

    Encourage your friends to help each other grow
    Questions and comments are generally answered within two days

    . + +![comment](index.assets/comment.gif) + +!!! quote "" + +

    "Chase the wind and catch the moon and don't stop, the flat barren place is the spring mountains"

    +

    Cheer together!

    + +--- + +

    Procedure

    + +Two years ago, I shared the "Sword to Offer" series on LeetCode, which was loved and supported by many students. During the communication with readers, one of the most common questions I received was "how to start learning algorithms". I have gradually developed a strong interest in this question. + +Brushing up on problems with a blank slate seems to be the most popular method, which is simple, direct and effective. Brushing questions is like playing the game of "minesweeper", students with strong self-learning ability can successfully remove the mines one by one, while students with insufficient foundation are likely to be blown up all over their heads and retreat in frustration. Reading through textbooks and books is also a common practice, but for job-seeking students, the graduation season, submitting resumes, and preparing for written tests and interviews have taken up most of their energy, and heavy books often become a daunting challenge. + +If you are facing similar struggles, then luckily this book has found you. This book is my answer to this problem, if not the optimal solution, at least a positive attempt. Although this book is not enough for you to get an offer directly, it will guide you to explore the "knowledge map" of data structures and algorithms, bring you to understand the shapes, sizes, and distributions of different "landmines", and let you master various "demining methods". demining methods". With these skills, I believe you can be more comfortable to deal with the questions and read the literature, and gradually build up a complete knowledge system. + +

    The Author

    + +Yudong Jin([Krahets](https://leetcode.cn/u/jyd/)), Senior Algorithmic Engineer of Big Factory, Master of Shanghai Jiaotong University. LeetCode network reads the highest number of bloggers, published by the ["Diagramming of Algorithmic Data Structures"](https://leetcode.cn/leetbook/detail/illustration-of-algorithm/) has been subscribed to 270,000 books. + +--- + +

    Acknowledgments

    + +This book has grown through the combined efforts of many contributors in the open source community. Thank you to each of the contributors who invested their time and effort to make this book better through their selfless contributions, they are (in order of GitHub auto-generation): + +

    + + + +

    + +The code in this book were reviewed by Gonglja, gvenusleo, justin-tse, krahets, night-cruise, nuomi1, Reanon, sjinzh (in alphabetical order). We thank them for their time and effort in ensuring that the code is standardized and consistent across languages. + +
    + + + + + + + + + + + + +
    Gonglja
    Gonglja

    C, C++
    gvenusleo
    gvenusleo

    Dart
    hpstory
    hpstory

    C#
    justin-tse
    justin-tse

    JS, TS
    krahets
    krahets

    Java, Python
    night-cruise
    night-cruise

    Rust
    nuomi1
    nuomi1

    Swift
    Reanon
    Reanon

    Go, C
    sjinzh
    sjinzh

    Rust, Zig
    +
    + +--- diff --git a/mkdocs-en.yml b/mkdocs-en.yml new file mode 100644 index 0000000000..5cbb58caf1 --- /dev/null +++ b/mkdocs-en.yml @@ -0,0 +1,164 @@ +# Config inheritance +INHERIT: mkdocs.yml + +# Project information +site_name: Hello Algo +site_url: https://www.hello-algo.com/en/ +site_description: 动画图解、一键运行的数据结构与算法教程 +docs_dir: build/docs-en +site_dir: site/en +# Repository +edit_uri: tree/main/docs-en +version: 1.0.0b6 + +# Configuration +theme: + language: en + font: + text: Roboto + code: Fira Code + +extra: + status: + new: Recently Added + +# Page tree +nav: + - Chapter 0. Preface: + # [icon: material/book-open-outline] + - chapter_preface/index.md + - 0.1 The Book: chapter_preface/about_the_book.md + - 0.2 How to Read: chapter_preface/suggestions.md + - 0.3 Summary: chapter_preface/summary.md + - Chapter 1. Introduction to Algorithms: + # [icon: material/calculator-variant-outline] + - chapter_introduction/index.md + - 1.1 Algorithms Are Everywhere: chapter_introduction/algorithms_are_everywhere.md + - 1.2 What is Algorithms: chapter_introduction/what_is_dsa.md + - 1.3 Summary: chapter_introduction/summary.md + - Chapter 2. Computational Complexity: + # [icon: material/timer-sand] + - chapter_computational_complexity/index.md + - 2.1 Evaluating Algorithm Efficiency: chapter_computational_complexity/performance_evaluation.md + - 2.2 Iteration and Recursion: chapter_computational_complexity/iteration_and_recursion.md + - 2.3 Time Complexity: chapter_computational_complexity/time_complexity.md + - 2.4 Space Complexity: chapter_computational_complexity/space_complexity.md + - 2.5 Summary: chapter_computational_complexity/summary.md + - Chapter 3. Data Structure: + # [icon: material/shape-outline] + - chapter_data_structure/index.md + - 3.1 Classification of Data Structure: chapter_data_structure/classification_of_data_structure.md + - 3.2 Basic Data Types: chapter_data_structure/basic_data_types.md + - 3.3 Number Encoding *: chapter_data_structure/number_encoding.md + - 3.4 Character Encoding *: chapter_data_structure/character_encoding.md + - 3.5 Summary: chapter_data_structure/summary.md + - Chapter 4. Array and Linked List: + # [icon: material/view-list-outline] + - chapter_array_and_linkedlist/index.md + - 4.1 Array: chapter_array_and_linkedlist/array.md + - 4.2 Linked List: chapter_array_and_linkedlist/linked_list.md + - 4.3 List: chapter_array_and_linkedlist/list.md + - 4.4 Summary: chapter_array_and_linkedlist/summary.md + - Chapter 5. Stack and Queue: + # [icon: material/stack-overflow] + - chapter_stack_and_queue/index.md + - 5.1 Stack: chapter_stack_and_queue/stack.md + - 5.2 Queue: chapter_stack_and_queue/queue.md + - 5.3 Double-ended Queue: chapter_stack_and_queue/deque.md + - 5.4 Summary: chapter_stack_and_queue/summary.md + - Chapter 6. Hash Table: + # [icon: material/table-search] + - chapter_hashing/index.md + - 6.1 Hash Table: chapter_hashing/hash_map.md + - 6.2 Hash Collision: chapter_hashing/hash_collision.md + - 6.3 Hash Algorithm: chapter_hashing/hash_algorithm.md + - 6.4 Summary: chapter_hashing/summary.md + - Chapter 7. Tree: + # [icon: material/graph-outline] + - chapter_tree/index.md + - 7.1 Binary Tree: chapter_tree/binary_tree.md + - 7.2 Binary Tree Traversal: chapter_tree/binary_tree_traversal.md + - 7.3 Array Representation of Tree: chapter_tree/array_representation_of_tree.md + - 7.4 Binary Search Tree: chapter_tree/binary_search_tree.md + - 7.5 AVL Tree *: chapter_tree/avl_tree.md + - 7.6 Summary: chapter_tree/summary.md + - Chapter 8. Heap: + # [icon: material/family-tree] + - chapter_heap/index.md + - 8.1 Heap: chapter_heap/heap.md + - 8.2 Building a Heap: chapter_heap/build_heap.md + - 8.3 Top-K Problem: chapter_heap/top_k.md + - 8.4 Summary: chapter_heap/summary.md + - Chapter 9. Graph: + # [icon: material/graphql] + - chapter_graph/index.md + - 9.1 Graph: chapter_graph/graph.md + - 9.2 Basic Graph Operations: chapter_graph/graph_operations.md + - 9.3 Graph Traversal: chapter_graph/graph_traversal.md + - 9.4 Summary: chapter_graph/summary.md + - Chapter 10. Searching: + # [icon: material/text-search] + - chapter_searching/index.md + - 10.1 Binary Search: chapter_searching/binary_search.md + - 10.2 Binary Search Insertion Point: chapter_searching/binary_search_insertion.md + - 10.3 Binary Search Boundaries: chapter_searching/binary_search_edge.md + - 10.4 Hashing Optimization Strategy: chapter_searching/replace_linear_by_hashing.md + - 10.5 Revisiting Search Algorithms: chapter_searching/searching_algorithm_revisited.md + - 10.6 Summary: chapter_searching/summary.md + - Chapter 11. Sorting: + # [icon: material/sort-ascending] + - chapter_sorting/index.md + - 11.1 Sorting Algorithms: chapter_sorting/sorting_algorithm.md + - 11.2 Selection Sort: chapter_sorting/selection_sort.md + - 11.3 Bubble Sort: chapter_sorting/bubble_sort.md + - 11.4 Insertion Sort: chapter_sorting/insertion_sort.md + - 11.5 Quick Sort: chapter_sorting/quick_sort.md + - 11.6 Merge Sort: chapter_sorting/merge_sort.md + - 11.7 Heap Sort: chapter_sorting/heap_sort.md + - 11.8 Bucket Sort: chapter_sorting/bucket_sort.md + - 11.9 Counting Sort: chapter_sorting/counting_sort.md + - 11.10 Radix Sort: chapter_sorting/radix_sort.md + - 11.11 Summary: chapter_sorting/summary.md + - Chapter 12. Divide and Conquer: + # [icon: material/set-split] + - chapter_divide_and_conquer/index.md + - 12.1 Divide and Conquer Algorithm: chapter_divide_and_conquer/divide_and_conquer.md + - 12.2 Divide and Conquer Search Strategy: chapter_divide_and_conquer/binary_search_recur.md + - 12.3 Building Tree Problems: chapter_divide_and_conquer/build_binary_tree_problem.md + - 12.4 Tower of Hanoi Problem: chapter_divide_and_conquer/hanota_problem.md + - 12.5 Summary: chapter_divide_and_conquer/summary.md + - Chapter 13. Backtracking: + # [icon: material/map-marker-path] + - chapter_backtracking/index.md + - 13.1 Backtracking Algorithm: chapter_backtracking/backtracking_algorithm.md + - 13.2 Permutations Problem: chapter_backtracking/permutations_problem.md + - 13.3 Subset Sum Problem: chapter_backtracking/subset_sum_problem.md + - 13.4 N-Queens Problem: chapter_backtracking/n_queens_problem.md + - 13.5 Summary: chapter_backtracking/summary.md + - Chapter 14. Dynamic Programming: + # [icon: material/table-pivot] + - chapter_dynamic_programming/index.md + - 14.1 Introduction to Dynamic Programming: chapter_dynamic_programming/intro_to_dynamic_programming.md + - 14.2 Features of DP Problems: chapter_dynamic_programming/dp_problem_features.md + - 14.3 DP Solution Approach: chapter_dynamic_programming/dp_solution_pipeline.md + - 14.4 0-1 Knapsack Problem: chapter_dynamic_programming/knapsack_problem.md + - 14.5 Unbounded Knapsack Problem: chapter_dynamic_programming/unbounded_knapsack_problem.md + - 14.6 Edit Distance Problem: chapter_dynamic_programming/edit_distance_problem.md + - 14.7 Summary: chapter_dynamic_programming/summary.md + - Chapter 15. Greedy: + # [icon: material/head-heart-outline] + - chapter_greedy/index.md + - 15.1 Greedy Algorithm: chapter_greedy/greedy_algorithm.md + - 15.2 Fractional Knapsack Problem: chapter_greedy/fractional_knapsack_problem.md + - 15.3 Maximum Capacity Problem: chapter_greedy/max_capacity_problem.md + - 15.4 Maximum Product Cutting Problem: chapter_greedy/max_product_cutting_problem.md + - 15.5 Summary: chapter_greedy/summary.md + - Chapter 16. Appendix: + # [icon: material/help-circle-outline] + - chapter_appendix/index.md + - 16.1 Installation of Programming Environment: chapter_appendix/installation.md + - 16.2 Contributing to the Creation: chapter_appendix/contribution.md + # [status: new] + - 16.3   Terminology: chapter_appendix/terminology.md + - References: + - chapter_reference/index.md diff --git a/mkdocs.yml b/mkdocs.yml index 41d4825306..40a6f6eaa7 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -71,9 +71,9 @@ extra: - name: 中文 link: / lang: zh - # - name: English - # link: /en/ - # lang: en + - name: English + link: /en/ + lang: en social: - icon: fontawesome/brands/github link: https://github.com/krahets From a6fc570bb7dabbf2c5fa4aed8c2b526ca41ccaab Mon Sep 17 00:00:00 2001 From: krahets Date: Thu, 2 Nov 2023 03:08:58 +0800 Subject: [PATCH 02/30] Update Dockerfile A few translation improvements. --- Dockerfile | 11 +++++++---- docs-en/chapter_array_and_linkedlist/linked_list.md | 2 +- .../iteration_and_recursion.md | 2 +- .../classification_of_data_structure.md | 2 +- mkdocs-en.yml | 2 +- 5 files changed, 11 insertions(+), 8 deletions(-) diff --git a/Dockerfile b/Dockerfile index 9febad0b62..58a77553e8 100644 --- a/Dockerfile +++ b/Dockerfile @@ -5,12 +5,15 @@ RUN pip install -i https://pypi.tuna.tsinghua.edu.cn/simple mkdocs-material==9.4 WORKDIR /app -COPY docs /app/build/docs COPY overrides /app/build/overrides +COPY docs /app/build/docs +COPY docs-en /app/build/docs-en COPY mkdocs.yml /app/mkdocs.yml +COPY mkdocs-en.yml /app/mkdocs-en.yml -RUN mkdocs build +RUN mkdocs build -f mkdocs.yml +RUN mkdocs build -f mkdocs-en.yml +WORKDIR /app/site EXPOSE 8000 - -CMD ["mkdocs", "serve", "-a", "0.0.0.0:8000"] +CMD ["python", "-m", "http.server", "8000"] diff --git a/docs-en/chapter_array_and_linkedlist/linked_list.md b/docs-en/chapter_array_and_linkedlist/linked_list.md index 1c79f0d49f..ca73518888 100644 --- a/docs-en/chapter_array_and_linkedlist/linked_list.md +++ b/docs-en/chapter_array_and_linkedlist/linked_list.md @@ -433,7 +433,7 @@ Traverses the linked list, finds the node whose value is `target`, and outputs t [file]{linked_list}-[class]{}-[func]{find} ``` -## Arrays Vs. Linked Lists +## Arrays vs. Linked Lists The table below summarizes and compares the various properties and operational efficiencies of arrays and linked lists. Since they use two opposite storage strategies, the various properties and operational efficiencies are also characterized by opposites. diff --git a/docs-en/chapter_computational_complexity/iteration_and_recursion.md b/docs-en/chapter_computational_complexity/iteration_and_recursion.md index ac80c2baf4..0878e09bfc 100644 --- a/docs-en/chapter_computational_complexity/iteration_and_recursion.md +++ b/docs-en/chapter_computational_complexity/iteration_and_recursion.md @@ -1,4 +1,4 @@ -# Iteration Vs. Recursion +# Iteration vs. Recursion In data structures and algorithms, it is common to repeat a task, which is closely related to the complexity of the algorithm. There are two basic program structures that we usually use to repeat a task: iteration and recursion. diff --git a/docs-en/chapter_data_structure/classification_of_data_structure.md b/docs-en/chapter_data_structure/classification_of_data_structure.md index 012e21e6db..a5778f608c 100644 --- a/docs-en/chapter_data_structure/classification_of_data_structure.md +++ b/docs-en/chapter_data_structure/classification_of_data_structure.md @@ -19,7 +19,7 @@ Nonlinear data structures can be further categorized into tree and graph structu - **Tree structure**: tree, heap, hash table with one-to-many relationships between elements. - **Graph**: graph with many-to-many relationships between elements. -## Physical Structure: Continuous Vs. Dispersed +## Physical Structure: Continuous vs. Dispersed In computers, memory and hard disk are the two main storage hardware devices. Hard disks are used primarily for long-term data storage, and are larger (usually up to the terabyte degree) and slower. Memory is used for temporary storage of data while running a program, and is faster, but has a smaller capacity (usually at the GB level). diff --git a/mkdocs-en.yml b/mkdocs-en.yml index 5cbb58caf1..f0209ce482 100644 --- a/mkdocs-en.yml +++ b/mkdocs-en.yml @@ -4,7 +4,7 @@ INHERIT: mkdocs.yml # Project information site_name: Hello Algo site_url: https://www.hello-algo.com/en/ -site_description: 动画图解、一键运行的数据结构与算法教程 +site_description: "Data Structures and Algorithms: Animated Illutrations and Off-the-Shelf Code" docs_dir: build/docs-en site_dir: site/en # Repository From 44d9da89d9b1827511c47d30b435e4c596698b13 Mon Sep 17 00:00:00 2001 From: krahets Date: Thu, 2 Nov 2023 03:35:01 +0800 Subject: [PATCH 03/30] Fix a badge logo. --- docs-en/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs-en/index.md b/docs-en/index.md index dfdb1e5257..07a6cb1ca2 100644 --- a/docs-en/index.md +++ b/docs-en/index.md @@ -22,7 +22,7 @@ hide:   - GitHub contributors + GitHub contributors

    From a17236528aa9ff3cb4776b5eddaefbfc41e46800 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Fri, 3 Nov 2023 04:08:04 +0800 Subject: [PATCH 04/30] Fix EN translation of chapter_appendix/terminology.md (#913) --- docs-en/chapter_appendix/terminology.md | 216 ++++++++++++------------ 1 file changed, 108 insertions(+), 108 deletions(-) diff --git a/docs-en/chapter_appendix/terminology.md b/docs-en/chapter_appendix/terminology.md index e3bc3a08b3..bf1b92208f 100644 --- a/docs-en/chapter_appendix/terminology.md +++ b/docs-en/chapter_appendix/terminology.md @@ -4,111 +4,111 @@ The table below lists important terms that appear in the book. It is recommended

    Table   Important Terms for Data Structures and Algorithms

    -| Chinese | English | -| ------------------------------ | ------------------------------ | -| algorithm | algorithm | -| data structure | data structure | -| asymptotic complexity analysis | asymptotic complexity analysis | -| time complexity | time complexity | -| space complexity | space complexity | -| iteration | iteration | -| recursion | recursion | -| tail recursion | tail recursion | -| recursion tree | recursion tree | -| big $O$ notation | big-$O$ notation | -| asymptotic upper bound | asymptotic upper bound | asymptotic upper bound | asymptotic upper bound | -| true form | -| inverse code | 1's complement code | -| 2's complement code | 2's complement code | -| array | array | -| index | index | -| linked lists | linked list | -| linked list nodes | list node | -| lists | list | -| dynamic arrays | dynamic array | -| stack | stack | -| queues | queue | -| double-ended queue | double-ended queue | double-ended queue | double-ended queue | double-ended queue | double-ended queue | double-ended queue | double-ended queue | double-ended queue | double-ended queue | -| hash table | hash table | -| bucket | bucket | -| hash functions | hash function | -| hash collision | hash collision | -| load factor | load factor | load factor | load factor | -| separate chaining | separate chaining | separate chaining | -| open addressing | open addressing | -| linear probing | linear probing | linear probing | -| lazy deletion | lazy deletion | -| binary tree | binary tree | -| tree nodes | tree node | -| left-child node | left-child node | -| right-child node | right-child node | -| parent node | parent node | -| left subtree | left subtree | -| right subtree | right subtree | -| root node | root node | -| leaf node | leaf node | -| edge | edge | edge | edge | -| level | level | -| degree | degree | degree | degree | degree | -| height | height | -| depth | depth | -| perfect binary tree | perfect binary tree | -| complete binary tree | complete binary tree | -| full binary tree | full binary tree | -| balanced binary trees | balanced binary tree | -| avl tree | avl tree | avl tree | avl tree | -| red-black tree | red-black tree | -| level-order traversal | level-order traversal | level-order traversal | level-order traversal | -| breadth-first traversal | breadth-first traversal | -| depth-first traversal | depth-first traversal | depth-first traversal | -| binary search tree | binary search tree | -| balanced binary search tree | balanced binary search tree | -| balance factor | balance factor | -| heap | heap | heap | heap | heap | heap | heap | heap | heap | heap | heap | -| max heap | max heap | -| min heap | min heap | -| priority queue | priority queue | -| heapify | heapify | heapify | heapify | heapify | heapify | heapify | heapify | heapify | heapify | heapify | heapify | -| graph | graph | -| vertex | vertex | -| undirected graph | undirected graph | -| directed graphs | directed graphs | -| connected graph | connected graph | -| connected graph | disconnected graph | -| weighted graph | weighted graph | weighted graph | weighted graph | -| adjacency | adjacency | -| path | path | -| in-degree | in-degree | in-degree | -| out-degree | out-degree | out-degree | -| adjacency matrix | adjacency matrix | -| adjacency list | adjacency list | adjacency list | adjacency list | -| breadth-first search | breadth-first search | -| depth-first search | depth-first search | -| binary search | binary search | -| searching algorithm | searching algorithm | -| sorting algorithm | sorting algorithm | -| selection sort | selection sort | -| bubble sort | bubble sort | -| insertion sort | insertion sort | -| quick sort | quick sort | -| merge sort | merge sort | -| heap sort | heap sort | -| bucket sort | bucket sort | -| counting sort | counting sort | -| radix sort | radix sort | radix sort | -| divide and conquer | -| hanota problem | hanota problem | -| backtracking algorithm | backtracking algorithm | -| constraints | constraints | -| solution | solution | -| state | state | -| pruning | pruning | -| permutations problem | permutations problem | -| subset-sum problem | subset-sum problem | -| n-queens problem | N-queens problem | -| dynamic programming | dynamic programming | -| initial state | initial state | -| state-trasition equation | state-trasition equation | -| knapsack problem | knapsack problem | -| edit distance problem | edit distance problem | -| greedy algorithm | greedy algorithm | +| 中文 | English | +| -------------- | ------------------------------ | +| 算法 | algorithm | +| 数据结构 | data structure | +| 渐近复杂度分析 | asymptotic complexity analysis | +| 时间复杂度 | time complexity | +| 空间复杂度 | space complexity | +| 迭代 | iteration | +| 递归 | recursion | +| 尾递归 | tail recursion | +| 递归树 | recursion tree | +| 大 $O$ 记号 | big-$O$ notation | +| 渐近上界 | asymptotic upper bound | +| 原码 | true form | +| 反码 | 1's complement code | +| 补码 | 2's complement code | +| 数组 | array | +| 索引 | index | +| 链表 | linked list | +| 链表节点 | list node | +| 列表 | list | +| 动态数组 | dynamic array | +| 栈 | stack | +| 队列 | queue | +| 双向队列 | double-ended queue | +| 哈希表 | hash table | +| 桶 | bucket | +| 哈希函数 | hash function | +| 哈希冲突 | hash collision | +| 负载因子 | load factor | +| 链式地址 | separate chaining | +| 开放寻址 | open addressing | +| 线性探测 | linear probing | +| 懒删除 | lazy deletion | +| 二叉树 | binary tree | +| 树节点 | tree node | +| 左子节点 | left-child node | +| 右子节点 | right-child node | +| 父节点 | parent node | +| 左子树 | left subtree | +| 右子树 | right subtree | +| 根节点 | root node | +| 叶节点 | leaf node | +| 边 | edge | +| 层 | level | +| 度 | degree | +| 高度 | height | +| 深度 | depth | +| 完美二叉树 | perfect binary tree | +| 完全二叉树 | complete binary tree | +| 完满二叉树 | full binary tree | +| 平衡二叉树 | balanced binary tree | +| AVL 树 | AVL tree | +| 红黑树 | red-black tree | +| 层序遍历 | level-order traversal | +| 广度优先遍历 | breadth-first traversal | +| 深度优先遍历 | depth-first traversal | +| 二叉搜索树 | binary search tree | +| 平衡二叉搜索树 | balanced binary search tree | +| 平衡因子 | balance factor | +| 堆 | heap | +| 大顶堆 | max heap | +| 小顶堆 | min heap | +| 优先队列 | priority queue | +| 堆化 | heapify | +| 图 | graph | +| 顶点 | vertex | +| 无向图 | undirected graph | +| 有向图 | directed graph | +| 连通图 | connected graph | +| 非连通图 | disconnected graph | +| 有权图 | weighted graph | +| 邻接 | adjacency | +| 路径 | path | +| 入度 | in-degree | +| 出度 | out-degree | +| 邻接矩阵 | adjacency matrix | +| 邻接表 | adjacency list | +| 广度优先搜索 | breadth-first search | +| 深度优先搜索 | depth-first search | +| 二分查找 | binary search | +| 搜索算法 | searching algorithm | +| 排序算法 | sorting algorithm | +| 选择排序 | selection sort | +| 冒泡排序 | bubble sort | +| 插入排序 | insertion sort | +| 快速排序 | quick sort | +| 归并排序 | merge sort | +| 堆排序 | heap sort | +| 桶排序 | bucket sort | +| 计数排序 | counting sort | +| 基数排序 | radix sort | +| 分治 | divide and conquer | +| 汉诺塔问题 | hanota problem | +| 回溯算法 | backtracking algorithm | +| 约束 | constraint | +| 解 | solution | +| 状态 | state | +| 剪枝 | pruning | +| 全排列问题 | permutations problem | +| 子集和问题 | subset-sum problem | +| N 皇后问题 | N-queens problem | +| 动态规划 | dynamic programming | +| 初始状态 | initial state | +| 状态转移方程 | state-trasition equation | +| 背包问题 | knapsack problem | +| 编辑距离问题 | edit distance problem | +| 贪心算法 | greedy algorithm | From ddec2de9ef1dbad756df5ffa7051574780283d92 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Fri, 3 Nov 2023 05:52:36 +0800 Subject: [PATCH 05/30] Update README.md --- README.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/README.md b/README.md index ec1587ef69..912d907b54 100644 --- a/README.md +++ b/README.md @@ -32,6 +32,10 @@ This project aims to create an open-source, free, and beginner-friendly tutorial - The source code can be run with a single click, helping readers improve their programming skills, understand the working principles of algorithms, and the underlying implementation of data structures. - We encourage readers to learn together, ask questions, and comment. Replies are usually received within two days. +> **Important** +> +> We are working on Chinese-to-English translation. For more information please see [#914](https://github.com/krahets/hello-algo/issues/914). + If this book is helpful to you, please give it a Star :star: in the top right corner of the page to show your support. Thank you! ## Praise From c82b2e7abb1c7ddca0d9c7b35e92b7a97325343e Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Fri, 3 Nov 2023 15:38:32 +0800 Subject: [PATCH 06/30] Update README.md --- README.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/README.md b/README.md index 912d907b54..e6aacacd46 100644 --- a/README.md +++ b/README.md @@ -24,6 +24,10 @@

    +> **Important** +> +> We are working on Chinese-to-English translation. For more information please see [#914](https://github.com/krahets/hello-algo/issues/914). + ## The Book This project aims to create an open-source, free, and beginner-friendly tutorial on data structures and algorithms. @@ -32,10 +36,6 @@ This project aims to create an open-source, free, and beginner-friendly tutorial - The source code can be run with a single click, helping readers improve their programming skills, understand the working principles of algorithms, and the underlying implementation of data structures. - We encourage readers to learn together, ask questions, and comment. Replies are usually received within two days. -> **Important** -> -> We are working on Chinese-to-English translation. For more information please see [#914](https://github.com/krahets/hello-algo/issues/914). - If this book is helpful to you, please give it a Star :star: in the top right corner of the page to show your support. Thank you! ## Praise From a0f140a48ff2cf94dbc42900e6004a0efc099edd Mon Sep 17 00:00:00 2001 From: Yuelin Xin Date: Sat, 11 Nov 2023 12:24:40 +0000 Subject: [PATCH 07/30] translation: Refined the automated translation of README (#932) * refined the automated translation of README * Update index.md * Update mkdocs-en.yml --------- Co-authored-by: Yudong Jin --- README.md | 60 +++++++++++++++++++++++++++++------------------- docs-en/index.md | 2 +- mkdocs-en.yml | 2 +- 3 files changed, 39 insertions(+), 25 deletions(-) diff --git a/README.md b/README.md index e6aacacd46..c2beb24ff0 100644 --- a/README.md +++ b/README.md @@ -7,58 +7,72 @@

    hello-algo-typing-svg
    - Data Structures and Algorithms: Animated Illutrations and Off-the-Shelf Code + Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code

    - - + - - + +

    + +

    + +

    - - + + + + + + + + + + + + +

    > **Important** > > We are working on Chinese-to-English translation. For more information please see [#914](https://github.com/krahets/hello-algo/issues/914). -## The Book +## About -This project aims to create an open-source, free, and beginner-friendly tutorial on data structures and algorithms. +This project aims to create a free, open-source, and beginner-friendly crash course for data structures and algorithms. -- The book uses animated illustrations to make the content clear, easy to understand, and provides a smooth learning curve, guiding beginners through the knowledge map of data structures and algorithms. -- The source code can be run with a single click, helping readers improve their programming skills, understand the working principles of algorithms, and the underlying implementation of data structures. -- We encourage readers to learn together, ask questions, and comment. Replies are usually received within two days. +- Animated illustrations, easy-to-understand content, and a smooth learning curve help beginners explore the "knowledge map" of data structures and algorithms. +- Run code with just one click, helping readers improve their programming skills and understand the working principle of algorithms and the underlying implementation of data structures. +- We encourage readers to help each other. Questions and comments are usually replied to within two days. -If this book is helpful to you, please give it a Star :star: in the top right corner of the page to show your support. Thank you! +If you find this book helpful, please give it a Star :star: to support us, thank you! -## Praise +## Recommendations -> “An accessible and easy-to-understand introduction to data structures and algorithms, guiding readers to learn actively and hands-on. Highly recommended for beginners in algorithms.” +> "An easy-to-understand introductory book on data structures and algorithms, which guides readers to learn by thinking and hands-on, strongly recommended for algorithm beginners." > -> **—— Junhui Deng, Professor of Computer Science at Tsinghua University** +> **—— Junhui Deng, Professor of Computer Science, Tsinghua University** -> “If I had 'Hello Algorithm' when I was learning data structures and algorithms, it would have been 10 times easier!” +> "If I had *Hello Algorithm* when I was learning data structures and algorithms, it would have been 10 times easier!" > -> **—— Mu Li, Senior Principal Scientist at Amazon** +> **—— Mu Li, Senior Principal Scientist, Amazon** ## Contribution This open-source book is continuously being updated, and we welcome your participation in this project to provide better learning content for our readers. -- [Content Correction](https://www.hello-algo.com/chapter_appendix/contribution/): Please help us correct or point out issues in the comments section such as grammatical errors, missing content, ambiguities, invalid links, or code bugs. -- [Code Translation](https://github.com/krahets/hello-algo/issues/15): We look forward to your contributions of code in various programming languages. We currently support Python, Java, C++, Go, JavaScript, and 11 other programming languages. -- [Whole Book Translation](https://github.com/krahets/hello-algo/tree/en): We invite you to join our Chinese-to-English translation team. Members are mainly from computer-related majors, English majors, and native English speakers. +- [Content Correction](https://www.hello-algo.com/chapter_appendix/contribution/): Please help us correct or point out mistakes in the comments section such as grammatical errors, missing content, ambiguities, invalid links, or code bugs. +- [Code Translation](https://github.com/krahets/hello-algo/issues/15): We look forward to your contributions of code in various programming languages. We currently support Python, Java, C++, Go, JavaScript, and 12 other programming languages. +- [Book Translation](https://github.com/krahets/hello-algo/tree/en): We would love to invite you to join our Chinese-to-English translation team. Members are mainly from computer science related majors, English majors, and native English speakers. -We welcome your valuable suggestions and feedback. If you have any questions, please submit Issues or contact us via WeChat at `krahets-jyd`. +We welcome your valuable suggestions and feedback. If you have any questions, please submit Issues or reach out via WeChat: `krahets-jyd`. -Thanks to each contributor of this open-source book. It is their selfless dedication that has made this book better. They are: +We would like to dedicate our thank to all the contributors of this book. It is their selfless dedication that has made this book better. They are:

    diff --git a/docs-en/index.md b/docs-en/index.md index 07a6cb1ca2..b39090eec9 100644 --- a/docs-en/index.md +++ b/docs-en/index.md @@ -14,7 +14,7 @@ hide:

    Hello Algo

    -

    Data Structures and Algorithms: Animated Illutrations and Off-the-Shelf Code

    +

    Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code

    diff --git a/mkdocs-en.yml b/mkdocs-en.yml index f0209ce482..5ab4de665d 100644 --- a/mkdocs-en.yml +++ b/mkdocs-en.yml @@ -4,7 +4,7 @@ INHERIT: mkdocs.yml # Project information site_name: Hello Algo site_url: https://www.hello-algo.com/en/ -site_description: "Data Structures and Algorithms: Animated Illutrations and Off-the-Shelf Code" +site_description: "Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code" docs_dir: build/docs-en site_dir: site/en # Repository From ef5cc2908179105c7803cbc91b832ca0bbdd724a Mon Sep 17 00:00:00 2001 From: Phoenix Xie Date: Mon, 13 Nov 2023 16:47:08 +1030 Subject: [PATCH 08/30] translate: Embellish chapter_computational_complexity/index.md (#940) --- docs-en/chapter_computational_complexity/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/docs-en/chapter_computational_complexity/index.md b/docs-en/chapter_computational_complexity/index.md index f6b817a3f4..ce38a2b4d6 100644 --- a/docs-en/chapter_computational_complexity/index.md +++ b/docs-en/chapter_computational_complexity/index.md @@ -8,6 +8,6 @@ !!! abstract - Complexity analysis is like a space-time guide in the vast algorithmic universe. + Complexity analysis is like a space-time guide in the vast universe of algorithms. - It leads us to delve deeper into the two dimensions of time and space to find a more elegant solution. + It leads us to explore deeply in the dimensions of time and space, in search of more elegant solutions. From abe55db00e846afa7f419e30c70283af5a929ab3 Mon Sep 17 00:00:00 2001 From: Phoenix Xie Date: Wed, 15 Nov 2023 23:13:02 +1030 Subject: [PATCH 09/30] translation: Update chapter_computational_complexity/performance_evaluation.md (#943) * Update performance_evaluation.md * Update performance_evaluation.md * Update performance_evaluation.md change 'methods' to 'approaches' on line 15 * Update performance_evaluation.md on line 21, change the sentence to 'the results could be the opposite on another computer with different specifications.' * Update performance_evaluation.md delete two short sentence on line 5 and 6 * Update performance_evaluation.md change `unavoidable` to `inevitable` on line 48 * Update performance_evaluation.md small changes on line 23 --- .../performance_evaluation.md | 51 ++++++++++--------- 1 file changed, 26 insertions(+), 25 deletions(-) diff --git a/docs-en/chapter_computational_complexity/performance_evaluation.md b/docs-en/chapter_computational_complexity/performance_evaluation.md index c1237b3705..a6af765399 100644 --- a/docs-en/chapter_computational_complexity/performance_evaluation.md +++ b/docs-en/chapter_computational_complexity/performance_evaluation.md @@ -1,48 +1,49 @@ # Evaluation Of Algorithm Efficiency -In the algorithm design, we pursue the following two levels successively. +In algorithm design, we aim to achieve two goals in succession: -1. **Finding the solution to the problem**: the algorithm needs to reliably find the correct solution to the problem within a specified input range. -2. **Seek optimal solutions**: there may be multiple solutions to the same problem and we want to find the most efficient algorithm possible. +1. **Finding a Solution to the Problem**: The algorithm needs to reliably find the correct solution within the specified input range. +2. **Seeking the Optimal Solution**: There may be multiple ways to solve the same problem, and our goal is to find the most efficient algorithm possible. -That is to say, under the premise of being able to solve the problem, algorithm efficiency has become the main evaluation index for measuring the merits of algorithms, which includes the following two dimensions. +In other words, once the ability to solve the problem is established, the efficiency of the algorithm emerges as the main benchmark for assessing its quality, which includes the following two aspects. -- **Time efficiency**: how fast or slow the algorithm runs. -- **Space efficiency**: the amount of memory space occupied by the algorithm. +- **Time Efficiency**: The speed at which an algorithm runs. +- **Space Efficiency**: The amount of memory space the algorithm consumes. -In short, our goal is to design data structures and algorithms that are "fast and frugal". Evaluating the efficiency of algorithms effectively is crucial because only then can we compare various algorithms to guide the algorithm design and optimization process. +In short, our goal is to design data structures and algorithms that are both "fast and economical". Effectively evaluating algorithm efficiency is crucial, as it allows for the comparison of different algorithms and guides the design and optimization process. -There are two main types of efficiency assessment methods: practical testing, and theoretical estimation. +There are mainly two approaches for assessing efficiency: practical testing and theoretical estimation. -## Practical Tests +## Practical Testing -Suppose we now have algorithm `A` and algorithm `B` , both of which solve the same problem, and we now need to compare the efficiency of these two algorithms. The most direct way is to find a computer, run these two algorithms, and monitor and record their running time and memory usage. This type of evaluation reflects the real situation, but also has major limitations. +Let's consider a scenario where we have two algorithms, `A` and `B`, both capable of solving the same problem. To compare their efficiency, the most direct method is to use a computer to run both algorithms while monitoring and recording their execution time and memory usage. This approach provides a realistic assessment of their performance, but it also has significant limitations. -On the one hand, **it is difficult to exclude disturbing factors in the test environment**. Hardware configuration affects the performance of algorithms. For example, in a certain computer, the running time of algorithm `A` is shorter than that of algorithm `B`; however, in another computer with a different configuration, we may get the opposite test results. This means that we need to test on a variety of machines and count the average efficiency, which is unrealistic. +On one hand, it's challenging to eliminate the interference of the test environment. Hardware configurations can significantly affect the performance of algorithms. For instance, on one computer, Algorithm `A` might run faster than Algorithm `B`, but the results could be the opposite on another computer with different specifications. This means we would need to conduct tests on a variety of machines and calculate an average efficiency, which is impractical. -On the other hand, **it is very resource-consuming to carry out a full test**. As the amount of input data varies, the algorithms show different efficiencies. For example, when the amount of input data is small, the running time of algorithm `A` is less than that of algorithm `B`; while the test results may be opposite when the amount of input data is large. Therefore, in order to obtain convincing conclusions, we need to test various sizes of input data, which requires a lot of computational resources. +Furthermore, conducting comprehensive tests is resource-intensive. The efficiency of algorithms can vary with different volumes of input data. For example, with smaller data sets, Algorithm A might run faster than Algorithm B; however, this could change with larger data sets. Therefore, to reach a convincing conclusion, it's necessary to test a range of data sizes, which requires excessive computational resources. -## Theoretical Estimates +## Theoretical Estimation -Since practical tests have large limitations, we can consider evaluating the efficiency of an algorithm by only a few calculations. This estimation method is called "asymptotic complexity analysis", or "complexity analysis" for short. +Given the significant limitations of practical testing, we can consider assessing algorithm efficiency solely through calculations. This method of estimation is known as 'asymptotic complexity analysis,' often simply referred to as 'complexity analysis. -The complexity analysis reflects the relationship between the time (space) resources required to run the algorithm and the size of the input data. **It describes the growing trend in the time and space required for the execution of an algorithm as the size of the input data increases**. This definition is a bit of a mouthful, and we can understand it by dividing it into three key points. +Complexity analysis illustrates the relationship between the time (and space) resources required by an algorithm and the size of its input data. **It describes the growing trend in the time and space required for the execution of an algorithm as the size of the input data increases**. This definition might sound a bit complex, so let's break it down into three key points for easier understanding. -- The terms "time and space resources" correspond to "time complexity" and "space complexity" respectively. -- "As the size of the input data increases" means that the complexity reflects the relationship between the efficiency of the algorithm operation and the volume of the input data. -- "Trends in the growth of time and space" means that the analysis of complexity is not concerned with specific values of running time or occupied space, but rather with the "speed" of the growth of time or space. +- In complexity analysis, 'time and space' directly relate to 'time complexity' and 'space complexity,' respectively. +- The statement "as the size of the input data increases" highlights that complexity analysis examines the interplay between the size of the input data and the algorithm's efficiency. +- Lastly, the phrase "the growing trend in time and space required" emphasizes that the focus of complexity analysis is not on the specific values of running time or space occupied, but on the rate at which these requirements increase with larger input sizes. -**Complexity analysis overcomes the drawbacks of practical testing methods in two ways**. +**Complexity analysis overcomes the drawbacks of practical testing methods in two key ways:**. -- It is independent of the test environment and the analysis results are applicable to all operating platforms. -- It can reflect the efficiency of algorithms under different data volumes, especially the performance of algorithms under large data volumes. +- It is independent of the testing environment, meaning the analysis results are applicable across all operating platforms. +- It effectively demonstrates the efficiency of algorithms with varying data volumes, particularly highlighting performance in large-scale data scenarios. !!! tip - If you're still confused about the concept of complexity, don't worry, we'll cover it in detail in subsequent chapters. + If you're still finding the concept of complexity confusing, don't worry. We will cover it in more detail in the subsequent chapters. -Complexity analysis provides a "yardstick" for evaluating the efficiency of an algorithm, allowing us to measure the time and space resources required to execute an algorithm and to compare the efficiency of different algorithms. +Complexity analysis provides us with a 'ruler' for evaluating the efficiency of algorithms, enabling us to measure the time and space resources required to execute a given algorithm and to compare the efficiency of different algorithms. -Complexity is a mathematical concept that may be abstract and relatively difficult for beginners to learn. From this point of view, complexity analysis may not be very suitable as the first introduction. However, when we discuss the characteristics of a particular data structure or algorithm, it is difficult to avoid analyzing its running speed and space usage. +Complexity is a mathematical concept that might seem abstract and somewhat challenging for beginners. From this perspective, introducing complexity analysis at the very beginning may not be the most suitable approach. However, when discussing the characteristics of a particular data structure or algorithm, analyzing its operational speed and space usage is often inevitable. + +Therefore, it is recommended that before diving deeply into data structures and algorithms, **one should first gain a basic understanding of complexity analysis. This foundational knowledge will facilitate the complexity analysis of simple algorithms.** -To summarize, it is recommended that you build a preliminary understanding of complexity analysis before you dive into data structures and algorithms **to be able to complete complexity analysis of simple algorithms**. From 77b7967e3b83c44d6aaad025c58fc99dfbe67e5a Mon Sep 17 00:00:00 2001 From: Sizhuo Long Date: Tue, 21 Nov 2023 21:36:07 +0800 Subject: [PATCH 10/30] translation: Update terminology and improve readability in preface summary (#954) * Update terminology and improve readability in preface summary This commit made a few adjustments in the 'summary.md' file for clearer and more accessible language. "Brushing tool library" was replaced with "Coding Toolkit" to better reflect common terminology. Also, advice for beginners in algorithm learning journey was reformulated to imply a more positive approach avoiding detours and common pitfalls. The section related to the discussion forum was rewritten to sound more inviting to readers. * Format --- docs-en/chapter_preface/summary.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/docs-en/chapter_preface/summary.md b/docs-en/chapter_preface/summary.md index 2ea2c18ff6..1f1b6c315f 100644 --- a/docs-en/chapter_preface/summary.md +++ b/docs-en/chapter_preface/summary.md @@ -1,8 +1,8 @@ # Summary -- The main audience of this book is beginners in algorithm. If you already have some basic knowledge, this book can help you systematically review your algorithm knowledge, and the source code in this book can also be used as a "brushing tool library". +- The main audience of this book is beginners in algorithm. If you already have some basic knowledge, this book can help you systematically review your algorithm knowledge, and the source code in this book can also be used as a "Coding Toolkit". - The book consists of three main sections, Complexity Analysis, Data Structures, and Algorithms, covering most of the topics in the field. -- For newcomers to algorithms, it is crucial to read an introductory book in the beginning stages to take the road less traveled. +- For newcomers to algorithms, it is crucial to read an introductory book in the beginning stages to avoid many detours or common pitfalls. - Animations and graphs within the book are usually used to introduce key points and difficult knowledge. These should be given more attention when reading the book. - Practice is the best way to learn programming. It is highly recommended that you run the source code and type in the code yourself. -- Each chapter in the web version of the book has a discussion forum, so feel free to share your doubts and insights. +- Each chapter in the web version of this book features a discussion forum, and you are welcome to share your questions and insights at any time. \ No newline at end of file From 22edeaa840fb68a757356ded578dcf733d354748 Mon Sep 17 00:00:00 2001 From: krahets Date: Tue, 21 Nov 2023 23:52:40 +0800 Subject: [PATCH 11/30] Optimize the translation of chapter_introduction/algorithms_are_everywhere. --- .../algorithms_are_everywhere.md | 46 +++++++++---------- 1 file changed, 23 insertions(+), 23 deletions(-) diff --git a/docs-en/chapter_introduction/algorithms_are_everywhere.md b/docs-en/chapter_introduction/algorithms_are_everywhere.md index 6c7d691a44..809b2b980e 100644 --- a/docs-en/chapter_introduction/algorithms_are_everywhere.md +++ b/docs-en/chapter_introduction/algorithms_are_everywhere.md @@ -1,14 +1,14 @@ # Algorithms Are Everywhere -When we hear the word "algorithm", we naturally think of math. In reality, however, many algorithms do not involve complex math, but rely more on basic logic, which can be found in our daily lives. +When we hear the word "algorithm", we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which is ubiquitous in our daily lives. -Before formally exploring algorithms, there is an interesting fact worth sharing:**You have already learned many algorithms without realizing it and are used to applying them in your daily life**. Below, I will give a few concrete examples to confirm this. +Before we formally discuss algorithms, an interesting fact is worth sharing: **you have already learned many algorithms unconsciously and have become accustomed to applying them in your daily life**. Below, I will give a few specific examples to prove this point. -**Example 1: Consulting the dictionary**. In a dictionary, each Chinese character corresponds to a Pinyin character, and the dictionary is organized by Pinyin alphabetical order. Suppose we need to look up a character whose pinyin initial letter is $r$, we usually realize it as shown in the figure below. +**Example 1: Looking Up a Dictionary**. In a dictionary, each Chinese character corresponds to a pinyin, and the dictionary is arranged in alphabetical order of pinyin. Suppose we need to find a character whose pinyin starts with the letter $r$. This is usually achieved in the following way: -1. Go through about half of the pages of the dictionary to see what the first letter of the page is, assuming the first letter is $m$ . -2. Since $r$ comes after $m$ in the pinyin alphabet, the first half of the dictionary is excluded and the search is narrowed down to the second half. -3. Repeat step `1.` and step `2.` until you find the page number with the initial letter $r$. +1. Open the dictionary to about halfway and check the first letter on the page, assuming it is $m$. +2. Since $r$ comes after $m$ in the pinyin alphabet, we exclude the first half of the dictionary and narrow the search to the second half. +3. Repeat steps `1.` and `2.` until you find the page where the pinyin starts with $r$. === "<1>" ![Dictionary search step](algorithms_are_everywhere.assets/binary_search_dictionary_step1.png) @@ -25,32 +25,32 @@ Before formally exploring algorithms, there is an interesting fact worth sharing === "<5>" ![binary_search_dictionary_step5](algorithms_are_everywhere.assets/binary_search_dictionary_step5.png) -Looking up the dictionary, a necessary skill for elementary school students, is actually the famous binary search algorithm. From the data structure point of view, we can regard the dictionary as a sorting algorithm, and from the algorithm point of view, we can regard the above series of operations as "binary search". +The skill of looking up a dictionary, essential for elementary school students, is actually the famous binary search algorithm. From the perspective of data structures, we can view the dictionary as a sorted "array"; from the perspective of algorithms, the series of operations in looking up a dictionary can be seen as "binary search". -**Example 2: Organize Poker**. When we play poker, we need to organize the playing cards so that they are arranged from smallest to largest in each game, the implementation process is shown in the figure below. +**Example 2: Organizing Playing Cards**. When playing cards, we need to arrange the cards in ascending order each game, as shown in the following process. -1. Divide the playing cards into "ordered" and "unordered" parts, and assume that the leftmost card in the initial state is already ordered. -2. Draw a playing card from the unordered section and insert it into the correct position in the ordered section; when finished, the leftmost 2 playing cards are in order. -3. Keep cycling through the steps `2.` , inserting a playing card from the unordered part to the ordered part each round until all cards are in order. +1. Divide the playing cards into "ordered" and "unordered" parts, assuming initially that the leftmost card is already ordered. +2. Take out a card from the unordered part and insert it into the correct position in the ordered part; after this, the leftmost two cards are in order. +3. Continue looping step `2.`, each round inserting one card from the unordered part into the ordered part, until all cards are ordered. -![Poker sorting steps](algorithms_are_everywhere.assets/playing_cards_sorting.png) +![Playing cards sorting process](algorithms_are_everywhere.assets/playing_cards_sorting.png) -The above method of organizing playing cards is essentially an "insertion sort" algorithm, which is very efficient when dealing with small data sets. Insertion sort can be found in the sorting library functions of many programming languages. +The above method of organizing playing cards is essentially the "insertion sort" algorithm, which is very efficient for small datasets. Many programming languages' sorting library functions include insertion sort. -**Example 3: Currency change**. Suppose we purchased $69$ yuan of goods at the supermarket and gave the cashier $100$ yuan, then the cashier needs to find us $31$ yuan. He will naturally complete the thinking shown in the figure below. +**Example 3: Making Change**. Suppose we buy goods worth $69$ at a supermarket and give the cashier $100$, then the cashier needs to give us $31$ in change. They would naturally complete the thought process as shown below. -1. Options are currencies with smaller denominations than $31$ yuan, including $1$ yuan, $5$ yuan, $10$ yuan, $20$ yuan. -2. Take the largest $20$ dollars from the available options, leaving $31 - 20 = 11$ dollars. -3. Take the largest $10$ from the remaining available options, leaving $11 - 10 = 1$. -4. Take the largest $1$ from the remaining available options, leaving $1 - 1 = 0$. -5. Complete the change and the solution is $20 + 10 + 1 = 31$. +1. The options are currencies smaller than $31$, including $1$, $5$, $10$, and $20$. +2. Take out the largest $20$ from the options, leaving $31 - 20 = 11$. +3. Take out the largest $10$ from the remaining options, leaving $11 - 10 = 1$. +4. Take out the largest $1$ from the remaining options, leaving $1 - 1 = 0$. +5. Complete the change-making, with the solution being $20 + 10 + 1 = 31$. -![Currency change process](algorithms_are_everywhere.assets/greedy_change.png) +![Change making process](algorithms_are_everywhere.assets/greedy_change.png) -In each of the above steps, we take the best option available at the time (using as large a denomination of currency as possible) and end up with a feasible solution for finding change. From a data structure and algorithm perspective, this approach is essentially a "greedy algorithm". +In the steps above, we make the best choice at each step (using the largest denomination possible), ultimately arriving at a feasible change-making solution. From the perspective of data structures and algorithms, this method is essentially a "greedy" algorithm. -Algorithms are essential to the solution of almost any problem, from cooking a dish to interstellar voyages. The advent of computers has allowed us to program data structures to be stored in memory while writing code that calls the CPU and GPU to execute algorithms. In this way, we are able to transfer the problems of our lives to computers and solve all kinds of complex problems in a more efficient way. +From cooking a dish to interstellar travel, almost all problem-solving is inseparable from algorithms. The advent of computers allows us to store data structures in memory and write code to call CPUs and GPUs to execute algorithms. In this way, we can transfer problems from life to computers, solving various complex issues more efficiently. !!! tip - If you still feel half-aware of concepts such as data structures, algorithms, arrays, and binary searches, read on as this book will guide you through the halls of knowledge of data structures and algorithms. + If you still feel only partially informed about concepts like data structures, algorithms, arrays, and binary search, please continue reading. This book will guide you into the hall of knowledge of data structures and algorithms. From 3b7953ce421c97b6fc85f44dc5cd8e84def4cfe6 Mon Sep 17 00:00:00 2001 From: krahets Date: Wed, 22 Nov 2023 00:19:44 +0800 Subject: [PATCH 12/30] Add .gitignore to Java subfolder. --- codes/java/.gitignore | 1 + 1 file changed, 1 insertion(+) create mode 100644 codes/java/.gitignore diff --git a/codes/java/.gitignore b/codes/java/.gitignore new file mode 100644 index 0000000000..378eac25d3 --- /dev/null +++ b/codes/java/.gitignore @@ -0,0 +1 @@ +build From f0c4afe53b319ec4a452c7a2389cd8d8a354211b Mon Sep 17 00:00:00 2001 From: krahets Date: Mon, 4 Dec 2023 18:28:38 +0800 Subject: [PATCH 13/30] Update the button assets. --- README.md | 4 ++-- docs-en/index.assets/btn_download_code_dark.png | Bin 2878 -> 0 bytes docs-en/index.assets/btn_download_code_light.png | Bin 3165 -> 0 bytes docs-en/index.assets/btn_download_pdf_dark.png | Bin 2283 -> 0 bytes docs-en/index.assets/btn_download_pdf_dark.svg | 1 + docs-en/index.assets/btn_download_pdf_light.png | Bin 2526 -> 0 bytes docs-en/index.assets/btn_read_online_dark.png | Bin 3284 -> 0 bytes docs-en/index.assets/btn_read_online_dark.svg | 1 + docs-en/index.assets/btn_read_online_light.png | Bin 4061 -> 0 bytes 9 files changed, 4 insertions(+), 2 deletions(-) delete mode 100644 docs-en/index.assets/btn_download_code_dark.png delete mode 100644 docs-en/index.assets/btn_download_code_light.png delete mode 100644 docs-en/index.assets/btn_download_pdf_dark.png create mode 100644 docs-en/index.assets/btn_download_pdf_dark.svg delete mode 100644 docs-en/index.assets/btn_download_pdf_light.png delete mode 100644 docs-en/index.assets/btn_read_online_dark.png create mode 100644 docs-en/index.assets/btn_read_online_dark.svg delete mode 100644 docs-en/index.assets/btn_read_online_light.png diff --git a/README.md b/README.md index c2beb24ff0..5a39683947 100644 --- a/README.md +++ b/README.md @@ -12,9 +12,9 @@

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zHotidmWoRVV>!!YPUXpKCg;_+!cpGeAr_mZI%VLqq(_Ptm~)I@4fBvH>&?zzcuj&; z8r)nq^b5Pw`Un7-#{62MC+70=8@au{FkF%wUFo=~-nGq=MH47hY;57Mv&IVVTg{Z4 zr{*0SZLBgTSa5o9p!HXGFRw;Gj)S~gu|Mn~>?|*9A-Zek!y_euA}4e52T7~E?!Akp z^b~cCU1xwP)O+(?TuaPoBUbC-^jyRK_uU0YpL8HF?^bSJ_Z4e@!aWanPicr8q|8Ug zWT(Z473%k;IU_JgYwOSEHkSOA6tJV#ZI*D$6@?9O*dF|&EPys2=9&MtXdpVe90G35 zLp;2%u#5sv`~r3L0$63C)1HGZZisKK9q&YBf4pZr->xT>UugR7!)jUjVJjP~oWOt( zOhH>>6c)R$nof6p2G%aB+Q~$9|t-jN_vV{e6vn`(_-JOWKc)+L6=Xn5Ne{Aa} zVF1W`KJ8QIFLxVrDVD>I0MgGUt}bxHccDqE3ryxV-nP?OkdHg+v#G4+Vy)|hh}w*Z zg>v;cP1z9>bh6_oZ0duZFL~#y8sNzq2GeGLmRtmaV|%1-SpC-GZOh1d#Q zy<*Wor@&p{PKwlzZM=R{^pe9JB*(k)sh Date: Mon, 4 Dec 2023 18:29:51 +0800 Subject: [PATCH 14/30] Fix the callout --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 5a39683947..7b8b512ccb 100644 --- a/README.md +++ b/README.md @@ -38,7 +38,7 @@

    -> **Important** +> [!Important] > > We are working on Chinese-to-English translation. For more information please see [#914](https://github.com/krahets/hello-algo/issues/914). From f4e64fed37ff3cd25eb276abbea1605f8a26caa6 Mon Sep 17 00:00:00 2001 From: Spark Date: Thu, 7 Dec 2023 02:07:20 -0500 Subject: [PATCH 15/30] translation: chapter_computational_complexity/summary to en (#953) * translate chapter_computational_complexity/summary * minor format * Update summary.md with comment * Update summary.md * Update summary.md --- .../summary.md | 48 +++++++++---------- 1 file changed, 24 insertions(+), 24 deletions(-) diff --git a/docs-en/chapter_computational_complexity/summary.md b/docs-en/chapter_computational_complexity/summary.md index 1ffad3777c..6deb872ece 100644 --- a/docs-en/chapter_computational_complexity/summary.md +++ b/docs-en/chapter_computational_complexity/summary.md @@ -2,48 +2,48 @@ ### Highlights -**Evaluation of the efficiency of the algorithm** +**Evaluation of Algorithm Efficiency** -- Time efficiency and space efficiency are the two main evaluation metrics for measuring the merit of an algorithm. -- We can evaluate the efficiency of the algorithm through real-world testing, but it is difficult to eliminate the effect of the testing environment and it consumes a lot of computational resources. -- Complexity analysis overcomes the drawbacks of real-world testing, the results of the analysis are applicable to all operating platforms and reveal the efficiency of the algorithm at different data sizes. +- Time and space efficiency are the two leading evaluation indicators to measure an algorithm. +- We can evaluate the efficiency of an algorithm through real-world testing. Still, it isn't easy to eliminate the side effects from the testing environment, and it consumes a lot of computational resources. +- Complexity analysis overcomes the drawbacks of real-world testing. The analysis results can apply to all operating platforms and reveal the algorithm's efficiency under variant data scales. -**Time complexity** +**Time Complexity** -- Time complexity is used to measure the trend of the algorithm's running time with the growth of data volume, which can effectively evaluate the efficiency of the algorithm, but it may fail in some cases, such as when the input data volume is small or the time complexity is the same, it is not possible to accurately compare the advantages and disadvantages of the efficiency of the algorithm. -- The worst time complexity is denoted by a large $O$ symbol, which corresponds to the asymptotic upper bound of the function, reflecting the level of growth in the number of operations $T(n)$ as $n$ tends to positive infinity. -- Extrapolating the time complexity is done in two steps, first counting the number of operations and then determining the asymptotic upper bound. -- Common time complexities are $O(1)$, $O(\log n)$, $O(n)$, $O(n \log n)$, $O(n^2)$, $O(2^n)$, and $O(n!)$, etc., in order of their degree. -- The time complexity of certain algorithms is non-fixed and is related to the distribution of the input data. The time complexity is categorized into worst, best, and average time complexity, with the best time complexity hardly ever being used because the input data generally needs to satisfy stringent conditions to achieve the best case. -- The average time complexity reflects the efficiency of an algorithm's operation under random data inputs, and is closest to the performance of algorithms in real-world applications. Calculating the average time complexity requires a statistical input data distribution as well as a synthesized mathematical expectation. +- Time complexity is used to measure the trend of algorithm running time as the data size grows., which can effectively evaluate the algorithm's efficiency. However, it may fail in some cases, such as when the input volume is small or the time complexities are similar, making it difficult to precisely compare the efficiency of algorithms. +- The worst time complexity is denoted by big $O$ notation, which corresponds to the asymptotic upper bound of the function, reflecting the growth rate in the number of operations $T(n)$ as $n$ tends to positive infinity. +- The estimation of time complexity involves two steps: first, counting the number of operations, and then determining the asymptotic upper bound. +- Common time complexities, from lowest to highest, are $O(1)$, $O(\log n)$, $O(n)$, $O(n \log n)$, $O(n^2)$, $O(2^n)$, and $O(n!)$. +- The time complexity of certain algorithms is not fixed and depends on the distribution of the input data. The time complexity can be categorized into worst-case, best-case, and average. The best-case time complexity is rarely used because the input data must meet strict conditions to achieve the best-case. +- The average time complexity reflects the efficiency of an algorithm with random data inputs, which is closest to the performance of algorithms in real-world scenarios. Calculating the average time complexity requires statistical analysis of input data and a synthesized mathematical expectation. -**space complexity** +**Space Complexity** -- Space complexity acts similarly to time complexity and is used to measure the tendency of an algorithm's space occupation to grow with the amount of data. -- The memory space associated with the operation of an algorithm can be categorized into input space, staging space, and output space. Normally, the input space is not counted in the space complexity calculation. The transient space can be divided into instruction space, data space, and stack frame space, of which the stack frame space usually affects the space complexity only in the recursion function. -- We usually focus only on the worst-case space complexity, i.e., the space complexity of the statistical algorithm with the worst-case input data and the worst-case running time complexity. -- Common space complexities are $O(1)$, $O(\log n)$, $O(n)$, $O(n^2)$ and $O(2^n)$ in descending order. +- Space complexity serves a similar purpose to time complexity and is used to measure the trend of space occupied by an algorithm as the data volume increases. +- The memory space associated with the operation of an algorithm can be categorized into input space, temporary space, and output space. Normally, the input space is not considered when determining space complexity. The temporary space can be classified into instruction space, data space, and stack frame space, and the stack frame space usually only affects the space complexity for recursion functions. +- We mainly focus on the worst-case space complexity, which refers to the measurement of an algorithm's space usage when given the worst-case input data and during the worst-case execution scenario. +- Common space complexities are $O(1)$, $O(\log n)$, $O(n)$, $O(n^2)$ and $O(2^n)$ from lowest to highest. ### Q & A !!! question "Is the space complexity of tail recursion $O(1)$?" - Theoretically, the space complexity of a tail recursion function can be optimized to $O(1)$ . However, most programming languages (e.g. Java, Python, C++, Go, C#, etc.) do not support automatic optimization of tail recursion, so the space complexity is usually considered to be $O(n)$ . + Theoretically, the space complexity of a tail recursion function can be optimized to $O(1)$. However, most programming languages (e.g., Java, Python, C++, Go, C#, etc.) do not support auto-optimization for tail recursion, so the space complexity is usually considered as $O(n)$. !!! question "What is the difference between the terms function and method?" - A function can be executed independently and all arguments are passed explicitly. A method is associated with an object, is implicitly passed to the object that calls it, and can manipulate data contained in instances of the class. + A *function* can be executed independently, and all arguments are passed explicitly. A *method* is associated with an object and is implicitly passed to the object that calls it, allowing it to operate on the data contained within an instance of a class. - Here are a few common programming languages to illustrate. + Let's illustrate with a few common programming languages. - - C is a procedural programming language and has no object-oriented concepts, so it has only functions. However, we can simulate object-oriented programming by creating structures, and the functions associated with structures are equivalent to methods in other languages. - - Java and C# are object-oriented programming languages, and blocks of code (methods) are usually included as part of a class. A static method behaves like a function because it is bound to the class and cannot access specific instance variables. - - C++ and Python support both procedural programming (functions) and object-oriented programming (methods). + - C is a procedural programming language without object-oriented concepts, so it has only functions. However, we can simulate object-oriented programming by creating structures (struct), and the functions associated with structures are equivalent to methods in other languages. + - Java and C# are object-oriented programming languages, and blocks of code (methods) are typically part of a class. Static methods behave like a function because it is bound to the class and cannot access specific instance variables. + - Both C++ and Python support both procedural programming (functions) and object-oriented programming (methods). !!! question "Does the figure "Common Types of Space Complexity" reflect the absolute size of the occupied space?" No, that figure shows the space complexity, which reflects the growth trend, not the absolute size of the space occupied. - Suppose you take $n = 8$ , you may find that the values of each curve do not correspond to the function. This is because each curve contains a constant term that is used to compress the range of values to a visually comfortable range. + For example, if you take $n = 8$ , the values of each curve do not align with the function because each curve contains a constant term used to compress the range of values to a visually comfortable range. - In practice, since we usually don't know what the "constant term" complexity of each method is, it is generally impossible to choose the optimal solution for $n = 8$ based on complexity alone. But for $n = 8^5$ it's a good choice, when the growth trend is already dominant. + In practice, since we usually don't know each method's "constant term" complexity, it is generally impossible to choose the optimal solution for $n = 8$ based on complexity alone. But it's easier to choose for $n = 8^5$ as the growth trend is already dominant. From c5f5abf10c57ec69ef85fd73d778869ddd5c9849 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Mon, 11 Dec 2023 20:38:01 +0800 Subject: [PATCH 16/30] translation: chapter_introduction/what_is_dsa.md (#962) * Optimize translation of what_is_dsa.md * Update --- docs-en/chapter_introduction/what_is_dsa.md | 64 ++++++++++----------- 1 file changed, 32 insertions(+), 32 deletions(-) diff --git a/docs-en/chapter_introduction/what_is_dsa.md b/docs-en/chapter_introduction/what_is_dsa.md index 784af7023e..bb95fc4f9a 100644 --- a/docs-en/chapter_introduction/what_is_dsa.md +++ b/docs-en/chapter_introduction/what_is_dsa.md @@ -1,53 +1,53 @@ -# What is Algorithms +# What is an Algorithm -## Algorithm Definition +## Definition of an Algorithm -An "algorithm" is a set of instructions or operational steps to solve a particular problem in a limited time, which has the following characteristics. +An "algorithm" is a set of instructions or steps to solve a specific problem within a finite amount of time. It has the following characteristics: -- Problems are well defined and contain clear definitions of inputs and outputs. -- Feasible and able to be accomplished with limited steps, time and memory space. -- Each step has a defined meaning and the output is always the same for the same inputs and operating conditions. +- The problem is clearly defined, including unambiguous definitions of input and output. +- The algorithm is feasible, meaning it can be completed within a finite number of steps, time, and memory space. +- Each step has a definitive meaning. The output is consistently the same under the same inputs and conditions. -## Data Structure Definitions +## Definition of a Data Structure -"A data structure is a way of organizing and storing data in a computer and has the following design goals. +A "data structure" is a way of organizing and storing data in a computer, with the following design goals: -- Space occupation is minimized to save computer memory. -- Data manipulation is as fast as possible, covering data access, additions, deletions, updates, etc. -- Provide concise data representation and logical information in order to make the algorithm work efficiently. +- Minimize space occupancy to save computer memory. +- Make data operations as fast as possible, covering data access, addition, deletion, updating, etc. +- Provide concise data representation and logical information to enable efficient algorithm execution. -**Data structure design is a process full of tradeoffs**. If you want to improve in one area, you often need to compromise in another. Two examples are given below. +**Designing data structures is a balancing act, often requiring trade-offs**. If you want to improve in one aspect, you often need to compromise in another. Here are two examples: -- Compared to arrays, linked lists are more convenient for data addition and deletion operations, but at the expense of data access speed. -- Graphs provide richer logical information compared to linked lists, but take up more memory space. +- Compared to arrays, linked lists offer more convenience in data addition and deletion but sacrifice data access speed. +- Graphs, compared to linked lists, provide richer logical information but require more memory space. -## Relationship Between Data Structures And Algorithms +## Relationship Between Data Structures and Algorithms -As shown in the figure below, data structure and algorithm are highly correlated and tightly integrated in the following three ways. +As shown in the diagram below, data structures and algorithms are highly related and closely integrated, specifically in the following three aspects: -- Data structures are the cornerstone of algorithms. Data structures provide algorithms with structured storage of data, and methods for manipulating the data. -- Algorithms are the arena where data structures come into play. Data structures by themselves only store information about the data, and it is in combination with an algorithm that a particular problem can be solved. -- Algorithms can often be implemented based on different data structures, but the efficiency of execution can vary greatly, and choosing the right data structure is key. +- Data structures are the foundation of algorithms. They provide structured data storage and methods for manipulating data for algorithms. +- Algorithms are the stage where data structures come into play. The data structure alone only stores data information; it is through the application of algorithms that specific problems can be solved. +- Algorithms can often be implemented based on different data structures, but their execution efficiency can vary greatly. Choosing the right data structure is key. -![Relationship between data structure and algorithm](what_is_dsa.assets/relationship_between_data_structure_and_algorithm.png) +![Relationship between data structures and algorithms](what_is_dsa.assets/relationship_between_data_structure_and_algorithm.png) -Data structures and algorithms are like the building blocks shown in the figure below. A set of building blocks, in addition to containing many parts, comes with detailed assembly instructions. By following the instructions step by step, we can assemble a beautiful model of the blocks. +Data structures and algorithms can be likened to a set of building blocks, as illustrated in the figure below. A building block set includes numerous pieces, accompanied by detailed assembly instructions. Following these instructions step by step allows us to construct an intricate block model. -![assembling blocks](what_is_dsa.assets/assembling_blocks.jpg) +![Assembling blocks](what_is_dsa.assets/assembling_blocks.jpg) The detailed correspondence between the two is shown in the table below. -

    Table   Analogizing Data Structures and Algorithms to Building Blocks

    +

    Table   Comparing Data Structures and Algorithms to Building Blocks

    -| Data Structures and Algorithms | Building Blocks | -| ------------------------------ | ----------------------------------------------------------------------------- | -| Input data | Unassembled blocks | -| data structure | building block organization, including shape, size, and connections | -| algorithms | a series of operational steps to put the blocks together into the target form | -| Output Data | Building Block Models | +| Data Structures and Algorithms | Building Blocks | +| ------------------------------ | --------------------------------------------------------------- | +| Input data | Unassembled blocks | +| Data structure | Organization of blocks, including shape, size, connections, etc | +| Algorithm | A series of steps to assemble the blocks into the desired shape | +| Output data | Completed Block model | -It is worth stating that data structures and algorithms are independent of programming languages. Because of this, this book is able to provide implementations in multiple programming languages. +It's worth noting that data structures and algorithms are independent of programming languages. For this reason, this book is able to provide implementations in multiple programming languages. -!!! tip "agreed upon acronym" +!!! tip "Conventional Abbreviation" - In practice, we usually refer to "data structures and algorithms" as "algorithms". For example, the well-known LeetCode algorithm question actually examines both data structure and algorithm. + In real-life discussions, we often refer to "Data Structures and Algorithms" simply as "Algorithms". For example, the well-known LeetCode algorithm problems actually test both data structure and algorithm knowledge. From ac0e2bbe24bd7335d4174b3b2e4ac29cc5ceb0c6 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Thu, 21 Dec 2023 20:31:10 +0800 Subject: [PATCH 17/30] translation: chapter_introduction/summary.md (#963) * Translate chapter_introduction/summary.md * Update --- docs-en/chapter_introduction/summary.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/docs-en/chapter_introduction/summary.md b/docs-en/chapter_introduction/summary.md index ac56c258e5..acfa2d4a5d 100644 --- a/docs-en/chapter_introduction/summary.md +++ b/docs-en/chapter_introduction/summary.md @@ -1,9 +1,9 @@ # Summary -- Algorithms are ubiquitous in our daily lives and are not a distant knowledge. In fact, we have learned many algorithms without even realizing it to solve life's problems, big and small. -- The principle of looking up a dictionary is consistent with the binary search algorithm. The binary search algorithm embodies the important algorithmic idea of divide and conquer. -- The process of sorting poker is very similar to the insertion sort algorithm. The insertion sort algorithm is suitable for sorting small data sets. -- The steps in currency change are essentially greedy algorithms, taking the best option that currently seems to be available at each step. -- An algorithm is a set of instructions or operational steps to solve a specific problem in a limited amount of time, while a data structure is the way data is organized and stored in a computer. -- Data structures and algorithms are closely intertwined. The data structure is the cornerstone of the algorithm, while the algorithm is the arena in which the data structure is utilized. -- We can analogize data structures and algorithms to putting together building blocks, where the blocks represent the data, the shapes and connections of the blocks represent the data structure, and the steps in putting the blocks together correspond to the algorithm. +- Algorithms are ubiquitous in daily life and are not as inaccessible and complex as they might seem. In fact, we have already unconsciously learned many algorithms to solve various problems in life. +- The principle of looking up a word in a dictionary is consistent with the binary search algorithm. The binary search algorithm embodies the important algorithmic concept of divide and conquer. +- The process of organizing playing cards is very similar to the insertion sort algorithm. The insertion sort algorithm is suitable for sorting small datasets. +- The steps of making change in currency essentially follow the greedy algorithm, where each step involves making the best possible choice at the moment. +- An algorithm is a set of instructions or steps used to solve a specific problem within a finite amount of time, while a data structure is the way data is organized and stored in a computer. +- Data structures and algorithms are closely linked. Data structures are the foundation of algorithms, and algorithms are the stage to utilize the functions of data structures. +- We can liken data structures and algorithms to building blocks. The blocks represent data, the shape and connection method of the blocks represent data structures, and the steps of assembling the blocks correspond to algorithms. From bd8d47ec3f74030b7cdf8440fe55346a0d7a4042 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Thu, 21 Dec 2023 20:36:21 +0800 Subject: [PATCH 18/30] translation: Update README.md (#964) * Update en translation of README.md * Update README.md --- README.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/README.md b/README.md index 7b8b512ccb..dc856fe8b9 100644 --- a/README.md +++ b/README.md @@ -44,7 +44,7 @@ ## About -This project aims to create a free, open-source, and beginner-friendly crash course for data structures and algorithms. +This open-source project aims to create a free and beginner-friendly crash course for data structures and algorithms. - Animated illustrations, easy-to-understand content, and a smooth learning curve help beginners explore the "knowledge map" of data structures and algorithms. - Run code with just one click, helping readers improve their programming skills and understand the working principle of algorithms and the underlying implementation of data structures. @@ -52,27 +52,27 @@ This project aims to create a free, open-source, and beginner-friendly crash cou If you find this book helpful, please give it a Star :star: to support us, thank you! -## Recommendations +## Endorsements -> "An easy-to-understand introductory book on data structures and algorithms, which guides readers to learn by thinking and hands-on, strongly recommended for algorithm beginners." +> "An easy-to-understand introductory book on data structures and algorithms, which guides readers to learn by minds-on and hands-on. Strongly recommended for algorithm beginners!" > > **—— Junhui Deng, Professor of Computer Science, Tsinghua University** -> "If I had *Hello Algorithm* when I was learning data structures and algorithms, it would have been 10 times easier!" +> "If I had *Hello Algo* when I was learning data structures and algorithms, it would have been 10 times easier!" > > **—— Mu Li, Senior Principal Scientist, Amazon** ## Contribution -This open-source book is continuously being updated, and we welcome your participation in this project to provide better learning content for our readers. +This open-source book is continuously being updated, and we welcome your participation in this project to provide better learning content for readers. - [Content Correction](https://www.hello-algo.com/chapter_appendix/contribution/): Please help us correct or point out mistakes in the comments section such as grammatical errors, missing content, ambiguities, invalid links, or code bugs. -- [Code Translation](https://github.com/krahets/hello-algo/issues/15): We look forward to your contributions of code in various programming languages. We currently support Python, Java, C++, Go, JavaScript, and 12 other programming languages. -- [Book Translation](https://github.com/krahets/hello-algo/tree/en): We would love to invite you to join our Chinese-to-English translation team. Members are mainly from computer science related majors, English majors, and native English speakers. +- [Code Transpilation](https://github.com/krahets/hello-algo/issues/15): We look forward to your contributions in various programming languages. We currently support 12 languages including Python, Java, C++, Go, and JavaScript. +- [Chinese-to-English](https://github.com/krahets/hello-algo/tree/en): We would love to invite you to join our translation team. The members are mainly from computer-science-related fields, English majors, and native English speakers. We welcome your valuable suggestions and feedback. If you have any questions, please submit Issues or reach out via WeChat: `krahets-jyd`. -We would like to dedicate our thank to all the contributors of this book. It is their selfless dedication that has made this book better. They are: +We would like to dedicate our thanks to all the contributors of this book. It is their selfless dedication that has made this book better. They are:

    From a8f0d8938974899f98d1746c0c55baa32fd64cbe Mon Sep 17 00:00:00 2001 From: Thomas Date: Thu, 21 Dec 2023 23:39:42 +1100 Subject: [PATCH 19/30] translation: update space_complexity.md (#970) * update space_complexity.md * the rest of translation piece * Update space_complexity.md --------- Co-authored-by: ThomasQiu Co-authored-by: Yudong Jin --- .../space_complexity.md | 192 +++++++++--------- 1 file changed, 96 insertions(+), 96 deletions(-) diff --git a/docs-en/chapter_computational_complexity/space_complexity.md b/docs-en/chapter_computational_complexity/space_complexity.md index c6f774680c..5dadce191f 100644 --- a/docs-en/chapter_computational_complexity/space_complexity.md +++ b/docs-en/chapter_computational_complexity/space_complexity.md @@ -1,24 +1,24 @@ # Space Complexity -"Space complexity is a measure of how the amount of memory space an algorithm occupies grows as the amount of data grows. This concept is very similar to time complexity, just replace "runtime" with "memory space". +The space complexity is used to measure the growth trend of memory consumption as the scale of data increases for an algorithm solution. This concept is analogous to time complexity by replacing "runtime" with "memory space". ## Algorithmic Correlation Space -The main types of memory space used by algorithms during operation include the following. +The memory space used by algorithms during its execution include the following types. -- **Input Space**: used to store the input data for the algorithm. -- **Staging Space**: used to store data such as variables, objects, function contexts, etc. of the algorithm during runtime. -- **Output Space**: used to store the output data of the algorithm. +- **Input Space**: Used to store the input data for the algorithm. +- **Temporary Space**: Used to store variables, objects, function contexts, and other data of the algorithm during runtime. +- **Output Space**: Used to store the output data of the algorithm. -In general, the statistical range of space complexity is "staging space" plus "output space". +In general, the "Input Space" is excluded from the statistics of space complexity. -The staging space can be further divided into three sections. +The **Temporary Space** can be further divided into three parts. -- **Staging Data**: used to save various constants, variables, objects, etc. during the running of the algorithm. -- **Stack frame space**: used to hold the context data of the called function. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is freed when the function returns. -- **Instruction space**: used to hold compiled program instructions, usually ignored in practical statistics. +- **Temporary Data**: Used to store various constants, variables, objects, etc., during the the algorithm's execution. +- **Stack Frame Space**: Used to hold the context data of the called function. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is freed when the function returns. +- **Instruction Space**: Used to hold compiled program instructions, usually ignored in practical statistics. -When analyzing the space complexity of a piece of program, **we usually count three parts**: transient data, stack frame space and output data. +When analyzing the space complexity of a piece of program, **three parts are usually taken into account: Temporary Data, Stack Frame Space and Output Data**. ![Associated spaces used by the algorithm](space_complexity.assets/space_types.png) @@ -28,20 +28,20 @@ When analyzing the space complexity of a piece of program, **we usually count th class Node: """Classes"""" def __init__(self, x: int): - self.val: int = x # node value - self.next: Node | None = None # reference to the next node + self.val: int = x # node value + self.next: Node | None = None # reference to the next node def function() -> int: - """"Function""""" + """"Functions""""" # Perform certain operations... return 0 - def algorithm(n) -> int: # input data - A = 0 # Temporary storage of data (constant, usually in uppercase) - b = 0 # Temporary data (variable) - node = Node(0) # temporary data (object) - c = function() # Stack frame space (call function) - return A + b + c # output data + def algorithm(n) -> int: # input data + A = 0 # temporary data (constant, usually in uppercase) + b = 0 # temporary data (variable) + node = Node(0) # temporary data (object) + c = function() # Stack frame space (call function) + return A + b + c # output data ``` === "C++" @@ -60,12 +60,12 @@ When analyzing the space complexity of a piece of program, **we usually count th return 0; } - int algorithm(int n) { // input data - const int a = 0; // temporary data (constant) - int b = 0; // temporary data (variable) - Node* node = new Node(0); // temporary data (object) - int c = func(); // stack frame space (call function) - return a + b + c; // output data + int algorithm(int n) { // input data + const int a = 0; // temporary data (constant) + int b = 0; // temporary data (variable) + Node* node = new Node(0); // temporary data (object) + int c = func(); // stack frame space (call function) + return a + b + c; // output data } ``` @@ -85,12 +85,12 @@ When analyzing the space complexity of a piece of program, **we usually count th return 0; } - int algorithm(int n) { // input data - final int a = 0; // temporary data (constant) - int b = 0; // temporary data (variable) - Node node = new Node(0); // temporary data (object) - int c = function(); // stack frame space (call function) - return a + b + c; // output data + int algorithm(int n) { // input data + final int a = 0; // temporary data (constant) + int b = 0; // temporary data (variable) + Node node = new Node(0); // temporary data (object) + int c = function(); // stack frame space (call function) + return a + b + c; // output data } ``` @@ -110,12 +110,12 @@ When analyzing the space complexity of a piece of program, **we usually count th return 0; } - int Algorithm(int n) { // input data - const int a = 0; // temporary data (constant) - int b = 0; // temporary data (variable) + int Algorithm(int n) { // input data + const int a = 0; // temporary data (constant) + int b = 0; // temporary data (variable) Node node = new(0); // temporary data (object) int c = Function(); // stack frame space (call function) - return a + b + c; // output data + return a + b + c; // output data } ``` @@ -140,11 +140,11 @@ When analyzing the space complexity of a piece of program, **we usually count th } func algorithm(n int) int { // input data - const a = 0 // temporary data (constant) - b := 0 // temporary storage of data (variable) - newNode(0) // temporary data (object) - c := function() // stack frame space (call function) - return a + b + c // output data + const a = 0 // temporary data (constant) + b := 0 // temporary storage of data (variable) + newNode(0) // temporary data (object) + c := function() // stack frame space (call function) + return a + b + c // output data } ``` @@ -168,11 +168,11 @@ When analyzing the space complexity of a piece of program, **we usually count th } func algorithm(n: Int) -> Int { // input data - let a = 0 // temporary data (constant) - var b = 0 // temporary data (variable) - let node = Node(x: 0) // temporary data (object) - let c = function() // stack frame space (call function) - return a + b + c // output data + let a = 0 // temporary data (constant) + var b = 0 // temporary data (variable) + let node = Node(x: 0) // temporary data (object) + let c = function() // stack frame space (call function) + return a + b + c // output data } ``` @@ -185,7 +185,7 @@ When analyzing the space complexity of a piece of program, **we usually count th next; constructor(val) { this.val = val === undefined ? 0 : val; // node value - this.next = null; // reference to the next node + this.next = null; // reference to the next node } } @@ -195,12 +195,12 @@ When analyzing the space complexity of a piece of program, **we usually count th return 0; } - function algorithm(n) { // input data - const a = 0; // temporary data (constant) - let b = 0; // temporary data (variable) - const node = new Node(0); // temporary data (object) - const c = constFunc(); // Stack frame space (calling function) - return a + b + c; // output data + function algorithm(n) { // input data + const a = 0; // temporary data (constant) + let b = 0; // temporary data (variable) + const node = new Node(0); // temporary data (object) + const c = constFunc(); // Stack frame space (calling function) + return a + b + c; // output data } ``` @@ -213,7 +213,7 @@ When analyzing the space complexity of a piece of program, **we usually count th next: Node | null; constructor(val?: number) { this.val = val === undefined ? 0 : val; // node value - this.next = null; // reference to the next node + this.next = null; // reference to the next node } } @@ -224,11 +224,11 @@ When analyzing the space complexity of a piece of program, **we usually count th } function algorithm(n: number): number { // input data - const a = 0; // temporary data (constant) - let b = 0; // temporary data (variable) - const node = new Node(0); // temporary data (object) - const c = constFunc(); // Stack frame space (calling function) - return a + b + c; // output data + const a = 0; // temporary data (constant) + let b = 0; // temporary data (variable) + const node = new Node(0); // temporary data (object) + const c = constFunc(); // Stack frame space (calling function) + return a + b + c; // output data } ``` @@ -248,12 +248,12 @@ When analyzing the space complexity of a piece of program, **we usually count th return 0; } - int algorithm(int n) { // input data - const int a = 0; // temporary data (constant) - int b = 0; // temporary data (variable) - Node node = Node(0); // temporary data (object) - int c = function(); // stack frame space (call function) - return a + b + c; // output data + int algorithm(int n) { // input data + const int a = 0; // temporary data (constant) + int b = 0; // temporary data (variable) + Node node = Node(0); // temporary data (object) + int c = function(); // stack frame space (call function) + return a + b + c; // output data } ``` @@ -282,12 +282,12 @@ When analyzing the space complexity of a piece of program, **we usually count th return 0; } - fn algorithm(n: i32) -> i32 { // input data - const a: i32 = 0; // temporary data (constant) - let mut b = 0; // temporary data (variable) - let node = Node::new(0); // temporary data (object) - let c = function(); // stack frame space (call function) - return a + b + c; // output data + fn algorithm(n: i32) -> i32 { // input data + const a: i32 = 0; // temporary data (constant) + let mut b = 0; // temporary data (variable) + let node = Node::new(0); // temporary data (object) + let c = function(); // stack frame space (call function) + return a + b + c; // output data } ``` @@ -300,11 +300,11 @@ When analyzing the space complexity of a piece of program, **we usually count th return 0; } - int algorithm(int n) { // input data - const int a = 0; // temporary data (constant) - int b = 0; // temporary data (variable) - int c = func(); // stack frame space (call function) - return a + b + c; // output data + int algorithm(int n) { // input data + const int a = 0; // temporary data (constant) + int b = 0; // temporary data (variable) + int c = func(); // stack frame space (call function) + return a + b + c; // output data } ``` @@ -314,16 +314,16 @@ When analyzing the space complexity of a piece of program, **we usually count th ``` -## Method Of Projection +## Calculation Method -Space complexity is derived in much the same way as time complexity, simply by changing the statistic from "number of operations" to "amount of space used". +The calculation method for space complexity is pretty similar to time complexity, with the only difference being that the focus shifts from "operation count" to "space usage size". -And unlike time complexity, **we usually focus only on the worst space complexity**. This is because memory space is a hard requirement and we have to make sure that there is enough memory space reserved under all input data. +On top of that, unlike time complexity, **we usually only focus on the worst-case space complexity**. This is because memory space is a hard requirement, and we have to make sure that there is enough memory space reserved for all possibilities incurred by input data. -Looking at the following code, the "worst" in worst space complexity has two levels of meaning. +Looking at the following code, the "worst" in worst-case space complexity has two layers of meaning. -1. **whichever is the worst input data**: when $n < 10$, the space complexity is $O(1)$; however, when $n > 10$, the initialized array `nums` occupies $O(n)$ space; thus the worst space complexity is $O(n)$ . -2. **in terms of peak memory during the run of the algorithm**: for example, the program occupies $O(1)$ space until the last line is executed; when the array `nums` is initialized, the program occupies $O(n)$ space; thus the worst space complexity is $O(n)$ . +1. **Based on the worst-case input data**: when $n < 10$, the space complexity is $O(1)$; however, when $n > 10$, the initialized array `nums` occupies $O(n)$ space; thus the worst-case space complexity is $O(n)$. +2. **Based on the peak memory during algorithm execution**: for example, the program occupies $O(1)$ space until the last line is executed; when the array `nums` is initialized, the program occupies $O(n)$ space; thus the worst-case space complexity is $O(n)$. === "Python" @@ -460,10 +460,10 @@ Looking at the following code, the "worst" in worst space complexity has two lev ``` -**In recursion functions, care needs to be taken to count the stack frame space**. For example in the following code: +**In recursion functions, it is important to take into count the measurement of stack frame space**. For example in the following code: - The function `loop()` calls $n$ times `function()` in a loop, and each round of `function()` returns and frees stack frame space, so the space complexity is still $O(1)$. -- The recursion function `recur()` will have $n$ unreturned `recur()` at the same time during runtime, thus occupying $O(n)$ of stack frame space. +- The recursion function `recur()` will have $n$ unreturned `recur()` during runtime, thus occupying $O(n)$ of stack frame space. === "Python" @@ -702,7 +702,7 @@ Looking at the following code, the "worst" in worst space complexity has two lev ## Common Types -Let the input data size be $n$ , the figure below shows the common types of space complexity (ordered from low to high). +Assuming the input data size is $n$, the figure illustrates common types of space complexity (ordered from low to high). $$ \begin{aligned} @@ -715,9 +715,9 @@ $$ ### Constant Order $O(1)$ -Constant orders are common for constants, variables, and objects whose quantity is unrelated to the size of the input data $n$. +Constant order is common for constants, variables, and objects whose quantity is unrelated to the size of the input data $n$. -Note that memory occupied by initializing a variable or calling a function in a loop is freed when it enters the next loop, so there is no accumulation of occupied space and the space complexity remains $O(1)$ : +It is important to note that memory occupied by initializing a variable or calling a function in a loop is released once the next iteration begins. Therefore, there is no accumulation of occupied space and the space complexity remains $O(1)$ : ```src [file]{space_complexity}-[class]{}-[func]{constant} @@ -725,7 +725,7 @@ Note that memory occupied by initializing a variable or calling a function in a ### Linear Order $O(N)$ -Linear orders are commonly found in arrays, linked lists, stacks, queues, etc. where the number of elements is proportional to $n$: +Linear order is commonly found in arrays, linked lists, stacks, queues, and similar structures where the number of elements is proportional to $n$: ```src [file]{space_complexity}-[class]{}-[func]{linear} @@ -739,15 +739,15 @@ As shown in the figure below, the depth of recursion for this function is $n$, w ![Linear order space complexity generated by recursion function](space_complexity.assets/space_complexity_recursive_linear.png) -### Squared Order $O(N^2)$ +### Quadratic Order $O(N^2)$ -The squared order is common in matrices and graphs, where the number of elements is squared with $n$: +Quadratic order is common in matrices and graphs, where the number of elements is in a square relationship with $n$: ```src [file]{space_complexity}-[class]{}-[func]{quadratic} ``` -As shown in the figure below, the recursion depth of this function is $n$ , and an array is initialized in each recursion function with lengths $n$, $n-1$, $\dots$, $2$, $1$ , and an average length of $n / 2$ , thus occupying $O(n^2)$ space overall: +As shown in the figure below, the recursion depth of this function is $n$, and an array is initialized in each recursion function with lengths $n$, $n-1$, $\dots$, $2$, $1$, and an average length of $n / 2$, thus occupying $O(n^2)$ space overall: ```src [file]{space_complexity}-[class]{}-[func]{quadratic_recur} @@ -767,14 +767,14 @@ Exponential order is common in binary trees. Looking at the figure below, a "ful ### Logarithmic Order $O(\Log N)$ -Logarithmic order is commonly used in divide and conquer algorithms. For example, in a merge sort, an array of length $n$ is input, and each round of recursion divides the array in half from its midpoint to form a recursion tree of height $\log n$, using $O(\log n)$ stack frame space. +Logarithmic order is commonly used in divide and conquer algorithms. For example, in a merge sort, given an array of length $n$ as the input, each round of recursion divides the array in half from its midpoint to form a recursion tree of height $\log n$, using $O(\log n)$ stack frame space. -Another example is to convert a number into a string, input a positive integer $n$ which has the number of bits $\log_{10} n + 1$, i.e., the corresponding length of the string is $\log_{10} n + 1$, so the space complexity is $O(\log_{10} n + 1) = O(\log n)$. +Another example is to convert a number into a string. Given a positive integer $n$ with a digit count of $\log_{10} n + 1$, the corresponding string length is $\log_{10} n + 1$. Therefore, the space complexity is $O(\log_{10} n + 1) = O(\log n)$. ## Weighing Time And Space -Ideally, we would like to optimize both the time complexity and the space complexity of an algorithm. In practice, however, optimizing both time complexity and space complexity at the same time is usually very difficult. +Ideally, we would like to optimize both the time complexity and the space complexity of an algorithm. However, in reality, simultaneously optimizing time and space complexity is often challenging. -**Reducing time complexity usually comes at the cost of increasing space complexity, and vice versa**. We refer to the idea of sacrificing memory space to increase the speed of an algorithm as "space for time"; the opposite is called "time for space". +**Reducing time complexity usually comes at the expense of increasing space complexity, and vice versa**. The approach of sacrificing memory space to improve algorithm speed is known as "trading space for time", while the opposite is called "trading time for space". -The choice of thinking depends on what we value more. In most cases, time is more valuable than space, so "space for time" is usually the more common strategy. Of course, it is also important to control the space complexity in case of large amount of data. +The choice between these approaches depends on which aspect we prioritize. In most cases, time is more valuable than space, so "trading space for time" is usually the more common strategy. Of course, in situations with large data volumes, controlling space complexity is also crucial. From 41b6ba3a4d0717225af060e2e2aeda8ecc8322a6 Mon Sep 17 00:00:00 2001 From: K3v123 <123932560+K3v123@users.noreply.github.com> Date: Fri, 22 Dec 2023 01:41:02 +1300 Subject: [PATCH 20/30] translation: Update chapter_introduction/index.md (#971) * Update index.md sorry, first time doing this... now this is the final change. changes: title of the chapter is shorter. refined the abstract. * Update index.md --------- Co-authored-by: Yudong Jin --- docs-en/chapter_introduction/index.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/docs-en/chapter_introduction/index.md b/docs-en/chapter_introduction/index.md index 9d4fa14b83..2f61cea8c4 100644 --- a/docs-en/chapter_introduction/index.md +++ b/docs-en/chapter_introduction/index.md @@ -1,4 +1,4 @@ -# Getting To Know The Algorithm For The First Time +# Introduction to Algorithms

    @@ -8,6 +8,6 @@ !!! abstract - A young girl dances, intertwining with the data, a melody of algorithms fluttering from her skirts. + A graceful maiden dances, intertwined with the data, her skirt swaying to the melody of algorithms. - She invites you to dance, so follow her lead and step into the world of algorithms full of logic and beauty. + She invites you to a dance, follow her steps, and enter the world of algorithms full of logic and beauty. From 7b60f1ebaed8704823b6235d471c611db41ee50e Mon Sep 17 00:00:00 2001 From: Jin <36914748+yanedie@users.noreply.github.com> Date: Thu, 21 Dec 2023 20:46:37 +0800 Subject: [PATCH 21/30] translation: Update chapter_data_structure/classification_of_data_structure.md (#980) * update classification_of_data_structure.md * Update classification_of_data_structure.md --------- Co-authored-by: Yudong Jin --- .../classification_of_data_structure.md | 36 ++++++++++--------- 1 file changed, 19 insertions(+), 17 deletions(-) diff --git a/docs-en/chapter_data_structure/classification_of_data_structure.md b/docs-en/chapter_data_structure/classification_of_data_structure.md index a5778f608c..4ec268b1db 100644 --- a/docs-en/chapter_data_structure/classification_of_data_structure.md +++ b/docs-en/chapter_data_structure/classification_of_data_structure.md @@ -1,46 +1,48 @@ # Classification Of Data Structures -Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs, which can be categorized in terms of both "logical structure" and "physical structure" dimensions. +Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs. They can be divided into two categories: logical structure and physical structure. -## Logical Structures: Linear And Nonlinear +## Logical Structures: Linear And Non-linear -**Logical structures reveal logical relationships between data elements**. In arrays and linked lists, data are arranged sequentially in order, reflecting the linear relationship between data; while in trees, data are arranged hierarchically from the top down, showing the derived relationship between ancestors and descendants; and graphs are composed of nodes and edges, reflecting the complex network relationship. +**Logical structures reveal logical relationships between data elements**. In arrays and linked lists, data are arranged in sequential order, reflecting the linear relationship between data; while in trees, data are arranged hierarchically from the top down, showing the derived relationship between ancestors and descendants; and graphs are composed of nodes and edges, reflecting the complex network relationship. -As shown in the figure below, logical structures can be categorized into "linear" and "non-linear". Linear structures are more intuitive, meaning that the data are arranged linearly in terms of logical relationships; non-linear structures, on the other hand, are arranged non-linearly. +As shown in the figure below, logical structures can further be divided into "linear data structure" and "non-linear data structure". Linear data structures are more intuitive, meaning that the data are arranged linearly in terms of logical relationships; non-linear data structures, on the other hand, are arranged non-linearly. - **Linear data structures**: arrays, linked lists, stacks, queues, hash tables. - **Nonlinear data structures**: trees, heaps, graphs, hash tables. ![Linear and nonlinear data structures](classification_of_data_structure.assets/classification_logic_structure.png) -Nonlinear data structures can be further categorized into tree and graph structures. +Non-linear data structures can be further divided into tree and graph structures. -- **Linear structures**: arrays, linked lists, queues, stacks, hash tables, with one-to-one sequential relationships between elements. -- **Tree structure**: tree, heap, hash table with one-to-many relationships between elements. -- **Graph**: graph with many-to-many relationships between elements. +- **Linear structures**: arrays, linked lists, queues, stacks, hash tables, with one-to-one sequential relationship between elements. +- **Tree structure**: tree, heap, hash table, with one-to-many relationship between elements. +- **Graph**: graph with many-to-many relationship between elements. ## Physical Structure: Continuous vs. Dispersed -In computers, memory and hard disk are the two main storage hardware devices. Hard disks are used primarily for long-term data storage, and are larger (usually up to the terabyte degree) and slower. Memory is used for temporary storage of data while running a program, and is faster, but has a smaller capacity (usually at the GB level). +**When an algorithm is running, the data being processed is stored in memory**. The figure below shows a computer memory module where each black square represents a memory space. We can think of the memory as a giant Excel sheet in which each cell can store data of a certain size. -**During the operation of the algorithm, the relevant data are stored in memory**. The figure below shows a computer memory stick where each black square contains a block of memory space. We can think of the memory as a giant Excel sheet in which each cell can store data of a certain size, and in which all data is stored while the algorithm is running. - -**The system accesses the data at the target location by means of a memory address**. As shown in the figure below, the computer assigns numbers to each cell in the table according to specific rules, ensuring that each memory space has a unique memory address. With these addresses, the program can access the data in memory. +**The system accesses the data at the target location by means of a memory address**. As shown in the figure below, the computer assigns a unique identifier to each cell in the table according to specific rules, ensuring that each memory space has a unique memory address. With these addresses, the program can access the data in memory. ![memory_strip, memory_space, memory_address](classification_of_data_structure.assets/computer_memory_location.png) -Memory is a shared resource for all programs, and when a block of memory is occupied by one program, it cannot be used by other programs at the same time. **Therefore, memory resources are an important consideration in the design of data structures and algorithms**. For example, the peak memory occupied by an algorithm should not exceed the remaining free memory of the system; if there is a lack of consecutive large chunks of memory space, then the data structure chosen must be able to be stored in a decentralized memory space. +!!! tip + + It is worth noting that comparing memory to the Excel sheet is a simplified analogy. The actual memory working mechanism is more complicated, involving the concepts of address, space, memory management, cache mechanism, virtual and physical memory. + +Memory is a shared resource for all programs, and when a block of memory is occupied by one program, it cannot be used by other programs at the same time. **Therefore, considering memory resources is crucial in designing data structures and algorithms**. For example, the algorithm's peak memory usage should not exceed the remaining free memory of the system; if there is a lack of contiguous memory blocks, then the data structure chosen must be able to be stored in non-contiguous memory blocks. -As shown in the figure below, **Physical structure reflects the way data is stored in computer memory and can be categorized into continuous space storage (arrays) and distributed space storage (linked lists)**. The physical structure determines from the bottom up the methods of accessing, updating, adding, deleting, and other operations on the data, while presenting complementary characteristics in terms of time efficiency and space efficiency. +As shown in the figure below, **Physical structure reflects the way data is stored in computer memory and it can be divided into consecutive space storage (arrays) and distributed space storage (linked lists)**. The physical structure determines how data is accessed, updated, added, deleted, etc. Logical and physical structure complement each other in terms of time efficiency and space efficiency. ![continuous vs. decentralized spatial storage](classification_of_data_structure.assets/classification_phisical_structure.png) **It is worth stating that all data structures are implemented based on arrays, linked lists, or a combination of the two**. For example, stacks and queues can be implemented using both arrays and linked lists; and implementations of hash tables may contain both arrays and linked lists. -- **Based on arrays can be realized**: stacks, queues, hash tables, trees, heaps, graphs, matrices, tensors (arrays of dimension $\geq 3$), and so on. -- **Can be implemented based on linked lists**: stacks, queues, hash tables, trees, heaps, graphs, etc. +- **Array-based structures**: stacks, queues, hash tables, trees, heaps, graphs, matrices, tensors (arrays of dimension $\geq 3$), and so on. +- **Linked list-based structures**: stacks, queues, hash tables, trees, heaps, graphs, etc. -Data structures based on arrays are also known as "static data structures", which means that they are not changeable in length after initialization. In contrast, data structures based on linked lists are called "dynamic data structures", meaning that their length can be adjusted during program execution after initialization. +Data structures based on arrays are also known as "static data structures", which means that such structures' length remains constant after initialization. In contrast, data structures based on linked lists are called "dynamic data structures", meaning that their length can be adjusted during program execution after initialization. !!! tip From 42041301885b7ba9c1ef6999a57df3a483664959 Mon Sep 17 00:00:00 2001 From: K3v123 <123932560+K3v123@users.noreply.github.com> Date: Fri, 22 Dec 2023 01:48:50 +1300 Subject: [PATCH 22/30] translation: Update chapter_introduction/algorithms_are_everywhere.md (#972) * Update algorithms_are_everywhere.md changed or refined parts of the words and sentences including tips. Some of them I didnt change that much because im worried that it might not meet the requirement of accuracy. some other ones i changed a lot to make it sound better, but also kind of following the same wording as the CN version * Update algorithms_are_everywhere.md re-edited the dictionary part from Piyin to just normal Eng dictionary. again thank you very much hpstory for you suggestion. * Update algorithms_are_everywhere.md --------- Co-authored-by: Yudong Jin --- .../algorithms_are_everywhere.md | 22 +++++++++---------- 1 file changed, 11 insertions(+), 11 deletions(-) diff --git a/docs-en/chapter_introduction/algorithms_are_everywhere.md b/docs-en/chapter_introduction/algorithms_are_everywhere.md index 809b2b980e..47e0e2c630 100644 --- a/docs-en/chapter_introduction/algorithms_are_everywhere.md +++ b/docs-en/chapter_introduction/algorithms_are_everywhere.md @@ -4,11 +4,11 @@ When we hear the word "algorithm", we naturally think of mathematics. However, m Before we formally discuss algorithms, an interesting fact is worth sharing: **you have already learned many algorithms unconsciously and have become accustomed to applying them in your daily life**. Below, I will give a few specific examples to prove this point. -**Example 1: Looking Up a Dictionary**. In a dictionary, each Chinese character corresponds to a pinyin, and the dictionary is arranged in alphabetical order of pinyin. Suppose we need to find a character whose pinyin starts with the letter $r$. This is usually achieved in the following way: +**Example 1: Looking Up a Dictionary**. In a standard dictionary, each word corresponds to a phonetic transcription and the dictionary is organized alphabetically based on these transcriptions. Let's say we're looking for a word that begins with the letter $r$. This is typically done in the following way: -1. Open the dictionary to about halfway and check the first letter on the page, assuming it is $m$. -2. Since $r$ comes after $m$ in the pinyin alphabet, we exclude the first half of the dictionary and narrow the search to the second half. -3. Repeat steps `1.` and `2.` until you find the page where the pinyin starts with $r$. +1. Open the dictionary around its midpoint and note the first letter on that page, assuming it to be $m$. +2. Given the sequence of words following the initial letter $m$, estimate where words starting with the letter $r$ might be located within the alphabetical order. +3. Iterate steps `1.` and `2.` until you find the page where the word begins with the letter $r$. === "<1>" ![Dictionary search step](algorithms_are_everywhere.assets/binary_search_dictionary_step1.png) @@ -25,19 +25,19 @@ Before we formally discuss algorithms, an interesting fact is worth sharing: **y === "<5>" ![binary_search_dictionary_step5](algorithms_are_everywhere.assets/binary_search_dictionary_step5.png) -The skill of looking up a dictionary, essential for elementary school students, is actually the famous binary search algorithm. From the perspective of data structures, we can view the dictionary as a sorted "array"; from the perspective of algorithms, the series of operations in looking up a dictionary can be seen as "binary search". +The skill of looking up a dictionary, essential for elementary school students, is actually the renowned binary search algorithm. Through the lens of data structures, we can view the dictionary as a sorted "array"; while from an algorithmic perspective, the series of operations in looking up a dictionary can be seen as "binary search". **Example 2: Organizing Playing Cards**. When playing cards, we need to arrange the cards in ascending order each game, as shown in the following process. 1. Divide the playing cards into "ordered" and "unordered" parts, assuming initially that the leftmost card is already ordered. -2. Take out a card from the unordered part and insert it into the correct position in the ordered part; after this, the leftmost two cards are in order. -3. Continue looping step `2.`, each round inserting one card from the unordered part into the ordered part, until all cards are ordered. +2. Take out a card from the unordered part and insert it into the correct position in the ordered part; once completed, the leftmost two cards will be in an ordered sequence. +3. Continue the loop described in step `2.`, each iteration involving insertion of one card from the unordered segment into the ordered portion, until all cards are appropriately ordered. ![Playing cards sorting process](algorithms_are_everywhere.assets/playing_cards_sorting.png) The above method of organizing playing cards is essentially the "insertion sort" algorithm, which is very efficient for small datasets. Many programming languages' sorting library functions include insertion sort. -**Example 3: Making Change**. Suppose we buy goods worth $69$ at a supermarket and give the cashier $100$, then the cashier needs to give us $31$ in change. They would naturally complete the thought process as shown below. +**Example 3: Making Change**. Suppose we buy goods worth $69$ yuan at a supermarket and give the cashier $100$ yuan, then the cashier needs to give us $31$ yuan in change. They would naturally complete the thought process as shown below. 1. The options are currencies smaller than $31$, including $1$, $5$, $10$, and $20$. 2. Take out the largest $20$ from the options, leaving $31 - 20 = 11$. @@ -47,10 +47,10 @@ The above method of organizing playing cards is essentially the "insertion sort" ![Change making process](algorithms_are_everywhere.assets/greedy_change.png) -In the steps above, we make the best choice at each step (using the largest denomination possible), ultimately arriving at a feasible change-making solution. From the perspective of data structures and algorithms, this method is essentially a "greedy" algorithm. +In the aforementioned steps, at each stage, we make the optimal choice (utilizing the highest denomination possible), ultimately deriving at a feasible change-making approach. From the perspective of data structures and algorithms, this approach is essentially a "greedy" algorithm. -From cooking a dish to interstellar travel, almost all problem-solving is inseparable from algorithms. The advent of computers allows us to store data structures in memory and write code to call CPUs and GPUs to execute algorithms. In this way, we can transfer problems from life to computers, solving various complex issues more efficiently. +From preparing a dish to traversing interstellar realms, virtually every problem-solving endeavor relies on algorithms. The emergence of computers enables us to store data structures in memory and write code to call CPUs and GPUs to execute algorithms. Consequently, we can transfer real-life predicaments to computers, efficiently addressing a myriad of complex issues. !!! tip - If you still feel only partially informed about concepts like data structures, algorithms, arrays, and binary search, please continue reading. This book will guide you into the hall of knowledge of data structures and algorithms. + If concepts such as data structures, algorithms, arrays, and binary search still seem somewhat obsecure, I encourage you to continue reading. This book will gently guide you into the realm of understanding data structures and algorithms. 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    diff --git a/docs-en/chapter_appendix/installation.md b/docs-en/chapter_appendix/installation.md deleted file mode 100644 index c23d6bacf6..0000000000 --- a/docs-en/chapter_appendix/installation.md +++ /dev/null @@ -1,55 +0,0 @@ -# Programming Environment Installation - -### VSCode - -This book recommends using the open source lightweight VSCode as a local IDE, download and install [VSCode](https://code.visualstudio.com/). - -### The Java Environment - -1. Download and install [OpenJDK](https://jdk.java.net/18/) (version > JDK 9). -2. Search for `java` in VSCode's plugin marketplace and install Extension Pack for Java. - -### C/C++ Environment - -1. Windows systems need to install [MinGW](https://sourceforge.net/projects/mingw-w64/files/) ([Configuration Tutorial](https://blog.csdn.net/qq_33698226/article/details /129031241)), MacOS comes with Clang without installation. -2. Search for `c++` in VSCode's plugin marketplace and install the C/C++ Extension Pack. -3. (Optional) Open the Settings page and search for the `Clang_format_fallback Style` code formatting option set to `{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }`. - -### Python Environment - -1. Download and install [Miniconda3](https://docs.conda.io/en/latest/miniconda.html) . -2. Search for `python` in VSCode's plugin marketplace and install the Python Extension Pack. -3. (Optional) Type `pip install black` at the command line to install the code formatting tool. - -### Go Environment - -1. Download and install [go](https://go.dev/dl/). -2. Search for `go` in VSCode's plugin marketplace and install Go. -3. The shortcut key `Ctrl + Shift + P` calls out the command bar, type go, select `Go: Install/Update Tools`, check all the boxes and install it. - -### JavaScript Environment - -1. Download and install [node.js](https://nodejs.org/en/) . -2. Install JavaScript (ES6) code snippets by searching for `javascript` in VSCode's plugin marketplace. -3. (Optional) Install the code formatting tool by searching for `Prettier` in VSCode's plugin marketplace. - -### C## Environment - -1. Download and install [.Net 6.0](https://dotnet.microsoft.com/en-us/download) . -2. Install the C# Dev Kit by searching for `C# Dev Kit` in VSCode's plugin marketplace ([Configuration Tutorial](https://code.visualstudio.com/docs/csharp/get-started)). -3. Visual Studio is also available ([Installation Tutorial](https://learn.microsoft.com/zh-cn/visualstudio/install/install-visual-studio?view=vs-2022)). - -### Swift Environment - -1. Download and install [Swift](https://www.swift.org/download/). -2. Search for `swift` in VSCode's plugin marketplace and install [Swift for Visual Studio Code](https://marketplace.visualstudio.com/items?itemName=sswg.swift-lang). - -### Dart Environment - -1. Download and install [Dart](https://dart.dev/get-dart) . -2. Search for `dart` in VSCode's plugin marketplace and install [Dart](https:/marketplace.visualstudio.com/items?itemName=Dart-Code.dart-code). - -### Rust Environment - -1. Download and install [Rust](https://www.rust-lang.org/tools/install). -2. Search for `rust` in VSCode's plugin marketplace and install [rust-analyzer](https://marketplace.visualstudio.com/items?itemName=rust-lang.rust-analyzer). diff --git a/docs-en/chapter_appendix/terminology.md b/docs-en/chapter_appendix/terminology.md deleted file mode 100644 index bf1b92208f..0000000000 --- a/docs-en/chapter_appendix/terminology.md +++ /dev/null @@ -1,114 +0,0 @@ -# Glossary - -The table below lists important terms that appear in the book. It is recommended that you memorize both their Chinese and English names in order to read the English literature. - -

    Table   Important Terms for Data Structures and Algorithms

    - -| 中文 | English | -| -------------- | ------------------------------ | -| 算法 | algorithm | -| 数据结构 | data structure | -| 渐近复杂度分析 | asymptotic complexity analysis | -| 时间复杂度 | time complexity | -| 空间复杂度 | space complexity | -| 迭代 | iteration | -| 递归 | recursion | -| 尾递归 | tail recursion | -| 递归树 | recursion tree | -| 大 $O$ 记号 | big-$O$ notation | -| 渐近上界 | asymptotic upper bound | -| 原码 | true form | -| 反码 | 1's complement code | -| 补码 | 2's complement code | -| 数组 | array | -| 索引 | index | -| 链表 | linked list | -| 链表节点 | list node | -| 列表 | list | -| 动态数组 | dynamic array | -| 栈 | stack | -| 队列 | queue | -| 双向队列 | double-ended queue | -| 哈希表 | hash table | -| 桶 | bucket | -| 哈希函数 | hash function | -| 哈希冲突 | hash collision | -| 负载因子 | load factor | -| 链式地址 | separate chaining | -| 开放寻址 | open addressing | -| 线性探测 | linear probing | -| 懒删除 | lazy deletion | -| 二叉树 | binary tree | -| 树节点 | tree node | -| 左子节点 | left-child node | -| 右子节点 | right-child node | -| 父节点 | parent node | -| 左子树 | left subtree | -| 右子树 | right subtree | -| 根节点 | root node | -| 叶节点 | leaf node | -| 边 | edge | -| 层 | level | -| 度 | degree | -| 高度 | height | -| 深度 | depth | -| 完美二叉树 | perfect binary tree | -| 完全二叉树 | complete binary tree | -| 完满二叉树 | full binary tree | -| 平衡二叉树 | balanced binary tree | -| AVL 树 | AVL tree | -| 红黑树 | red-black tree | -| 层序遍历 | level-order traversal | -| 广度优先遍历 | breadth-first traversal | -| 深度优先遍历 | depth-first traversal | -| 二叉搜索树 | binary search tree | -| 平衡二叉搜索树 | balanced binary search tree | -| 平衡因子 | balance factor | -| 堆 | heap | -| 大顶堆 | max heap | -| 小顶堆 | min heap | -| 优先队列 | priority queue | -| 堆化 | heapify | -| 图 | graph | -| 顶点 | vertex | -| 无向图 | undirected graph | -| 有向图 | directed graph | -| 连通图 | connected graph | -| 非连通图 | disconnected graph | -| 有权图 | weighted graph | -| 邻接 | adjacency | -| 路径 | path | -| 入度 | in-degree | -| 出度 | out-degree | -| 邻接矩阵 | adjacency matrix | -| 邻接表 | adjacency list | -| 广度优先搜索 | breadth-first search | -| 深度优先搜索 | depth-first search | -| 二分查找 | binary search | -| 搜索算法 | searching algorithm | -| 排序算法 | sorting algorithm | -| 选择排序 | selection sort | -| 冒泡排序 | bubble sort | -| 插入排序 | insertion sort | -| 快速排序 | quick sort | -| 归并排序 | merge sort | -| 堆排序 | heap sort | -| 桶排序 | bucket sort | -| 计数排序 | counting sort | -| 基数排序 | radix sort | -| 分治 | divide and conquer | -| 汉诺塔问题 | hanota problem | -| 回溯算法 | backtracking algorithm | -| 约束 | constraint | -| 解 | solution | -| 状态 | state | -| 剪枝 | pruning | -| 全排列问题 | permutations problem | -| 子集和问题 | subset-sum problem | -| N 皇后问题 | N-queens problem | -| 动态规划 | dynamic programming | -| 初始状态 | initial state | -| 状态转移方程 | state-trasition equation | -| 背包问题 | knapsack problem | -| 编辑距离问题 | edit distance problem | -| 贪心算法 | greedy algorithm | diff --git a/docs-en/chapter_array_and_linkedlist/array.assets/array_definition.png b/docs-en/chapter_array_and_linkedlist/array.assets/array_definition.png deleted file mode 100644 index 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"C" - - ```c title="array.c" - /* Initialize the array */ - int arr[5] = { 0 }; // { 0, 0, 0, 0, 0 } - int nums[5] = { 1, 3, 2, 5, 4 }; - ``` - -=== "Zig" - - ```zig title="array.zig" - // Initialize the array. - var arr = [_]i32{0} **5; // { 0, 0, 0, 0, 0 } - var nums = [_]i32{ 1, 3, 2, 5, 4 }; - ``` - -### Accessing Elements - -Array elements are stored in contiguous memory space, which means that calculating the memory address of an array element is very easy. Given an array memory address (i.e., the first element memory address) and the index of an element, we can use the formula shown in the figure below to calculate to get the memory address of that element, and thus access this element directly. - -![Memory address calculation for array elements](array.assets/array_memory_location_calculation.png) - -Looking at the figure above, we see that the first element of the array is indexed at $0$, which seems counter-intuitive since it would be more natural to start counting at $1$. However, from the perspective of the address calculation formula, **the meaning of index is essentially the offset of the memory address**. The address offset of the first element is $0$, so it makes sense that it is indexed as $0$. - -Accessing elements in an array is very efficient, we can randomly access any element in the array in $O(1)$ time. - -```src -[file]{array}-[class]{}-[func]{random_access} -``` - -### Insert Element - -The elements of an array are "next to each other" in memory, and there is no room for any more data between them. As shown in the figure below, if you want to insert an element in the middle of an array, you need to move all the elements after that element back one place, and then assign the element to the index. - -![Example of inserting elements into an array](array.assets/array_insert_element.png) - -It is worth noting that since the length of the array is fixed, inserting an element will always result in the "loss" of the last element of the array. We leave the solution to this problem to the section on lists. - -```src -[file]{array}-[class]{}-[func]{insert} -``` - -### Delete Element - -Similarly, as shown in the figure below, if you want to delete the element at index $i$, you need to move all the elements after index $i$ forward by one bit. - -![Example of removing elements from an array](array.assets/array_remove_element.png) - -Note that after the deletion of the element is complete, the element at the end becomes "meaningless", so we don't need to modify it. - -```src -[file]{array}-[class]{}-[func]{remove} -``` - -In general, insertion and deletion operations on arrays have the following disadvantages. - -- **High time complexity**: the average time complexity of both insertion and deletion of arrays is $O(n)$ , where $n$ is the length of the array. -- **Lost Elements**: since the length of the array is immutable, elements out of the range of the length of the array are lost after insertion. -- **Memory Waste**: We can initialize a long array and use only the first part, so that when inserting data, the missing end elements are "meaningless", but this will also result in a partial waste of memory space. - -### Traversing Array - -In most programming languages, we can either traverse an array by index or traverse directly to get each element in the array. - -```src -[file]{array}-[class]{}-[func]{traverse} -``` - -### Find An Element - -Finding the specified element in an array requires traversing the array, determining whether the element values match in each round, and outputting the corresponding index if they do. - -Because arrays are linear data structures, the above lookup operation is called "linear lookup". - -```src -[file]{array}-[class]{}-[func]{find} -``` - -### Expanded Array - -In complex system environments, it is difficult for a program to guarantee that the memory space after an array is available, thus making it impossible to safely extend the array capacity. Therefore in most programming languages, **the length of an array is immutable**. - -If we want to expand the array, we need to recreate a larger array and copy the elements of the original array to the new one in turn. This is a $O(n)$ operation, which is time-consuming when the array is very large. - -```src -[file]{array}-[class]{}-[func]{extend} -``` - -## Arrays Benefits And Limitations - -Arrays are stored in contiguous memory spaces with elements of the same type. This practice contains a wealth of a priori information that can be used by the system to optimize the operational efficiency of the data structure. - -- **Space efficient**: Arrays allocate contiguous blocks of memory for data without additional structural overhead. -- **Supports random access**: Arrays allow any element to be accessed in $O(1)$ time. -- **Cache locality**: When an array element is accessed, the computer not only loads it, but also caches the rest of the data around it, thereby leveraging the cache to speed up the execution of subsequent operations. - -Continuous space storage is a double-edged sword with the following disadvantages. - -- **Insertion and deletion are inefficient**: Insertion and deletion operations require moving a large number of elements when there are many elements in the array. -- **Length immutable**: Arrays have a fixed length after initialization, and expanding an array requires copying all the data to the new array, which has a lot of overhead. -- **Space Waste**: If an array is allocated a larger size than is actually needed, the extra space is wasted. - -## Typical Applications Of Arrays - -Arrays are a basic and common data structure that are frequently used in various types of algorithms as well as for implementing complex data structures. - -- **Random Access**: if we want to randomize some samples, then we can store them in an array and generate a random sequence to achieve random sampling of samples based on the index. -- **Sorting and Searching**: Arrays are the most commonly used data structure for sorting and searching algorithms. Quick sort, merge sort, binary search, etc. are all performed primarily on arrays. -- **Lookup Tables**: When we need to find an element quickly or need to find the correspondence of an element, we can use an array as a lookup table. If we want to implement a character to ASCII mapping, we can use the ASCII value of the character as the index and store the corresponding element in the corresponding position in the array. -- **Machine Learning**: neural networks make heavy use of linear algebra operations between vectors, matrices, and tensors, which are constructed as arrays. Arrays are the most commonly used data structure in neural network programming. -- **Data Structure Implementation**: arrays can be used to implement data structures such as stacks, queues, hash tables, heaps, graphs, and so on. For example, the adjacency matrix representation of a graph is actually a two-dimensional array. diff --git a/docs-en/chapter_array_and_linkedlist/index.md b/docs-en/chapter_array_and_linkedlist/index.md deleted file mode 100644 index d4b62790cf..0000000000 --- a/docs-en/chapter_array_and_linkedlist/index.md +++ /dev/null @@ -1,13 +0,0 @@ -# Arrays And Linked Lists - -

    - -![arrays and linked lists](../assets/covers/chapter_array_and_linkedlist.jpg){ width="600" } - -
    - -!!! abstract - - The world of data structures is like a thick brick wall. - - The bricks of the array are neatly arranged, tightly fitting one by one. 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z0V7a_^RQ4C3N1ln-c%YFI~_z93uoTFk^{0X$8-NAu{s-?%{OsHo4m6-P=^UB#&`s3 zGz*yLJl^&Vo_g>@aQ~Vv3)g?{k(+^8veH}8)zBEV_E2BqJkFb}xx(&s&0r)FA8T+{ zJc4D2lk5hQ|2$|aGiX&!j8&m}|K=ct5$SQ0nTpEW$EYQ66jydth_87M=x7>hyi{|D F{eO}}(pvxk diff --git a/docs-en/chapter_array_and_linkedlist/linked_list.md b/docs-en/chapter_array_and_linkedlist/linked_list.md deleted file mode 100644 index ca73518888..0000000000 --- a/docs-en/chapter_array_and_linkedlist/linked_list.md +++ /dev/null @@ -1,671 +0,0 @@ -# Linked List - -Memory space is a common resource for all programs, and in a complex system operating environment, free memory space may be scattered all over memory. We know that the memory space for storing an array must be contiguous, and when the array is very large, the memory may not be able to provide such a large contiguous space. This is where the flexibility of linked lists comes into play. - -A "linked list linked list" is a linear data structure in which each element is a node object and each node is connected by a "reference". The reference records the memory address of the next node, which can be accessed from the current node to the next node. - -Linked lists are designed so that individual nodes can be stored scattered throughout memory, and their memory addresses are needlessly contiguous. - -![linked list definitions and storage methods](linked_list.assets/linkedlist_definition.png) - -Looking at the figure above, a linked list is composed of "node node" objects. Each node contains two pieces of data: the node's "value" and a "reference" to the next node. - -- The first node of a linked list is called the "head node" and the last node is called the "tail node". -- The tail node points to "null", which is denoted as $\text{null}$, $\text{nullptr}$, and $\text{None}$ in Java, C++, and Python, respectively. -- In pointer-enabled languages such as C, C++, Go, and Rust, the above "reference" should be replaced with "pointer". - -As shown in the following code, a linked list node `ListNode` holds an additional reference (pointer) in addition to the value. Therefore, for the same amount of data, **linked lists take up more memory space than arrays**. - -=== "Python" - - ```python title="" - class ListNode: - """Class of linked lists nodes"""" - def __init__(self, val: int): - self.val: int = val # node value - self.next: ListNode | None = None # reference to the next node - ``` - -=== "C++" - - ```cpp title="" - /* Linked list node structure */ - struct ListNode { - int val; // node value - ListNode *next; // Pointer to the next node. - ListNode(int x) : val(x), next(nullptr) {} // constructor - }; - ``` - -=== "Java" - - ```java title="" - /* Linked list node class */ - class ListNode { - int val; // node value - ListNode next; // Reference to the next node. - ListNode(int x) { val = x; } // Constructor - } - ``` - -=== "C#" - - ```csharp title="" - /* Linked list node class */ - class ListNode { - int val; // node value - ListNode next; // Reference to the next node. - ListNode(int x) => val = x; //constructor - } - ``` - -=== "Go" - - ```go title="" - /* Linked list node structure */ - type ListNode struct { - Val int // Node value - Next *ListNode // Pointer to the next node. - } - - // The NewListNode constructor, which creates a new linked list. - func NewListNode(val int) *ListNode { - return &ListNode{ - Val: val, - Next: nil, - } - } - ``` - -=== "Swift" - - ```swift title="" - /* Linked list node class */ - class ListNode { - var val: Int // node value - var next: ListNode? // Reference to the next node. - - init(x: Int) { // constructor - val = x - } - } - ``` - -=== "JS" - - ```javascript title="" - /* Linked list node class */ - class ListNode { - constructor(val, next) { - this.val = (val === undefined ? 0 : val); // node value - this.next = (next === undefined ? null : next); // reference to next node - } - } - ``` - -=== "TS" - - ```typescript title="" - /* Linked list node class */ - class ListNode { - val: number; - next: ListNode | null; - constructor(val?: number, next?: ListNode | null) { - this.val = val === undefined ? 0 : val; // node value - this.next = next === undefined ? null : next; // reference to the next node - } - } - ``` - -=== "Dart" - - ```dart title="" - /* Linked list node class */ - class ListNode { - int val; // node value - ListNode? next; // Reference to the next node. - ListNode(this.val, [this.next]); // constructor - } - ``` - -=== "Rust" - - ```rust title="" - use std::rc::Rc; - use std::cell::RefCell; - /* Linked list node class */ - #[derive(Debug)] - struct ListNode { - val: i32, // node value - next: Option>>, // pointer to next node - } - ``` - -=== "C" - - ```c title="" - /* Linked list node structure */ - typedef struct ListNode { - int val; // node value - struct ListNode *next; // Pointer to the next node. - } ListNode; - - /* Constructors */ - ListNode *newListNode(int val) { - ListNode *node, *next; - node = (ListNode *) malloc(sizeof(ListNode)); - node->val = val; - node->next = NULL; - return node; - } - ``` - -=== "Zig" - - ```zig title="" - // Linked list node class - pub fn ListNode(comptime T: type) type { - return struct { - const Self = @This(); - - val: T = 0, // node value - next: ?*Self = null, // pointer to the next node - - // Constructor - pub fn init(self: *Self, x: i32) void { - self.val = x; - self.next = null; - } - }; - } - ``` - -## Common Operations On Linked Lists - -### Initialize A Linked List - -Creating a linked list is a two-step process. The first step is to initialize each node object, and the second step is to build a reference-pointing relationship. After initialization, we can start from the head node of the linked list node, through the reference pointing `next` to access all the nodes in turn. - -=== "Python" - - ```python title="linked_list.py" - # Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 - # Initialize individual nodes - n0 = ListNode(1) - n1 = ListNode(3) - n2 = ListNode(2) - n3 = ListNode(5) - n4 = ListNode(4) - # Construct references to point to - n0.next = n1 - n1.next = n2 - n2.next = n3 - n3.next = n4 - ``` - -=== "C++" - - ```cpp title="linked_list.cpp" - /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ - // Initialize each node - ListNode* n0 = new ListNode(1); - ListNode* n1 = new ListNode(3); - ListNode* n2 = new ListNode(2); - ListNode* n3 = new ListNode(5); - ListNode* n4 = new ListNode(4); - // Construct references to point to - n0->next = n1; - n1->next = n2; - n2->next = n3; - n3->next = n4; - ``` - -=== "Java" - - ```java title="linked_list.java" - /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ - // Initialize each node - ListNode n0 = new ListNode(1); - ListNode n1 = new ListNode(3); - ListNode n2 = new ListNode(2); - ListNode n3 = new ListNode(5); - ListNode n4 = new ListNode(4); - // Construct references to point to - n0.next = n1; - n1.next = n2; - n2.next = n3; - n3.next = n4; - ``` - -=== "C#" - - ```csharp title="linked_list.cs" - /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ - // Initialize each node - ListNode n0 = new(1); - ListNode n1 = new(3); - ListNode n2 = new(2); - ListNode n3 = new(5); - ListNode n4 = new(4); - // Construct references to point to - n0.next = n1; - n1.next = n2; - n2.next = n3; - n3.next = n4; - ``` - -=== "Go" - - ```go title="linked_list.go" - /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ - // Initialize each node - n0 := NewListNode(1) - n1 := NewListNode(3) - n2 := NewListNode(2) - n3 := NewListNode(5) - n4 := NewListNode(4) - // Construct references to point to - n0.Next = n1 - n1.Next = n2 - n2.Next = n3 - n3.Next = n4 - ``` - -=== "Swift" - - ```swift title="linked_list.swift" - /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ - // Initialize each node - let n0 = ListNode(x: 1) - let n1 = ListNode(x: 3) - let n2 = ListNode(x: 2) - let n3 = ListNode(x: 5) - let n4 = ListNode(x: 4) - // Construct references to point to - n0.next = n1 - n1.next = n2 - n2.next = n3 - n3.next = n4 - ``` - -=== "JS" - - ```javascript title="linked_list.js" - /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ - // Initialize each node - const n0 = new ListNode(1); - const n1 = new ListNode(3); - const n2 = new ListNode(2); - const n3 = new ListNode(5); - const n4 = new ListNode(4); - // Construct references to point to - n0.next = n1; - n1.next = n2; - n2.next = n3; - n3.next = n4; - ``` - -=== "TS" - - ```typescript title="linked_list.ts" - /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ - // Initialize each node - const n0 = new ListNode(1); - const n1 = new ListNode(3); - const n2 = new ListNode(2); - const n3 = new ListNode(5); - const n4 = new ListNode(4); - // Construct references to point to - n0.next = n1; - n1.next = n2; - n2.next = n3; - n3.next = n4; - ``` - -=== "Dart" - - ```dart title="linked_list.dart" - /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */\ - // Initialize each node - ListNode n0 = ListNode(1); - ListNode n1 = ListNode(3); - ListNode n2 = ListNode(2); - ListNode n3 = ListNode(5); - ListNode n4 = ListNode(4); - // Construct references to point to - n0.next = n1; - n1.next = n2; - n2.next = n3; - n3.next = n4; - ``` - -=== "Rust" - - ```rust title="linked_list.rs" - /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ - // Initialize each node - let n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None })); - let n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None })); - let n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None })); - let n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None })); - let n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None })); - - // Construct references to point to - n0.borrow_mut().next = Some(n1.clone()); - n1.borrow_mut().next = Some(n2.clone()); - n2.borrow_mut().next = Some(n3.clone()); - n3.borrow_mut().next = Some(n4.clone()); - ``` - -=== "C" - - ```c title="linked_list.c" - /* Initialize linked list 1 -> 3 -> 2 -> 5 -> 4 */ - // Initialize each node - ListNode* n0 = newListNode(1); - ListNode* n1 = newListNode(3); - ListNode* n2 = newListNode(2); - ListNode* n3 = newListNode(5); - ListNode* n4 = newListNode(4); - // Construct references to point to - n0->next = n1; - n1->next = n2; - n2->next = n3; - n3->next = n4; - ``` - -=== "Zig" - - ```zig title="linked_list.zig" - // Initialize the linked list. - // Initialize each node - var n0 = inc.ListNode(i32){.val = 1}; - var n1 = inc.ListNode(i32){.val = 3}; - var n2 = inc.ListNode(i32){.val = 2}; - var n3 = inc.ListNode(i32){.val = 5}; - var n4 = inc.ListNode(i32){.val = 4}; - // Construct references to point to - n0.next = &n1; - n1.next = &n2; - n2.next = &n3; - n3.next = &n4; - ``` - -An array as a whole is a variable, e.g., the array `nums` contains the elements `nums[0]` and `nums[1]`, etc., whereas a linked list consists of a number of separate node objects. **We usually take the head node as a surrogate for a linked list**. For example, the linked list in the above code can be notated as a linked list `n0` . - -### Insert Node - -Inserting nodes in a linked list is very easy. As shown in the figure below, suppose we want to insert a new node `P` between two neighboring nodes `n0` and `n1` , **then we just need to change the two node references (pointers)**with a time complexity of $O(1)$ . - -In contrast, inserting elements into an array has a time complexity of $O(n)$ , which is less efficient with large amounts of data. - -![Example of a linked list insert node](linked_list.assets/linkedlist_insert_node.png) - -```src -[file]{linked_list}-[class]{}-[func]{insert} -``` - -### Delete Node - -As shown in the figure below, it is also very easy to delete nodes in a linked list **just by changing the reference (pointer) to a node**. - -Note that although the node `P` still points to `n1` after the deletion operation is completed, it is not actually possible to traverse the linked list to reach `P`, which means that `P` no longer belongs to the linked list. - -![linked list remove node](linked_list.assets/linkedlist_remove_node.png) - -```src -[file]{linked_list}-[class]{}-[func]{remove} -``` - -### Access Nodes - -**It is less efficient to access nodes in a linked list**. As mentioned in the previous section, we can access any element in the array in $O(1)$ time. This is not the case for linked lists, where the program needs to start from the head node and traverse backward one by one until it finds the target node. In other words, accessing the $i$th node of the linked list requires a loop of $i - 1$ rounds, with a time complexity of $O(n)$ . - -```src -[file]{linked_list}-[class]{}-[func]{access} -``` - -### Find The Node - -Traverses the linked list, finds the node whose value is `target`, and outputs the index of the node in the linked list. This process is also a linear lookup. - -```src -[file]{linked_list}-[class]{}-[func]{find} -``` - -## Arrays vs. Linked Lists - -The table below summarizes and compares the various properties and operational efficiencies of arrays and linked lists. Since they use two opposite storage strategies, the various properties and operational efficiencies are also characterized by opposites. - -

    Table   Efficiency of Arrays vs. Linked Lists

    - -| | arrays | linked lists | -| ------------------ | ------------------------------------ | ------------------------ | -| Storage Methods | Continuous Memory Space | Distributed Memory Space | -| Cache Localization | Friendly | Unfriendly | -| Capacity Expansion | Non-variable Length | Flexible Expansion | -| Memory Efficiency | Less memory usage, some space wasted | More memory usage | -| access elements | $O(1)$ | $O(n)$ | -| Add Element | $O(n)$ | $O(1)$ | -| Delete Element | $O(n)$ | $O(1)$ | - -## Common Linked List Types - -As shown in the figure below, there are three common types of linked lists. - -- **Unidirectional linked list**: i.e., the ordinary linked list introduced above. The nodes of a unidirectional linked list contain two pieces of data, the value and a reference to the next node. We will refer to the first node as the head node and the last node as the tail node, which points to null $\text{None}$ . -- **Ring linked list**: If we make the tail node of a unidirectional linked list point to the head node (i.e., the head and tail are connected), we get a ring linked list. In a ring linked list node any node can be considered as the head node. -- **Bidirectional linked lists**: In contrast to unidirectional linked lists, bidirectional linked lists record references in both directions. The node definition of a bi-directional linked list contains references (pointers) to both the successor node (next node) and the predecessor node (previous node). Compared to unidirectional linked lists, bi-directional linked lists are more flexible and can be traversed in both directions, but accordingly require more memory space. - -=== "Python" - - ```python title="" - class ListNode: - """Bidirectional linked list node class"""" - def __init__(self, val: int): - self.val: int = val # node value - self.next: ListNode | None = None # reference to the successor node - self.prev: ListNode | None = None # reference to a predecessor node - ``` - -=== "C++" - - ```cpp title="" - /* Bidirectional linked list node structure */ - struct ListNode { - int val; // node value - ListNode *next; // Pointer to the successor node. - ListNode *prev; // pointer to the predecessor node - ListNode(int x) : val(x), next(nullptr), prev(nullptr) {} // constructor - }; - ``` - -=== "Java" - - ```java title="" - /* Bidirectional linked list node class */ - class ListNode { - int val; // node value - ListNode next; // Reference to the next node. - ListNode prev; // reference to the predecessor node - ListNode(int x) { val = x; } // Constructor - } - ``` - -=== "C#" - - ```csharp title="" - /* Bidirectional linked list node class */ - class ListNode { - int val; // node value - ListNode next; // Reference to the next node. - ListNode prev; // reference to the predecessor node - ListNode(int x) => val = x; // constructor - } - ``` - -=== "Go" - - ```go title="" - /* Bidirectional linked list node structure */ - type DoublyListNode struct { - Val int // Node value - Next *DoublyListNode // Pointer to the successor node. - Prev *DoublyListNode // pointer to the predecessor node - } - - // NewDoublyListNode initialization - func NewDoublyListNode(val int) *DoublyListNode { - return &DoublyListNode{ - Val: val, - Next: nil, - Prev: nil, - } - } - ``` - -=== "Swift" - - ```swift title="" - /* Bidirectional linked list node class */ - class ListNode { - var val: Int // node value - var next: ListNode? // Reference to the next node. - var prev: ListNode? // Reference to the predecessor node. - - init(x: Int) { // constructor - val = x - } - } - ``` - -=== "JS" - - ```javascript title="" - /* Bidirectional linked list node class */ - class ListNode { - constructor(val, next, prev) { - this.val = val === undefined ? 0 : val; // node value - this.next = next === undefined ? null : next; // reference to the successor node - this.prev = prev === undefined ? null : prev; // reference to the predecessor node - } - } - ``` - -=== "TS" - - ```typescript title="" - /* Bidirectional linked list node class */ - class ListNode { - val: number; - next: ListNode | null; - prev: ListNode | null; - constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) { - this.val = val === undefined ? 0 : val; // node value - this.next = next === undefined ? null : next; // reference to the successor node - this.prev = prev === undefined ? null : prev; // reference to the predecessor node - } - } - ``` - -=== "Dart" - - ```dart title="" - /* Bidirectional linked list node class */ - class ListNode { - int val; // node value - ListNode next; // Reference to the next node. - ListNode prev; // reference to the predecessor node - ListNode(this.val, [this.next, this.prev]); // Constructor - } - ``` - -=== "Rust" - - ```rust title="" - use std::rc::Rc; - use std::cell::RefCell; - - /* Bidirectional linked list node type */ - #[derive(Debug)] - struct ListNode { - val: i32, // node value - next: Option>>, // pointer to successor node - prev: Option>>, // pointer to predecessor node - } - - /* Constructors */ - impl ListNode { - fn new(val: i32) -> Self { - ListNode { - val, - next: None, - prev: None, - } - } - } - ``` - -=== "C" - - ```c title="" - /* Bidirectional linked list node structure */ - typedef struct ListNode { - int val; // node value - struct ListNode *next; // Pointer to the successor node. - struct ListNode *prev; // pointer to the predecessor node - } ListNode; - - /* Constructors */ - ListNode *newListNode(int val) { - ListNode *node, *next; - node = (ListNode *) malloc(sizeof(ListNode)); - node->val = val; - node->next = NULL; - node->prev = NULL; - return node; - } - ``` - -=== "Zig" - - ```zig title="" - // Bidirectional linked list node class - pub fn ListNode(comptime T: type) type { - return struct { - const Self = @This(); - - val: T = 0, // node value - next: ?*Self = null, // pointer to the successor node - prev: ?*Self = null, // pointer to the predecessor node - - // Constructor - pub fn init(self: *Self, x: i32) void { - self.val = x; - self.next = null; - self.prev = null; - } - }; - } - ``` - -![Common linked list types](linked_list.assets/linkedlist_common_types.png) - -## Typical Applications Of Linked Lists - -One-way linked lists are commonly used to implement data structures such as stacks, queues, hash tables and graphs. - -- **Stack vs. Queue**: when both insertion and deletion operations are performed at one end of a linked list, it exhibits a FIFO characteristic, corresponding to a stack; when insertion operations are performed at one end of a linked list and deletion operations are performed at the other end of a linked list, it exhibits a FIFO characteristic, corresponding to a queue. -- **Hash table**: the linked address method is one of the dominant solutions to hash collisions, in which all conflicting elements are put into a linked list. -- **Graph**: An adjacency list is a common way of representing a graph, in which each vertex of the graph is associated with a linked list, and each element of the linked list represents the other vertices connected to that vertex. - -Bidirectional linked lists are often used in scenarios where you need to quickly find the previous and next elements. - -- **Advanced Data Structures**: e.g. in red-black tree, B-tree, we need to access the parent node of a node, this can be done by keeping a reference to the parent node in the node, similar to a bidirectional linked list. -- **Browser History**: In web browsers, when a user clicks on the forward or back button, the browser needs to know the previous and next web pages that the user has visited. The property of bi-directional linked lists makes this operation easy. -- **LRU algorithm**: In a cache elimination algorithm (LRU), we need to quickly find the least recently used data as well as support adding and removing nodes quickly. This is where the use of bi-directional linked lists is very appropriate. - -Circular linked lists are often used in scenarios that require periodic operations, such as resource scheduling in operating systems. - -- **Time Slice Rotation Scheduling Algorithm**: In operating systems, the time slice rotation scheduling algorithm is a common CPU scheduling algorithm that involves looping over a set of processes. Each process is given a time slice and when the time slice runs out, the CPU switches to the next process. The operation of such a loop can then be realized by means of a circular linked list. -- **Data Buffers**: In some data buffer implementations, circular linked lists may also be used. For example, in audio and video players, the data stream may be divided into multiple buffer blocks and put into a circular linked list for seamless playback. diff --git a/docs-en/chapter_array_and_linkedlist/list.md b/docs-en/chapter_array_and_linkedlist/list.md deleted file mode 100644 index 45e792c44e..0000000000 --- a/docs-en/chapter_array_and_linkedlist/list.md +++ /dev/null @@ -1,872 +0,0 @@ -# List - -"A list is an abstract data structure that represents an ordered collection of elements and supports operations such as accessing, modifying, adding, deleting, and traversing elements without the need to consider capacity constraints. Lists can be implemented based on linked lists or arrays. - -- A linked list can naturally be viewed as a list, which supports element addition, deletion, checking, and modification operations, and can be flexibly and dynamically expanded. -- Arrays also support adding, deleting, checking, and changing elements, but because their length is immutable, they can only be viewed as a list with a length limit. - -When implementing lists using arrays, **the immutable nature of the length causes lists to be less useful**. This is because we usually cannot determine in advance how much data we need to store, making it difficult to choose an appropriate list length. If the length is too small, it is likely to be unsatisfactory for use; if it is too large, it will result in wasted memory space. - -To solve this problem, we can use "dynamic array dynamic array" to implement the list. It inherits the advantages of arrays and can be dynamically expanded during the program. - -In fact, the standard library in many programming languages provides lists based on dynamic array implementations, such as `list` in Python, `ArrayList` in Java, `vector` in C++, and `List` in C#. For the rest of this discussion, we'll treat "lists" and "dynamic arrays" as equivalent concepts. - -## Common Operations On Lists - -### Initialize The List. - -We usually use the two initialization methods "without initial value" and "with initial value". - -=== "Python" - - ```python title="list.py" - # Initialize the list - # No initial value - nums1: list[int] = [] - # With initial values - nums: list[int] = [1, 3, 2, 5, 4] - ``` - -=== "C++" - - ```cpp title="list.cpp" - /* Initialize list */ - // Note that in C++, vectors are the nums described in this article. - // No initial value - vector nums1; - // With initial values - vector nums = { 1, 3, 2, 5, 4 }; - ``` - -=== "Java" - - ```java title="list.java" - /* Initialize list */ - // No initial value - List nums1 = new ArrayList<>(); - // with initial values (note that the element type of the array needs to be Integer[], the wrapper class for int[]) - Integer[] numbers = new Integer[] { 1, 3, 2, 5, 4 }; - List nums = new ArrayList<>(Arrays.asList(numbers)); - ``` - -=== "C#" - - ```csharp title="list.cs" - /* Initialize list */ - // No initial value - List nums1 = new(); - // With initial values - int[] numbers = new int[] { 1, 3, 2, 5, 4 }; - List nums = numbers.ToList(); - ``` - -=== "Go" - - ```go title="list_test.go" - /* Initialize list */ - // No initial value - nums1 := []int - // With initial values - nums := []int{1, 3, 2, 5, 4} - ``` - -=== "Swift" - - ```swift title="list.swift" - /* Initialize list */ - // No initial value - let nums1: [Int] = [] - // With initial values - var nums = [1, 3, 2, 5, 4] - ``` - -=== "JS" - - ```javascript title="list.js" - /* Initialize list */ - // No initial value - const nums1 = []; - // With initial values - const nums = [1, 3, 2, 5, 4]; - ``` - -=== "TS" - - ```typescript title="list.ts" - /* Initialize list */ - // No initial value - const nums1: number[] = []; - // With initial values - const nums: number[] = [1, 3, 2, 5, 4]; - ``` - -=== "Dart" - - ```dart title="list.dart" - /* Initialize list */ - // No initial value - List nums1 = []; - // With initial values - List nums = [1, 3, 2, 5, 4]; - ``` - -=== "Rust" - - ```rust title="list.rs" - /* Initialize list */ - // No initial value - let nums1: Vec = Vec::new(); - // With initial values - let nums: Vec = vec![1, 3, 2, 5, 4]; - ``` - -=== "C" - - ```c title="list.c" - // C does not provide a built-in dynamic array. - ``` - -=== "Zig" - - ```zig title="list.zig" - // Initialization list - var nums = std.ArrayList(i32).init(std.heap.page_allocator); - defer nums.deinit(); - try nums.appendSlice(&[_]i32{ 1, 3, 2, 5, 4 }); - ``` - -### Accessing Elements - -Lists are essentially arrays, so it is efficient to access and update elements in $O(1)$ time. - -=== "Python" - - ```python title="list.py" - # Access to elements - num: int = nums[1] # access the element at index 1 - - # Update elements - nums[1] = 0 # update element at index 1 to 0 - ``` - -=== "C++" - - ```cpp title="list.cpp" - /* Access to elements */ - int num = nums[1]; // access the element at index 1 - - /* Update elements */ - nums[1] = 0; // update element at index 1 to 0 - ``` - -=== "Java" - - ```java title="list.java" - /* Access to elements */ - int num = nums.get(1); // access the element at index 1 - - /* Update elements */ - nums.set(1, 0); // update the element at index 1 to 0 - ``` - -=== "C#" - - ```csharp title="list.cs" - /* Access to elements */ - int num = nums[1]; // access the element at index 1 - - /* Update elements */ - nums[1] = 0; // update element at index 1 to 0 - ``` - -=== "Go" - - ```go title="list_test.go" - /* Access to elements */ - num := nums[1] // access the element at index 1 - - /* Update elements */ - nums[1] = 0 // update the element at index 1 to 0 - ``` - -=== "Swift" - - ```swift title="list.swift" - /* Access to elements */ - let num = nums[1] // access the element at index 1 - - /* Update elements */ - nums[1] = 0 // update the element at index 1 to 0 - ``` - -=== "JS" - - ```javascript title="list.js" - /* Access to elements */ - const num = nums[1]; // access the element at index 1 - - /* Update elements */ - nums[1] = 0; // update element at index 1 to 0 - ``` - -=== "TS" - - ```typescript title="list.ts" - /* Access to elements */ - const num: number = nums[1]; // access the element at index 1 - - /* Update elements */ - nums[1] = 0; // update element at index 1 to 0 - ``` - -=== "Dart" - - ```dart title="list.dart" - /* Access to elements */ - int num = nums[1]; // access the element at index 1 - - /* Update elements */ - nums[1] = 0; // update element at index 1 to 0 - ``` - -=== "Rust" - - ```rust title="list.rs" - /* Access to elements */ - let num: i32 = nums[1]; // access the element at index 1 - /* Update elements */ - nums[1] = 0; // update element at index 1 to 0 - ``` - -=== "C" - - ```c title="list.c" - // C does not provide a built-in dynamic array. - ``` - -=== "Zig" - - ```zig title="list.zig" - // Accessing elements - var num = nums.items[1]; // access the element at index 1 - - // Update the element - nums.items[1] = 0; // update the element at index 1 to 0 - ``` - -### Inserting And Deleting Elements - -Compared to arrays, lists are free to add and remove elements. Adding elements to the end of the list has a time complexity of $O(1)$ , but inserting and deleting elements is still as efficient as an array with a time complexity of $O(n)$ . - -=== "Python" - - ```python title="list.py" - # Empty the list - nums.clear() - - # Add elements to the end - nums.append(1) - nums.append(3) - nums.append(2) - nums.append(5) - nums.append(4) - - # Insert elements in the middle - nums.insert(3, 6) # insert the number 6 at index 3 - - # Delete element - nums.pop(3) # remove the element at index 3 - ``` - -=== "C++" - - ```cpp title="list.cpp" - /* Empty list */ - nums.clear(); - - /* Add element at the end */ - nums.push_back(1); - nums.push_back(3); - nums.push_back(2); - nums.push_back(5); - nums.push_back(4); - - /* Insertion of elements in the middle */ - nums.insert(nums.begin() + 3, 6); // insert the number 6 at index 3 - - /* Delete element */ - nums.erase(nums.begin() + 3); // remove the element at index 3 - ``` - -=== "Java" - - ```java title="list.java" - /* Empty list */ - nums.clear(); - - /* Add element at the end */ - nums.add(1); - nums.add(3); - nums.add(2); - nums.add(5); - nums.add(4); - - /* Insertion of elements in the middle */ - nums.add(3, 6); // insert the number 6 at index 3 - - /* Delete element */ - nums.remove(3); // Remove the element at index 3. - ``` - -=== "C#" - - ```csharp title="list.cs" - /* Empty list */ - nums.Clear(); - - /* Add element at the end */ - nums.Add(1); - nums.Add(3); - nums.Add(2); - nums.Add(5); - nums.Add(4); - - /* Insertion of elements in the middle */ - nums.Insert(3, 6); - - /* Delete element */ - nums.RemoveAt(3); - ``` - -=== "Go" - - ```go title="list_test.go" - /* Empty list */ - nums = nil - - /* Add element at the end */ - nums = append(nums, 1) - nums = append(nums, 3) - nums = append(nums, 2) - nums = append(nums, 5) - nums = append(nums, 4) - - /* Insertion of elements in the middle */ - nums = append(nums[:3], append([]int{6}, nums[3:]...)...) // Insert the number 6 at index 3. - - /* Delete element */ - nums = append(nums[:3], nums[4:]...) // Delete the element at index 3. - ``` - -=== "Swift" - - ```swift title="list.swift" - /* Empty list */ - nums.removeAll() - - /* Add element at the end */ - nums.append(1) - nums.append(3) - nums.append(2) - nums.append(5) - nums.append(4) - - /* Insertion of elements in the middle */ - nums.insert(6, at: 3) // insert the number 6 at index 3 - - /* Delete element */ - nums.remove(at: 3) // remove the element at index 3 - ``` - -=== "JS" - - ```javascript title="list.js" - /* Empty list */ - nums.length = 0; - - /* Add element at the end */ - nums.push(1); - nums.push(3); - nums.push(2); - nums.push(5); - nums.push(4); - - /* Insertion of elements in the middle */ - nums.splice(3, 0, 6); - - /* Delete element */ - nums.splice(3, 1); - ``` - -=== "TS" - - ```typescript title="list.ts" - /* Empty list */ - nums.length = 0; - - /* Add element at the end */ - nums.push(1); - nums.push(3); - nums.push(2); - nums.push(5); - nums.push(4); - - /* Insertion of elements in the middle */ - nums.splice(3, 0, 6); - - /* Delete element */ - nums.splice(3, 1); - ``` - -=== "Dart" - - ```dart title="list.dart" - /* Empty list */ - nums.clear(); - - /* Add element at the end */ - nums.add(1); - nums.add(3); - nums.add(2); - nums.add(5); - nums.add(4); - - /* Insertion of elements in the middle */ - nums.insert(3, 6); // insert the number 6 at index 3 - - /* Delete element */ - nums.removeAt(3); // Remove the element at index 3. - ``` - -=== "Rust" - - ```rust title="list.rs" - /* Empty list */ - nums.clear(); - - /* Add element at the end */ - nums.push(1); - nums.push(3); - nums.push(2); - nums.push(5); - nums.push(4); - - /* Insertion of elements in the middle */ - nums.insert(3, 6); // insert the number 6 at index 3 - - /* Delete element */ - nums.remove(3); // Remove the element at index 3. - ``` - -=== "C" - - ```c title="list.c" - // C does not provide a built-in dynamic array. - ``` - -=== "Zig" - - ```zig title="list.zig" - // Empty the list. - nums.clearRetainingCapacity(); - - // Add elements to the end - try nums.append(1); - try nums.append(3); - try nums.append(2); - try nums.append(5); - try nums.append(4); - - // Insert elements in the middle - try nums.insert(3, 6); // insert the number 6 at index 3 - - // Delete the element - _ = nums.orderedRemove(3); // remove the element at index 3 - ``` - -### Traverse The List. - -As with arrays, lists can be traversed by index or by traversing the elements directly. - -=== "Python" - - ```python title="list.py" - # Iterate through the list by indexes - count = 0 - for i in range(len(nums)): - count += 1 - - # Iterate over the list elements directly - count = 0 - for num in nums: - count += 1 - ``` - -=== "C++" - - ```cpp title="list.cpp" - /* Iterate through the list by index */ - int count = 0; - for (int i = 0; i < nums.size(); i++) { - count++; - } - - /* Iterate directly over the list elements */ - count = 0; - for (int num : nums) { - count++; - } - ``` - -=== "Java" - - ```java title="list.java" - /* Iterate through the list by index */ - int count = 0; - for (int i = 0; i < nums.size(); i++) { - count++; - } - - /* Iterate directly over the list elements */ - count = 0; - for (int num : nums) { - count++; - } - ``` - -=== "C#" - - ```csharp title="list.cs" - /* Iterate through the list by index */ - int count = 0; - for (int i = 0; i < nums.Count; i++) { - count++; - } - - /* Iterate directly over the list elements */ - count = 0; - foreach (int num in nums) { - count++; - } - ``` - -=== "Go" - - ```go title="list_test.go" - /* Iterate through the list by index */ - count := 0 - for i := 0; i < len(nums); i++ { - count++ - } - - /* Iterate directly over the list elements */ - count = 0 - for range nums { - count++ - } - ``` - -=== "Swift" - - ```swift title="list.swift" - /* Iterate through the list by index */ - var count = 0 - for _ in nums.indices { - count += 1 - } - - /* Iterate directly over the list elements */ - count = 0 - for _ in nums { - count += 1 - } - ``` - -=== "JS" - - ```javascript title="list.js" - /* Iterate through the list by index */ - let count = 0; - for (let i = 0; i < nums.length; i++) { - count++; - } - - /* Iterate directly over the list elements */ - count = 0; - for (const num of nums) { - count++; - } - ``` - -=== "TS" - - ```typescript title="list.ts" - /* Iterate through the list by index */ - let count = 0; - for (let i = 0; i < nums.length; i++) { - count++; - } - - /* Iterate directly over the list elements */ - count = 0; - for (const num of nums) { - count++; - } - ``` - -=== "Dart" - - ```dart title="list.dart" - /* Iterate through the list by index */ - int count = 0; - for (int i = 0; i < nums.length; i++) { - count++; - } - - /* Iterate directly over the list elements */ - count = 0; - for (int num in nums) { - count++; - } - ``` - -=== "Rust" - - ```rust title="list.rs" - /* Iterate through the list by index */ - let mut count = 0; - for (index, value) in nums.iter().enumerate() { - count += 1; - } - - /* Iterate directly over the list elements */ - let mut count = 0; - for value in nums.iter() { - count += 1; - } - ``` - -=== "C" - - ```c title="list.c" - // C does not provide a built-in dynamic array. - ``` - -=== "Zig" - - ```zig title="list.zig" - // Iterate through the list by indexes - var count: i32 = 0; - var i: i32 = 0; - while (i < nums.items.len) : (i += 1) { - count += 1; - } - - // Iterate directly over the list elements. - count = 0; - for (nums.items) |_| { - count += 1; - } - ``` - -### Splice The List - -Given a new list `nums1` , we can splice that list onto the end of the original list. - -=== "Python" - - ```python title="list.py" - # Splicing two lists together - nums1: list[int] = [6, 8, 7, 10, 9] - nums += nums1 # splice list nums1 after nums - ``` - -=== "C++" - - ```cpp title="list.cpp" - /* Splice two lists */ - vector nums1 = { 6, 8, 7, 10, 9 }; - // Splice the list nums1 after nums. - nums.insert(nums.end(), nums1.begin(), nums1.end()); - ``` - -=== "Java" - - ```java title="list.java" - /* Splice two lists */ - List nums1 = new ArrayList<>(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 })); - nums.addAll(nums1); // splice the list nums1 after nums - ``` - -=== "C#" - - ```csharp title="list.cs" - /* Splice two lists */ - List nums1 = new() { 6, 8, 7, 10, 9 }; - nums.AddRange(nums1); // splice the list nums1 after nums - ``` - -=== "Go" - - ```go title="list_test.go" - /* Splice two lists */ - nums1 := []int{6, 8, 7, 10, 9} - nums = append(nums, nums1...) // Append the list nums1 to the list after nums. - ``` - -=== "Swift" - - ```swift title="list.swift" - /* Splice two lists */ - let nums1 = [6, 8, 7, 10, 9] - nums.append(contentsOf: nums1) // splice list nums1 after nums - ``` - -=== "JS" - - ```javascript title="list.js" - /* Splice two lists */ - const nums1 = [6, 8, 7, 10, 9]; - nums.push(...nums1); // splice the list nums1 after nums - ``` - -=== "TS" - - ```typescript title="list.ts" - /* Splice two lists */ - const nums1: number[] = [6, 8, 7, 10, 9]; - nums.push(...nums1); // splice the list nums1 after nums - ``` - -=== "Dart" - - ```dart title="list.dart" - /* Splice two lists */ - List nums1 = [6, 8, 7, 10, 9]; - nums.addAll(nums1); // splice the list nums1 after nums - ``` - -=== "Rust" - - ```rust title="list.rs" - /* Splice two lists */ - let nums1: Vec = vec![6, 8, 7, 10, 9]; - nums.extend(nums1); - ``` - -=== "C" - - ```c title="list.c" - // C does not provide a built-in dynamic array. - ``` - -=== "Zig" - - ```zig title="list.zig" - // Splicing two lists together - var nums1 = std.ArrayList(i32).init(std.heap.page_allocator); - defer nums1.deinit(); - try nums1.appendSlice(&[_]i32{ 6, 8, 7, 10, 9 }); - try nums.insertSlice(nums.items.len, nums1.items); // splice list nums1 after nums - ``` - -### Sort The List - -Once the list is sorting algorithm is complete, we can use the "binary search" and "double pointer" algorithms that are often examined in array-based algorithmic problems. - -=== "Python" - - ```python title="list.py" - # Sort the list - nums.sort() # After sorting, list elements are ordered from smallest to largest - ``` - -=== "C++" - - ```cpp title="list.cpp" - /* Sort list */ - sort(nums.begin(), nums.end()); // after sorting, the list elements are sorted from smallest to largest - ``` - -=== "Java" - - ```java title="list.java" - /* Sort list */ - Collections.sort(nums); // after sorting, the list elements are ordered from smallest to largest - ``` - -=== "C#" - - ```csharp title="list.cs" - /* Sort list */ - nums.Sort(); // After sorting, the list elements are ordered from smallest to largest - ``` - -=== "Go" - - ```go title="list_test.go" - /* Sort list */ - sort.Ints(nums) // After sorting, the list elements are sorted from smallest to largest - ``` - -=== "Swift" - - ```swift title="list.swift" - /* Sort list */ - nums.sort() // After sorting, the list elements are arranged from smallest to largest - ``` - -=== "JS" - - ```javascript title="list.js" - /* Sort list */ - nums.sort((a, b) => a - b); // after sorting, the list elements are ordered from smallest to largest - ``` - -=== "TS" - - ```typescript title="list.ts" - /* Sort list */ - nums.sort((a, b) => a - b); // after sorting, the list elements are ordered from smallest to largest - ``` - -=== "Dart" - - ```dart title="list.dart" - /* Sort list */ - nums.sort(); // after sorting, the list elements are sorted from smallest to largest - ``` - -=== "Rust" - - ```rust title="list.rs" - /* Sort list */ - nums.sort(); // after sorting, the list elements are sorted from smallest to largest - ``` - -=== "C" - - ```c title="list.c" - // C does not provide a built-in dynamic array. - ``` - -=== "Zig" - - ```zig title="list.zig" - // Sort the list - std.sort.sort(i32, nums.items, {}, comptime std.sort.asc(i32)); - ``` - -## List Implementations - -Many programming languages provide built-in lists, such as Java, C++, Python, and so on. Their implementation is more complex, and the setting of each parameter is also very delicate, such as the initial capacity, expansion multiplier and so on. Interested readers can check the source code to learn. - -To deepen the solution of how lists work, let's try to implement a simple version of a list, including the following three key designs. - -- **Initial Capacity**: pick a reasonable initial capacity for the array. In this example, we choose 10 as the initial capacity. -- **Quantity Record**: declares a variable `size` that is used to record the current number of elements in the list and is updated in real time as elements are inserted and deleted. Based on this variable, we can locate the tail of the list, as well as determine if it needs to be expanded. -- **Expansion Mechanism**: If the list capacity is full when an element is inserted, it needs to be expanded. First, a larger array is created based on the expansion factor, and then all the elements of the current array are moved to the new array in turn. In this example, we specify that the array should be expanded to twice its previous size each time. - -```src -[file]{my_list}-[class]{my_list}-[func]{} -``` diff --git a/docs-en/chapter_array_and_linkedlist/summary.md b/docs-en/chapter_array_and_linkedlist/summary.md deleted file mode 100644 index 2f6010195d..0000000000 --- a/docs-en/chapter_array_and_linkedlist/summary.md +++ /dev/null @@ -1,78 +0,0 @@ -# Summary - -### Highlights - -- Arrays and linked lists are two basic data structures that represent two ways of storing data in computer memory: continuous space storage and decentralized space storage, respectively. The characteristics of the two present complementary properties. -- Arrays support random access and take up less memory; however, inserting and deleting elements is inefficient and the length is immutable after initialization. -- The linked list achieves efficient node insertion and deletion by changing references (pointers) and can flexibly adjust its length; however, node access is inefficient and occupies more memory. -- Common types of linked lists include unidirectional linked lists, circular linked lists, and bidirectional linked lists, which have their own application scenarios. -- A list is an ordered collection of elements that supports adding, deleting, checking and changing, usually based on a dynamic array, which retains the advantages of arrays and can be flexibly adjusted in length. -- The advent of lists has dramatically increased the utility of arrays, but may result in some wasted memory space. - -### Q & A - -!!! Question "Does storing an array on the stack versus storing it on the heap have an effect on time efficiency and space efficiency?" - - Arrays stored on both the stack and the heap are stored in contiguous memory space, and the efficiency of data manipulation is essentially the same. However, the stack and heap have their own characteristics that lead to the following differences. - - 1. Allocation and release efficiency: the stack is a smaller piece of memory, and allocation is done automatically by the compiler; whereas heap memory is relatively larger, can be allocated dynamically in code, and is more prone to fragmentation. As a result, allocation and release operations on the heap are usually slower than on the stack. - 2. Size Limit: The stack memory is relatively small, and the size of the heap is generally limited by the available memory. Therefore heap is more suitable for storing large arrays. - 3. Flexibility: the size of an array on the stack needs to be determined at compile time, while the size of an array on the heap can be determined dynamically at runtime. - -!!! question "Why do arrays require elements of the same type, but in linked lists the same type is not emphasized?" - - A linked list consists of nodes, which are connected to each other by references (pointers), and each node can store different types of data, such as int, double, string, object, and so on. - - In contrast, the elements of an array must be of the same type so that the offset can be calculated to obtain the corresponding element position. For example, if the array contains both int and long types, and a single element occupies 4 bytes and 8 bytes respectively, then the following formula cannot be used to calculate the offset, because the array contains elements of both lengths. - - ```shell - # Element memory address = array memory address + element length * element index - ``` - -!!! question "Do I need to set `P.next` to $\text{None}$ after deleting a node?" - - It is possible to leave `P.next` unchanged. From the point of view of the linked list, traversing from the head node to the tail node no longer encounters `P`. This means that the node `P` has been deleted from the linked list, and the node `P` will not be affected by where it points. - - From a garbage collection point of view, for languages like Java, Python, Go, etc. that have automatic garbage collection, whether or not a node `P` is reclaimed depends on whether or not a reference to it still exists, not on the value of `P.next`. In languages like C and C++, we need to manually free node memory. - -!!! question "The time complexity of insertion and deletion operations in a linked list is $O(1)$ . But before adding and deleting, you need to $O(n)$ find the element, so why is the time complexity not $O(n)$?" - - It is true that it is $O(n)$ if the elements are looked up first and then deleted. However, the advantage of $O(1)$ addition and deletion of linked lists can be realized in other applications. For example, a double-ended queue is suitable for implementation using a linked list, where we maintain a pointer variable always pointing to the head node, the tail node, and each insertion and deletion operation is $O(1)$ . - -!!! question "In the figure "Definition and storage of a linked list", does the light blue pointer to the storage node take up a block of memory? Or does it take up half of the memory address and half of the node value?" - - The graphs in the text are only qualitative representations; quantitative representations need to be analyzed on a case-by-case basis. - - - Different types of node values take up different amounts of space, such as int, long, double, and instance objects. - - The amount of memory space occupied by pointer variables depends on the operating system and compilation environment used, mostly 8 bytes or 4 bytes. - -!!! question "Is adding an element to the end of the list $O(1)$ at all times?" - - If you add an element beyond the length of the list, you need to expand the list before adding it. The system will request a new block of memory and carry over all the elements of the original list, at which point the time complexity will be $O(n)$ . - -!!! question ""The advent of lists greatly enhances the utility of arrays, but the side effect is that they cause some memory space to be wasted", where space wasted is the memory taken up by extra added variables such as capacity, length, and expansion multiplier?" - - There are two main implications of wasted space here: on the one hand, lists are set to an initial length, and we don't necessarily need to use that much. On the other hand, in order to prevent frequent expansions, expansions are usually multiplied by a factor, like $\times 1.5$ . This also leaves a lot of empty spaces, and we usually can't fill them completely. - -!!! question "After initializing `n = [1, 2, 3]` in Python, the addresses of these 3 elements are linked, but initializing `m = [2, 1, 3]` reveals that the ids of each of these elements are not contiguous, but are the same as those in `n`. These elements are not contiguous, so is `m` still an array?" - - If we replace the list elements with linked list nodes `n = [n1, n2, n3, n4, n5]` , normally these five node objects are also scattered in memory. However, given a list index, we can still get the node memory address in $O(1)$ time to access the corresponding node. This is because the references to the nodes are stored in the array, not the nodes themselves. - - Unlike many languages, in Python numbers are also wrapped as objects, and what is stored in the list is not the number itself, but a reference to the number. As a result, you'll find two arrays of identical numbers with the same id, and the memory addresses of those numbers are needlessly contiguous. - -!!! question "C++ STL inside `std::list` already implements bi-directional linked lists, but it seems that some algorithm books don't really use this directly, is there some limitation?" - - On the one hand, we tend to favor the use of arrays for implementing algorithms and use linked lists only when necessary for two main reasons. - - - Space overhead: `std::list` usually takes more space than `std::vector` since each element requires two extra pointers (one for the previous element and one for the next). - - Cache-unfriendly: `std::list` has a low utilization of cache because the data is not stored continuously. In general, `std::vector` has better performance. - - On the other hand, the cases where it is necessary to use linked lists are mainly binary trees and graphs. Stacks and queues tend to use the `stack` and `queue` provided by the programming language rather than linked lists. - -!!! question "Does the initialize list `res = [0] * self.size()` operation cause each element of `res` to reference the same address?" - - It won't. But two-dimensional arrays can have this problem, for example initializing a two-dimensional list `res = [[0] * self.size()]` references the same list `[0]` multiple times. - -!!! question "In deleting a node, is it necessary to break the reference pointing between that node and its successor?" - - From a data structure and algorithm (do-it-yourself) point of view, it doesn't matter if you don't disconnect, as long as you make sure the logic of the program is correct. From the point of view of the standard library, it is safer and more logical to disconnect. 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ztE1q3YUX`Nz&Vi`YxD~rDVM(gVYT~D{|ECgmr|GvyYZ}BVQgKn-aXG@Mv1!P3RN(p z9`*UszUp+(Q!<3bFCd<8cxV#Z@y(ne^ijYOoTX$^^E=mkY%sAsdA}{-^=B%a{O%Pb zd9(fpNg_ne!e+CZ#wVI~#6hFnTwzSUatDDaiq?f_J5e^=S;$)l`8-?Tg??xCy3yYV zzJ&vm61rYXVLGx@1X*%YTUw-=v?`ghcT)>T7MDWZ5F_X_$`P@3G>eg zO&X>O>E&ZuyzrpWSV#))3T>==wC4J1NUXgT@!mFN!522?IQKj64AG4@)9H%xCk}46 zi1uN;FEIwD-}}fc_>xL2;79ccU{trC+`Lk{sku9EL#~+rzV6SfZ<~WXzCyW*(hZjg za&3Fr5hD1r{tEc0X>XtrW;bvj(VQaQ<`)^XUFM=A0;w+p&p5h%4hVrUpo2PoC)7S~ zl|s!uMoAJPt?=e+ceKF?)>Xv7QEO#p8Gjf5oe(AAa(M$xznBm?=ODZv(bY}{0J8LQ z`7QHc?FkzkY(M~M3;M=(M(%B`6+A>o34@{-t2KapaTuK`r2gz@gaU?luvO{?+j8ay zMXPv$mr4O%%)*2g`Jx80!QTrhQO14GN+vMFn z6>&P`!@mvy9d9xss&U^>d$gU0n4SO01F8U7b1(q-;V&M8n)!f*24vvZWRzFtM8Y(5 zmK;E%XQpq|_&pAGV8}msw#v;=A(&nxib`7v20}7uOr+@apT<$3KCufZSX0ZLq(lp9Trd(UbgAp|Q~g9*ARc5NH_ zn04q43^>wGV6(jn@5|m4^J)lk`l@8p4r?ZvKci)zBoG>O2T(yTO&^b_)AqDd&#&BR zDk&D@iA_bJ%SH$^qydz?T0`KSKL34pV^Gn>AWNhcvsR81!&^Y6#u(0y-yuh=Tyw!d z&it=3N^Y}A5k8%ZrI#3`Awul-LIVZO=k>zP(WLWcs4~;Ypby vnL!QjZQ_vfBM>XoODuuh)g+P!58h)SLS+`u2<}J!+eE4" - ![try and backtrack](backtracking_algorithm.assets/preorder_find_paths_step1.png) - -=== "<2>" - ![preorder_find_paths_step2](backtracking_algorithm.assets/preorder_find_paths_step2.png) - -=== "<3>" - ![preorder_find_paths_step3](backtracking_algorithm.assets/preorder_find_paths_step3.png) - -=== "<4>" - ![preorder_find_paths_step4](backtracking_algorithm.assets/preorder_find_paths_step4.png) - -=== "<5>" - ![preorder_find_paths_step5](backtracking_algorithm.assets/preorder_find_paths_step5.png) - -=== "<6>" - ![preorder_find_paths_step6](backtracking_algorithm.assets/preorder_find_paths_step6.png) - -=== "<7>" - ![preorder_find_paths_step7](backtracking_algorithm.assets/preorder_find_paths_step7.png) - -=== "<8>" - ![preorder_find_paths_step8](backtracking_algorithm.assets/preorder_find_paths_step8.png) - -=== "<9>" - ![preorder_find_paths_step9](backtracking_algorithm.assets/preorder_find_paths_step9.png) - -=== "<10>" - ![preorder_find_paths_step10](backtracking_algorithm.assets/preorder_find_paths_step10.png) - -=== "<11>" - ![preorder_find_paths_step11](backtracking_algorithm.assets/preorder_find_paths_step11.png) - -## Pruning - -Complex backtracking problems usually contain one or more constraints, **Constraints can often be used for "pruning"**. - -!!! question "Example 3" - - Search for all nodes in the binary tree with value $7$, return the path from the root node to these nodes, **and require that the path does not contain nodes with value $3$**. - -In order to satisfy the above constraints, **we need to add the pruning operation**: if a node with the value $3$ is encountered during the search process, it is returned early and the search is stopped from continuing. - -```src -[file]{preorder_traversal_iii_compact}-[class]{}-[func]{pre_order} -``` - -Pruning is a very visual term. As shown in the figure below, during the search process, **we "cut off" the search branches that do not satisfy the constraints** to avoid many meaningless attempts, thus improving the search efficiency. - -![pruning based on constraints](backtracking_algorithm.assets/preorder_find_constrained_paths.png) - -## Framework Code - -Next, we try to distill the main framework of "try, fallback, pruning" of backtracking to improve the generality of the code. - -In the following framework code, `state` denotes the current state of the problem, and `choices` denotes the choices that can be made in the current state. - -=== "Python" - - ```python title="" - def backtrack(state: State, choices: list[choice], res: list[state]): - """Backtracking algorithm framework"""" - # Determine if it's a solution - if is_solution(state): - # Recorded solutions - record_solution(state, res) - # Stop continuing the search - return - # Iterate over all choices - for choice in choices: - # Pruning: determining whether a choice is legal or not - if is_valid(state, choice): - # Trying: making choices, updating state - make_choice(state, choice) - backtrack(state, choices, res) - # Fallback: undo the selection and revert to the previous state - undo_choice(state, choice) - ``` - -=== "C++" - - ```cpp title="" - /* Backtracking algorithm framework */ - void backtrack(State *state, vector &choices, vector &res) { - // Determine whether it is a solution or not - if (isSolution(state)) { - // Record the solution - recordSolution(state, res); - // Stop continuing the search - return; - } - // Iterate over all choices - for (Choice choice : choices) { - // Pruning: determining whether a choice is legal or not - if (isValid(state, choice)) { - // Try: make a choice, update the state - makeChoice(state, choice); - backtrack(state, choices, res); - // Fallback: undo the selection and revert to the previous state - undoChoice(state, choice); - } - } - } - ``` - -=== "Java" - - ```java title="" - /* Backtracking algorithm framework */ - void backtrack(State state, List choices, List res) { - // Determine whether it is a solution or not - if (isSolution(state)) { - // Record the solution - recordSolution(state, res); - // Stop continuing the search - return; - } - // Iterate over all choices - for (Choice choice : choices) { - // Pruning: determining whether a choice is legal or not - if (isValid(state, choice)) { - // Try: make a choice, update the state - makeChoice(state, choice); - backtrack(state, choices, res); - // Fallback: undo the selection and revert to the previous state - undoChoice(state, choice); - } - } - } - ``` - -=== "C#" - - ```csharp title="" - /* Backtracking algorithm framework */ - void Backtrack(State state, List choices, List res) { - // Determine whether it is a solution or not - if (IsSolution(state)) { - // Record the solution - RecordSolution(state, res); - // Stop continuing the search - return; - } - // Iterate over all choices - foreach (Choice choice in choices) { - // Pruning: determining whether a choice is legal or not - if (IsValid(state, choice)) { - // Try: make a choice, update the state - MakeChoice(state, choice); - Backtrack(state, choices, res); - // Fallback: undo the selection and revert to the previous state - UndoChoice(state, choice); - } - } - } - ``` - -=== "Go" - - ```go title="" - /* Backtracking algorithm framework */ - func backtrack(state *State, choices []Choice, res *[]State) { - // Determine whether it is a solution or not - if isSolution(state) { - // Record the solution - recordSolution(state, res) - // Stop continuing the search - return - } - // Iterate over all choices - for _, choice := range choices { - // Pruning: determining whether a choice is legal or not - if isValid(state, choice) { - // Try: make a choice, update the state - makeChoice(state, choice) - backtrack(state, choices, res) - // Fallback: undo the selection and revert to the previous state - undoChoice(state, choice) - } - } - } - ``` - -=== "Swift" - - ```swift title="" - /* Backtracking algorithm framework */ - func backtrack(state: inout State, choices: [Choice], res: inout [State]) { - // Determine whether it is a solution or not - if isSolution(state: state) { - // Record the solution - recordSolution(state: state, res: &res) - // Stop continuing the search - return - } - // Iterate over all choices - for choice in choices { - // Pruning: determining whether a choice is legal or not - if isValid(state: state, choice: choice) { - // Try: make a choice, update the state - makeChoice(state: &state, choice: choice) - backtrack(state: &state, choices: choices, res: &res) - // Fallback: undo the selection and revert to the previous state - undoChoice(state: &state, choice: choice) - } - } - } - ``` - -=== "JS" - - ```javascript title="" - /* Backtracking algorithm framework */ - function backtrack(state, choices, res) { - // Determine whether it is a solution or not - if (isSolution(state)) { - // Record the solution - recordSolution(state, res); - // Stop continuing the search - return; - } - // Iterate over all choices - for (let choice of choices) { - // Pruning: determining whether a choice is legal or not - if (isValid(state, choice)) { - // Try: make a choice, update the state - makeChoice(state, choice); - backtrack(state, choices, res); - // Fallback: undo the selection and revert to the previous state - undoChoice(state, choice); - } - } - } - ``` - -=== "TS" - - ```typescript title="" - /* Backtracking algorithm framework */ - function backtrack(state: State, choices: Choice[], res: State[]): void { - // Determine whether it is a solution or not - if (isSolution(state)) { - // Record the solution - recordSolution(state, res); - // Stop continuing the search - return; - } - // Iterate over all choices - for (let choice of choices) { - // Pruning: determining whether a choice is legal or not - if (isValid(state, choice)) { - // Try: make a choice, update the state - makeChoice(state, choice); - backtrack(state, choices, res); - // Fallback: undo the selection and revert to the previous state - undoChoice(state, choice); - } - } - } - ``` - -=== "Dart" - - ```dart title="" - /* Backtracking algorithm framework */ - void backtrack(State state, List, List res) { - // Determine whether it is a solution or not - if (isSolution(state)) { - // Record the solution - recordSolution(state, res); - // Stop continuing the search - return; - } - // Iterate over all choices - for (Choice choice in choices) { - // Pruning: determining whether a choice is legal or not - if (isValid(state, choice)) { - // Try: make a choice, update the state - makeChoice(state, choice); - backtrack(state, choices, res); - // Fallback: undo the selection and revert to the previous state - undoChoice(state, choice); - } - } - } - ``` - -=== "Rust" - - ```rust title="" - /* Backtracking algorithm framework */ - fn backtrack(state: &mut State, choices: &Vec, res: &mut Vec) { - // Determine whether it is a solution or not - if is_solution(state) { - // Record the solution - record_solution(state, res); - // Stop continuing the search - return; - } - // Iterate over all choices - for choice in choices { - // Pruning: determining whether a choice is legal or not - if is_valid(state, choice) { - // Try: make a choice, update the state - make_choice(state, choice); - backtrack(state, choices, res); - // Fallback: undo the selection and revert to the previous state - undo_choice(state, choice); - } - } - } - ``` - -=== "C" - - ```c title="" - /* Backtracking algorithm framework */ - void backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) { - // Determine if it's a solution. - if (isSolution(state)) { - // Record the solution - recordSolution(state, res, numRes); - // Stop continuing the search - return; - } - // Iterate over all choices - for (int i = 0; i < numChoices; i++) { - // Pruning: determining whether a choice is legal or not - if (isValid(state, &choices[i])) { - // Try: make a choice, update the state - makeChoice(state, &choices[i]); - backtrack(state, choices, numChoices, res, numRes); - // Fallback: undo the selection and revert to the previous state - undoChoice(state, &choices[i]); - } - } - } - ``` - -=== "Zig" - - ```zig title="" - - ``` - -Next, we solve example 3 based on the framework code. The state `state` is the node traversal path, the choices `choices` are the left-child nodes and right-child nodes of the current node, and the result `res` is the list of paths. - -```src -[file]{preorder_traversal_iii_template}-[class]{}-[func]{backtrack} -``` - -According to the question, we should continue the search after finding the node with the value $7$, **so we need to remove the `return` statement after recording the solution**. The figure below compares the search process of keeping or removing `return` statements. - -![Comparison of search process for retaining and removing return](backtracking_algorithm.assets/backtrack_remove_return_or_not.png) - -The code implementation based on the backtracking algorithm framework is more general, although it seems verbose, than the code implementation based on the preorder traversal. In fact, **many backtracking problems can be solved under this framework**. We just need to define `state` and `choices` according to the specific problem and implement the methods in the framework. - -## Commonly Used Terms - -In order to analyze the algorithm problem more clearly, we summarize the meaning of the common terms used in the backtracking algorithm and give corresponding examples against Example 3. - -

    Table   Common backtracking algorithm terminology

    - -| Nouns | Definitions | Example 3 | -| --------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------- | -| Solution Solution | A solution is an answer that satisfies a specific condition of the problem, there may be one or more | All paths from the root node to the node $7$ that satisfy the constraints | -| Constraint Constraint | A constraint is a condition in a problem that restricts the feasibility of a solution, usually used for pruning | The path does not contain a node $3$ | -| State State | The state indicates the situation of the problem at a given point in time, including the choices that have been made | The path of the nodes that are currently visited, i.e., the list of `path` nodes | -| Attempt | Attempt is the process of exploring the solution space based on the available choices, including making a choice, updating the state, and checking if it is a solution | Recursion accesses the left (right) child node, adds the node to the `path` , and determines if the node's value is $7$ | -| Backtracking | Backtracking means that when a state that does not satisfy the constraints is encountered, the choices made earlier are undone and return to the previous state | The search is terminated when the leaf node is crossed, the end node is accessed, and the node with the value $3$ is encountered, and the function returns | -| Pruning Pruning | Pruning is a method to avoid meaningless search paths based on problem characteristics and constraints, which improves the search efficiency | When a node with value $3$ is encountered, then terminate to continue the search | - -!!! tip - - The concepts of problem, solution, and state are generic and are covered in algorithms such as divide and conquer, backtracking, dynamic programming, and greedy algorithms. - -## Strengths And Limitations - -The backtracking algorithm is essentially a depth-first search algorithm that tries all possible solutions until it finds a solution that satisfies the conditions. The advantage of this method is that it is able to find all possible solutions and is highly efficient with reasonable pruning operations. - -However, when dealing with large-scale or complex problems, **backtracking algorithms may run with unacceptable efficiency**. - -- **Time**: backtracking algorithms usually need to traverse all possibilities of the state space, and the time complexity can be of exponential or multiplicative order. -- **Space**: the need to save the current state (e.g., path, auxiliary variables for pruning, etc.) in recursion calls can make space requirements large when the depth is large. - -Even so, **backtracking algorithms are still the best solution for certain search problems and constraint satisfaction problems**. For these problems, since it is impossible to predict which choices generate valid solutions, we must iterate over all possible choices. In this case, **the key is how to optimize for efficiency**. There are two common efficiency optimization methods. - -- **pruning**: saving time and space by avoiding searching for paths that will definitely not yield a solution. -- **Heuristic search**: some strategy or estimate is introduced into the search process so as to prioritize the search for paths that are most likely to yield valid solutions. - -## Backtracking Through Typical Examples - -Backtracking algorithms can be used to solve many search problems, constraint satisfaction problems and combinatorial optimization problems. - -**Search Problems**: the goal of this type of problem is to find solutions that satisfy specific conditions. - -- Permutations problem: Given a set, find all possible permutations of it. -- Subset-sum problem: Given a set and a target sum, find all subsets of the set whose sum is the target sum. -- Hanota problem: Given three columns and a series of disks of different sizes, it is required to move all the disks from one column to another, one at a time, and without placing the larger disk on top of the smaller disk. - -**Constraint Satisfaction Problems**: the goal of these problems is to find a solution that satisfies all the constraints. - -- $n$ Queens: Place $n$ queens on a $n \times n$ board so that they do not attack each other. -- Sudoku: Fill the grid of $9 \times 9$ with the numbers $1$ ~ $9$ so that the numbers in each row, column and each $3 \times 3$ subgrid are not repeated. -- Graph Coloring Problem: Given an undirected graph, color each vertex of the graph with the least number of colors such that adjacent vertices are different colors. - -**Combinatorial Optimization Problems**: the goal of these problems is to find an optimal solution in a combinatorial space that satisfies certain conditions. - -- 0-1 Knapsack problem: Given a set of items and a knapsack, each item has a certain value and weight, and it is required to select the items such that the total value is maximized within the capacity constraints of the knapsack. -- Traveler's problem: In a graph, find the shortest path from a point that visits all other points exactly once and then returns to the starting point. -- Maximum Group Problem: Given an undirected graph, find the largest complete subgraph where any two vertices in the subgraph are connected by an edge. - -Note that for many combinatorial optimization problems, backtracking is not an optimal solution. - -- 0-1 Knapsack problems are often solved using dynamic programming to achieve greater time efficiency. -- The traveler is a well-known NP-Hard problem, and commonly used solutions include genetic algorithms and ant colony algorithms. -- The maximal group problem is a classical problem in graph theory and can be solved with heuristic algorithms such as greedy algorithms. diff --git a/docs-en/chapter_backtracking/index.md b/docs-en/chapter_backtracking/index.md deleted file mode 100644 index 21a3e24f07..0000000000 --- a/docs-en/chapter_backtracking/index.md +++ /dev/null @@ -1,13 +0,0 @@ -# Backtracking - -
    - -![backtracking](../assets/covers/chapter_backtracking.jpg){ width="600" } - -
    - -!!! abstract - - Like explorers in a labyrinth, we may encounter difficulties along the way. - - The power of retrospection allows us to start over, keep trying, and eventually find the exit to the light. diff --git a/docs-en/chapter_backtracking/n_queens_problem.assets/n_queens_cols_diagonals.png b/docs-en/chapter_backtracking/n_queens_problem.assets/n_queens_cols_diagonals.png deleted file mode 100644 index cd290ea08555046593cf49c0527719d6de9eaf35..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 36176 zcmb??_d^q1(C%(R=)FskDn&#P1W^P6i1Z=?QlyCVB8c>2f^<+pI)Z?7DbkD7fFQji zy-RPQ_aqnI@4NRexcMa~dv>0A=9!sf_v}fSmWCo3F%vNW0A$KaPqYC5b~S_y5JIos zPBR59u0EV=sp%--aJc>beR%ffo1i)Ey3)%_3N2IHfI;n=o8H1 z=HA-IZdl6F!V)?uDe3GI+c~i1{&8V&Y1g)TXJ%%`KW?#Sad%>7r+0Wut8i;&d#|iv zJ+o|WdH=HL*G5p{znTg3-<{3%_4W5XZNIxW6LMFkrl!#Q7v-bqv8f%u65p}0vHri? zqbs{|Ihzjo_G@cvX+Kv(Tetu8ZPv7`cg*jNOl<2FZx0O(t!|#buH1fEw*7DQ>|4)P 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    Table   Full Arrangement Example

    - -| Input Array | All Arrangements | -| :---------- | :----------------------------------------------------------------- | -| $[1]$ | $[1]$ | -| $[1, 2]$ | $[1, 2], [2, 1]$ | -| $[1, 2, 3]$ | $[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]$ | - -## The Case Of Unequal Elements - -!!! question - - Inputs an array of integers, the array contains no duplicate elements and returns all possible permutations. - -From the backtracking algorithm point of view, **we can imagine the process of generating an arrangement as the result of a series of choices**. Suppose the input array is $[1, 2, 3]$ , and if we first select $1$, then $3$, and finally $2$ , we obtain the arrangement $[1, 3, 2]$ . Fallback means to undo a choice and continue to try other choices later. - -From the point of view of the backtracking code, the candidate set `choices` is all the elements in the input array, and the state `state` is the elements that have been selected until now. Note that each element is only allowed to be selected once, **so all elements in `state` should be unique**. - -As shown in the figure below, we can expand the search process into a recursion tree where each tree node represents the current state `state` . Starting from the root node, we reach the leaf nodes after three rounds of selection, and each leaf node corresponds to an arrangement. - -![fully-aligned recursion tree](permutations_problem.assets/permutations_i.png) - -### Repeat Selection Pruning - -In order to realize that each element is selected only once, we consider introducing a Boolean array `selected` , where `selected[i]` indicates whether `choices[i]` has been selected or not, and implement the following pruning operation based on it. - -- After making the selection `choice[i]`, we assign `selected[i]` to $\text{True}$ to represent that it has been selected. -- When traversing the choices list `choices`, skip all nodes that have already been selected, i.e. pruning. - -As shown in the figure below, suppose we choose 1 in the first round, 3 in the second round, and 2 in the third round, then we need to cut off the branch of element 1 in the second round, and the branches of element 1 and element 3 in the third round. - -![Example of fully-aligned pruning](permutations_problem.assets/permutations_i_pruning.png) - -Observing the figure above, we find that this pruning operation reduces the search space size from $O(n^n)$ to $O(n!)$ . - -### Code Implementation - -With the above information in mind, we can do the "fill-in-the-blanks" in the framework code. In order to reduce the number of lines of code, instead of implementing each function in the framework code separately, we will expand them in the `backtrack()` function. - -```src -[file]{permutations_i}-[class]{}-[func]{permutations_i} -``` - -## Consider The Case Of Equal Elements - -!!! question - - Input an array of integers, **The array may contain duplicate elements**. Return all non-duplicate permutations. - -Suppose the input array is $[1, 1, 2]$ . In order to easily distinguish between the two repeated elements $1$ , we denote the second $1$ as $\hat{1}$ . - -As shown in the figure below, half of the alignments generated by the above method are duplicates. - -![duplicate alignment](permutations_problem.assets/permutations_ii.png) - -So how do you remove duplicate alignments? Most directly, consider de-duplicating the alignments directly with the help of a hash table. However, this is not elegant enough, **because the search branch that generates the duplicate alignments is unnecessary and should be recognized and pruned in advance**, which further improves the efficiency of the algorithm. - -### Pruning Of Equivalent Elements - -Observe the figure below, in the first round, choosing $1$ or choosing $\hat{1}$ is equivalent, and all alignments generated under these two choices are duplicates. Therefore $\hat{1}$ should be pruned. - -Similarly, after selecting $2$ in the first round, $1$ and $\hat{1}$ in the second round of selection will also generate duplicate branches, so $\hat{1}$ in the second round should also be pruned. - -Essentially, **our goal is to ensure that multiple equal elements are selected only once in a given round of selection**. - -![duplicate alignment pruning](permutations_problem.assets/permutations_ii_pruning.png) - -### Code Implementation - -Building on the code from the previous question, we consider opening a hash table `duplicated` in each round of selection to keep track of the elements that have been tried in that round and pruning duplicates. - -```src -[file]{permutations_ii}-[class]{}-[func]{permutations_ii} -``` - -Assuming that the elements are not the same as each other two by two, there are $n$ a total of $n!$ permutations (factorials) of the elements; when recording the result, a list of length $n$ needs to be copied, using $O(n)$ time. **Therefore the time complexity is $O(n!n)$** . - -Maximum recursion depth is $n$, using $O(n)$ stack frame space. `selected` uses $O(n)$ space. There are at most $n$ `duplicated` at the same moment, using $O(n^2)$ space. **Therefore the space complexity is $O(n^2)$** . - -### Two Pruning Comparisons - -Note that although both `selected` and `duplicated` are used for pruning, they have different goals. - -- **Repeated selection pruning**: there is only one `selected` for the entire search. It records which elements are contained in the current state, and serves to prevent any of the elements in `choices` from recurring in `state`. -- **Equal element pruning**: each round of selection (i.e., each call to the `backtrack` function) contains a `duplicated` . It records which elements have been selected in the current round of traversal (i.e., the `for` loop), and serves to ensure that equal elements are selected only once. - -The figure below illustrates the range of validity of the two pruning conditions. 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znCXPXU}q>(=E$tli-u1C8+wM+%RI&v3xFMDpLnc1n_|bAL(Lxbfg#2EWXj$|)}8{j zxY2Z46UBK?3F0mpsebtCQ9wTgzAANW&SlZ}4So;Fx}}>8Q!h_xZu03^OL6`)Ea8V| zE-H$U&>P@pPBm@mrlxO6&1-?seg>WM>d2CqSITDS;7Sm@UwCJCWZ0zgA>APwn>Lb= z#5CF{i^6&@bjSPo!+K@RBHv$ykSsMFZo?oo3oqc&q5y6?9NMl~SmM`UedK3L+izlGqne>l5RCq^=bQ6nG()sK-}aeuIh)Vh4cG>M;-_@JQ`C61g3) - -| Type | Symbol | Occupancy | Minimum | Maximum | Default | -| -------------- | -------- | ---------------- | ------------------------ | ----------------------- | ----------------------------- | -| integer | `byte` | 1 byte | $-2^7$ ($-128$) | $2^7 - 1$ ($127$) | $0$ | -| | `short` | 2 bytes | $-2^{15}$ | $2^{15} - 1$ | $0$ | -| | `int` | 4 bytes | $-2^{31}$ | $2^{31} - 1$ | $0$ | -| | `long` | 8 bytes | $-2^{63}$ | $2^{63} - 1$ | $0$ | -| Floating Point | `float` | 4 bytes | $1.175 \times 10^{-38}$ | $3.403 \times 10^{38}$ | $0.0 f$ | -| | `double` | 8 bytes | $2.225 \times 10^{-308}$ | $1.798 \times 10^{308}$ | $0.0$ | -| Characters | `char` | 2 bytes / 1 byte | $0$ | $2^{16} - 1$ | $0$ | -| Boolean | `bool` | 1 byte | $\text{false}$ | $\text{true}$ | $\text{false}$ $\text{false}$ | - -For the above table, the following points need to be noted. - -- C and C++ do not explicitly specify the size of the base datatype, which varies by implementation and platform. The table above follows the LP64 [Data Model](https://en.cppreference.com/w/cpp/language/types#Properties), which is used on Unix 64-bit operating systems including Linux and macOS. -- The size of the character `char` is 1 byte in C and C++, and depends on a specific character encoding method in most programming languages, see the "Character Encoding" section for details. -- Even though it only takes 1 bit ($0$ or $1$) to represent a Boolean quantity, it is usually stored as 1 byte in memory. This is because modern computer CPUs usually use 1 byte as the smallest addressable memory unit. - -So what is the connection between basic data types and data structures? As we know, data structure is the way data is organized and stored in a computer. Its subject is "structure" rather than "data". - -If we want to represent "an array of numbers", we naturally think of using an array. This is because the linear structure of arrays allows us to represent the adjacency and order of numbers, but whether the contents are stored as integers `int`, decimals `float`, or characters `char` has nothing to do with the "data structure". - -In other words, **basic data types provide the "content type" of the data, while data structures provide the "organization" of the data**. For example, in the following code, we use the same data structure (array) to store and represent different basic data types, including `int`, `float`, `char`, `bool`, and so on. - -=== "Python" - - ```python title="" - # Initialize arrays with a variety of basic data types - numbers: list[int] = [0] * 5 - decimals: list[float] = [0.0] * 5 - # Python's characters should be treated as strings of length one - characters: list[str] = ['0'] * 5 - bools: list[bool] = [False] * 5 - # Python's lists are free to store all kinds of basic data types and object references. - data = [0, 0.0, 'a', False, ListNode(0)] - ``` - -=== "C++" - - ```cpp title="" - // Initialize arrays with multiple basic data types - int numbers[5]; - float decimals[5]; - char characters[5]; - bool bools[5]; - ``` - -=== "Java" - - ```java title="" - // Initialize arrays with multiple basic data types - int[] numbers = new int[5]; - float[] decimals = new float[5]; - char[] characters = new char[5]; - boolean[] bools = new boolean[5]; - ``` - -=== "C#" - - ```csharp title="" - // Initialize arrays with multiple basic data types - int[] numbers = new int[5]; - float[] decimals = new float[5]; - char[] characters = new char[5]; - bool[] bools = new bool[5]; - ``` - -=== "Go" - - ```go title="" - // Initialize arrays with multiple basic data types - var numbers = [5]int{} - var decimals = [5]float64{} - var characters = [5]byte{} - var bools = [5]bool{} - ``` - -=== "Swift" - - ```swift title="" - // Initialize arrays with multiple basic data types - let numbers = Array(repeating: Int(), count: 5) - let decimals = Array(repeating: Double(), count: 5) - let characters = Array(repeating: Character("a"), count: 5) - let bools = Array(repeating: Bool(), count: 5) - ``` - -=== "JS" - - ```javascript title="" - // JavaScript arrays are free to store a variety of basic data types and objects. - const array = [0, 0.0, 'a', false]; - ``` - -=== "TS" - - ```typescript title="" - // Initialize arrays with multiple basic data types - const numbers: number[] = []; - const characters: string[] = []; - const bools: boolean[] = []; - ``` - -=== "Dart" - - ```dart title="" - // Initialize arrays with multiple basic data types - List numbers = List.filled(5, 0); - List decimals = List.filled(5, 0.0); - List characters = List.filled(5, 'a'); - List bools = List.filled(5, false); - ``` - -=== "Rust" - - ```rust title="" - // Initialize arrays with multiple basic data types - let numbers: Vec = vec![0; 5]; - let decimals: Vec = vec![0.0, 5]; - let characters: Vec = vec!['0'; 5]; - let bools: Vec = vec![false; 5]; - ``` - -=== "C" - - ```c title="" - // Initialize arrays with multiple basic data types - int numbers[10]; - float decimals[10]; - char characters[10]; - bool bools[10]; - ``` - -=== "Zig" - - ```zig title="" - - ``` diff --git a/docs-en/chapter_data_structure/character_encoding.assets/ascii_table.png b/docs-en/chapter_data_structure/character_encoding.assets/ascii_table.png deleted file mode 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DRs2Z9 diff --git a/docs-en/chapter_data_structure/character_encoding.md b/docs-en/chapter_data_structure/character_encoding.md deleted file mode 100644 index efd7f87b51..0000000000 --- a/docs-en/chapter_data_structure/character_encoding.md +++ /dev/null @@ -1,87 +0,0 @@ -# Character Encoding * - -In computers, all data is stored as binary numbers, and the character `char` is no exception. In order to represent the characters, we need to establish a set of "character set", the provisions of each character and the one-to-one correspondence between the binary number. With the character set, the computer can look up the table to complete the conversion of binary numbers to characters. - -## Ascii Character Set - -"ASCII" is the earliest character set known as "American Standard Code for Information Interchange". It uses a 7-bit binary number (i.e. the lower 7 bits of a byte) to represent a character, and can represent up to 128 different characters. As shown in the figure below, the ASCII code consists of the upper and lower case letters of the English alphabet, the numbers 0 ~ 9, some punctuation marks, and some control characters (such as line breaks and tabs). - -![ASCII code](character_encoding.assets/ascii_table.png) - -However, **the ASCII code is only capable of representing the English language**. With the globalization of computers, a character set called "EASCII" was created that could represent more languages. It extends the 7-bit ASCII code to 8-bit and is capable of representing 256 different characters. - -Around the world, there are a number of EASCII character sets that are applicable to different regions. The first 128 characters of these character sets are standardized as ASCII, and the last 128 characters are defined differently to meet the needs of different languages. - -## Gbk Character Set - -It was later discovered that **EASCII still could not meet the number of characters required for many languages**. For example, there are about 100,000 Chinese characters, and several thousand of them are used in daily life. In 1980, the General Administration of National Standards of China released the "GB2312" character set, which contains 6,763 Chinese characters and basically meets the needs of computer processing of Chinese characters. - -However, GB2312 cannot handle some rare and traditional characters. The "GBK" character set is an extension of GB2312, which contains 21886 Chinese characters. In the GBK encoding scheme, ASCII characters are represented by one byte and Chinese characters are represented by two bytes. - -## Unicode Character Set - -With the booming development of computers, character sets and coding standards have blossomed, and this has created many problems. On the one hand, these character sets generally define only the characters of a particular language and cannot work properly in a multilingual environment. On the other hand, there are also multiple character set standards for the same language, so if two computers are installed with different encoding standards, the information will be garbled when transmitted. - -Researchers of that era thought:**Wouldn't the problem of cross-linguistic environments and garbled codes be solved if a character set complete enough to include all the languages and symbols of the world was introduced**? Driven by this idea, Unicode, a large and complete character set, was created. - -"Unicode" is the full name of "Unified Character Encoding", which can theoretically accommodate more than one million characters. It is committed to incorporating characters from all over the world into a unified character set, providing a common character set for processing and displaying various languages, and reducing the problem of garbled codes caused by different encoding standards. - -Since its release in 1991, Unicode has been expanding with new languages and characters. As of September 2022, Unicode contains 149,186 characters, including characters, symbols, and even emoticons for various languages. Within the large Unicode character set, commonly used characters take up 2 bytes, and some obscure characters take up 3 or even 4 bytes. - -Unicode is a character set standard that essentially assigns a number (called a "code point") to each character, **but it doesn't dictate how those code points are stored in a computer**. One wonders: how does the system resolve characters when Unicode code points of multiple lengths appear in the same text at the same time? For example, given a code of 2 bytes in length, how does the system recognize whether it is one 2-byte character or two 1-byte characters? - -A straightforward solution to the above problem **is to store all characters as equal length encodings**. As shown in the figure below, each character in "Hello" occupies 1 byte and each character in "algorithm" occupies 2 bytes. We can encode all the characters in "Hello Algo" as 2 bytes by filling the high bit with zero. This way, the system can parse a character every 2 bytes and recover the content of the phrase. - -![Unicode encoding example](character_encoding.assets/unicode_hello_algo.png) - -However, ASCII code has already proved to us that it takes only 1 byte to encode English. If we use the above scheme, the size of the English text will be twice the size of the ASCII encoding, which is a waste of memory space. Therefore, we need a more efficient way to encode Unicode. - -## Utf-8 Encoding - -Currently, UTF-8 has become the most widely used Unicode encoding method internationally. It is a variable-length encoding that uses between 1 and 4 bytes to represent a character, depending on the complexity of the character. 1 byte is needed for ASCII characters, 2 bytes for Latin and Greek alphabets, 3 bytes for commonly used Chinese characters, and 4 bytes for some other rare characters. - -The encoding rules for UTF-8 are not complicated and are divided into the following two cases. - -- For characters 1 byte long, set the highest bit to $0$ and the remaining 7 bits to Unicode code points. It is worth noting that ASCII characters occupy the first 128 code points in the Unicode character set. In other words, **UTF-8 encoding is backward compatible with ASCII**. This means that we can use UTF-8 to solve old ASCII text. -- For characters of length $n$ bytes (where $n > 1$), set the high $n$ bits of the first byte to $1$ and the $n + 1$ bits to $0$; from the second byte onwards, set the high 2 bits of each byte to $10$; and all the rest of the bits are used to fill in the Unicode code point of the character. - -The figure below shows the UTF-8 encoding of "Hello Algo". It is observed that since the highest $n$ bits are set to $1$, the system can read the number of the highest $1$ bits to parse out the length of the character as $n$. - -But why set the high 2 bits of all the rest of the bytes to $10$? In fact, this $10$ serves as a check digit. Assuming that the system starts parsing the text from an incorrect byte, the $10$ in the byte header can help the system to quickly recognize the exception. - -The reason why $10$ is used as a check digit is because under the UTF-8 encoding rules, it is impossible for any character to have the highest two digits as $10$. This conclusion can be proved by the inverse method: suppose the highest two bits of a character are $10$, that means the length of the character is $1$, which corresponds to the ASCII code. The highest bit of ASCII code should be $0$, which contradicts the assumption. - -![UTF-8 encoding example](character_encoding.assets/utf-8_hello_algo.png) - -In addition to UTF-8, there are two other common encodings. - -- **UTF-16 encoding**: 2 or 4 bytes are used to represent a character. All ASCII characters and common non-English characters are represented by 2 bytes; a few characters require 4 bytes. For 2-byte characters, UTF-16 encoding is equivalent to Unicode code points. -- **UTF-32 encoding**: each character uses 4 bytes. This means that UTF-32 will take up more space than UTF-8 and UTF-16, especially for text with a high proportion of ASCII characters. - -From a storage space point of view, using UTF-8 to represent English characters is very efficient because it takes only 1 byte; using UTF-16 to encode some non-English characters (e.g., Chinese) is more efficient because it takes only 2 bytes, whereas UTF-8 may take 3 bytes. - -From a compatibility standpoint, UTF-8 is the most versatile, and many tools and libraries prioritize UTF-8 support. - -## Character Encoding For Programming Languages - -For most of the programming languages in the past, the strings in the running program are encoded in UTF-16 or UTF-32 encoding of equal length. With equal-length encoding, we can treat strings as arrays, which has the following advantages. - -- **Random access**: UTF-16 encoded strings can be easily randomized. UTF-8 is a variable-length encoding, and to find the first $i$ character, we need to traverse from the beginning of the string to the first $i$ character, which takes $O(n)$ time. -- **Character Count**: Similar to random access, calculating the length of a UTF-16 string is an $O(1)$ operation. However, calculating the length of a UTF-8 encoded string requires traversing the entire string. -- **String Operations**: Many string operations (e.g., splitting, concatenation, insertion, deletion, etc.) are easier to perform on UTF-16 encoded strings. Performing these operations on UTF-8 encoded strings usually requires additional computation to ensure that invalid UTF-8 encodings are not produced. - -In fact, the design of character encoding schemes for programming languages is an interesting topic that involves many factors. - -- Java's `String` type uses UTF-16 encoding and takes up 2 bytes per character. This is because when the Java language was first designed, it was assumed that 16 bits would be enough to represent all possible characters. However, this was an incorrect judgment. The Unicode specification was later extended beyond 16 bits, so characters in Java may now be represented by a pair of 16-bit values (called a "proxy pair"). -- JavaScript and TypeScript strings are UTF-16 encoded for similar reasons to Java. When the JavaScript language was first introduced by Netscape in 1995, Unicode was in its relatively early stages, and a 16-bit encoding was sufficient to represent all Unicode characters. -- C# uses UTF-16 encoding, primarily because the .NET platform was designed by Microsoft, and many of Microsoft's technologies, including the Windows operating system, make extensive use of UTF-16 encoding. - -Due to the underestimation of the number of characters in the above programming languages, they have had to resort to "proxy pairs" to represent Unicode characters longer than 16 bits. This is a last resort. On the one hand, a character in a string containing a proxy pair may take up 2 or 4 bytes, thus losing the advantage of equal-length encoding. On the other hand, handling proxy pairs requires additional code, which increases programming complexity and Debugging difficulty. - -For these reasons, some programming languages have proposed a number of different coding schemes. - -- `str` in Python uses Unicode encoding and a flexible string representation where the length of the stored characters depends on the largest Unicode code point in the string. Each character takes 1 byte if the string is entirely ASCII, 2 bytes if there are characters outside the ASCII range but all within the Basic Multilingual Plane (BMP), and 4 bytes if there are characters outside the BMP. -- Go's `string` type is internally encoded in UTF-8. Go also provides the `rune` type, which is used to represent a single Unicode code point. -- The str and String types of the Rust language are internally encoded in UTF-8. Rust also provides the `char` type for representing individual Unicode code points. - -Note that the above discussion is all about how strings are stored in a programming language, **this is a different issue from how strings are stored in a file or transmitted over a network**. 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zmQu9c*WFN8nhimi?1!E#-)D|45Aho-{KHmr!0FnUgPsVfQJW9*#{!zA1atN)+MC3V zpPAdA*P`%)!IQp8VxO=9cqqTNeg22d6lKZlPw*kd#;lS`UI49B;rgI8b*)>onKHN{ rQh6|k4g}0iaoe+*tK$t|yvPp*WU5y2`^-@R%mbjIqN7}`2z&k?<&K|_ diff --git a/docs-en/chapter_data_structure/number_encoding.md b/docs-en/chapter_data_structure/number_encoding.md deleted file mode 100644 index 9155d4dfe9..0000000000 --- a/docs-en/chapter_data_structure/number_encoding.md +++ /dev/null @@ -1,150 +0,0 @@ -# Numeric Code * - -!!! note - - Throughout the book, the chapters with the * symbol in the title are optional. If you have limited time or find it difficult to understand, you can skip them and then attack them individually when you have finished the required chapters. - -## Integer Encoding - -In the table in the previous section, we found that all integer types are capable of representing negative numbers one more than positive numbers, e.g., the range of values for `byte` is $[-128, 127]$ . This phenomenon is rather counter-intuitive, and its intrinsic cause involves knowledge of true forms, inverse codes, and 2's complement codes. - -First of all, **it should be noted that numbers are stored in the computer as in the form of 2's complement code**. Before analyzing the reasons for this, we first give the definitions of the three. - -- **True form**: we consider the highest bit of the binary representation of a number as the sign bit, where $0$ denotes a positive number, $1$ denotes a negative number, and the rest of the bits denote the value of the number. -- **Inverse Code**: the inverse code of a positive number is the same as its true form, and the inverse code of a negative number is the inverse of all bits of its true form except the sign bit. -- **Complement code**: The complement code of a positive number is the same as its true form, and the complement code of a negative number is its inverse code plus $1$ . - -The figure below shows how to convert between true form, 1's complement code and 2's complement code. - -![Interconversion between true form, 1's complement code and 2's complement code](number_encoding.assets/1s_2s_complement.png) - -Although "true form" is the most intuitive, it has some limitations. On the one hand, **the true form of negative numbers cannot be used directly in arithmetic**. For example, if you calculate $1 + (-2)$ in true form, the result will be $-3$, which is obviously incorrect. - -$$ -\begin{aligned} -& 1 + (-2) \newline -& \rightarrow 0000 \; 0001 + 1000 \; 0010 \newline -& = 1000 \; 0011 \newline -& \rightarrow -3 -\end{aligned} -$$ - -In order to solve this problem, computers have introduced "1's complement code". If we first convert the original code to the inverse code, and calculate $1 + (-2)$ under the inverse code, and finally convert the result from the inverse code back to the original code, then we can get the true form $-1$. - -$$ -\begin{aligned} -& 1 + (-2) \newline -& \rightarrow 0000 \; 0001 \; \text{(true form)} + 1000 \; 0010 \; \text{(true form)} \newline -& = 0000 \; 0001 \; \text{(complement code)} + 1111 \; 1101 \; \text{(complement code)} \newline -& = 1111 \; 1110 \; \text{(1's complement code)} \newline -& = 1000 \; 0001 \; \text{(true form)} \newline -& \rightarrow -1 -\end{aligned} -$$ - -On the other hand, **the true form of the number zero has two representations $0$ and $-0$**. This means that the number zero corresponds to two different binary codes, and its may bring ambiguity. For example, in conditional judgment, if there is no distinction between positive and negative zeros, it may lead to wrong judgment results. If we want to deal with positive and negative zero ambiguities, we need to introduce additional judgment operations, which may reduce the computational efficiency of the computer. - -$$ -\begin{aligned} -+0 & \rightarrow 0000 \; 0000 \newline --0 & \rightarrow 1000 \; 0000 -\end{aligned} -$$ - -Like the true form, the inverse code also has the problem of positive and negative zero ambiguity, so the computer further introduces the "complement code 2's complement code". Let's first observe the conversion process of negative-zero true form, 1's complement code, and 2's complement code: - -$$ -\begin{aligned} --0 \rightarrow \; & 1000 \; 0000 \; \text{(true form)} \newline -= \; & 1111 \; 1111 \; \text{(complement code)} \newline -= 1 \; & 0000 \; 0000 \; \text{(complement code)} \newline -\end{aligned} -$$ - -Adding $1$ to the 1's complement code of negative zeros results in a rounding, but the length of the `byte` type is only 8 bits, so the $1$ that overflows to bit 9 is discarded. In other words, **the complement code for negative zeros is $0000 \; 0000$, which is the same as the complement code for positive zeros**. This means that there is only one zero in the 2's complement code representation, and the positive and negative zero ambiguity is thus resolved. - -There is one last puzzle: the range of the `byte` type is $[-128, 127]$, how do we get the extra negative number $-128$? We notice that all integers in the range $[-127, +127]$ have their true form, 1's complement code and 2's complement code, and the true form and 2's complement code can be converted to each other. - -However, **complement code $1000 \; 0000$ is an exception which does not have a corresponding true form code**. According to the conversion method, we get the true form of this complement code as $0000 \; 0000$ . This is clearly a contradiction, since that true form represents the number $0$ and its 2's complement code should be itself. The computer specifies that this particular 2's complement code $1000 \; 0000$ represents $-128$ . In fact, $(-1) + (-127)$ under the 2's complement code results in $-128$ . - -$$ -\begin{aligned} -& (-127) + (-1) \newline -& \rightarrow 1111 \; 1111 \; \text{(true form)} + 1000 \; 0001 \; \text{(true form)} \newline -& = 1000 \; 0000 \; \text{(complement code)} + 1111 \; 1110 \; \text{(complement code)} \newline -& = 1000 \; 0001 \; \text{(complement code)} + 1111 \; 1111 \; \text{(complement code)} \newline -& = 1000 \; 0000 \; \text{(complement code)} \newline -& \rightarrow -128 -\end{aligned} -$$ - -You may have noticed that all of the above calculations are additive operations. This implies an important fact:**The hardware circuits inside computers are designed primarily based on addition operations**. This is because addition operations are simpler to implement in hardware, easier to parallelize, and faster to perform compared to other operations such as multiplication, division, and subtraction. - -Note that this does not mean that computers can only do addition. **By combining addition with some basic logical operations, computers are able to perform a variety of other mathematical operations**. For example, calculating subtraction $a - b$ can be converted to calculating addition $a + (-b)$; calculating multiplication and division can be converted to calculating multiple additions or subtractions. - -We can now summarize the reasons why computers use complement codes: based on the 2's complement code representation, computers can use the same circuits and operations to handle addition of positive and negative numbers, do not need to design special hardware circuits to handle subtraction, and do not have to deal specifically with the ambiguity of positive and negative zeros. This greatly simplifies hardware design and improves arithmetic efficiency. - -The design of 2's complement code is very subtle, because of space relations we will introduce here, we recommend interested readers to further depth. - -## Floating Point Encoding - -You may notice that `int` and `float` are both 4 bytes long, but why is the range of `float` much larger than that of `int`? This is very counter-intuitive, because `float` needs to represent decimals, so its range should be smaller. - -In fact, **this is because floating-point numbers `float` use a different representation**. Remember that a 32-bit length binary number is: - -$$ -b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0 -$$ - -According to the IEEE 754 standard, a 32-bit length `float` consists of the following three parts. - -- Symbol bit $\mathrm{S}$ : Occupies 1 bit and corresponds to $b_{31}$. -- Exponential bits $\mathrm{E}$ : occupies 8 bits, corresponding to $b_{30} b_{29} \ldots b_{23}$. -- Fractional bits $\mathrm{N}$ : occupies 23 bits, corresponding to $b_{22} b_{21} \ldots b_0$. - -Calculates the value corresponding to a binary number `float`: - -$$ -\text {val} = (-1)^{b_{31}} \times 2^{\left(b_{30} b_{29} \ldots b_{23}\right)_2-127} \times\left(1 . b_{22} b_{21} \ldots b_0\right)_2 -$$ - -Converts to a formula in decimal: - -$$ -\text {val}=(-1)^{\mathrm{S}} \times 2^{\mathrm{E} -127} \times (1 + \mathrm{N}) -$$ - -The range of values of each of these: - -$$ -\begin{aligned} -\mathrm{S} \in & \{ 0, 1\}, \quad \mathrm{E} \in \{ 1, 2, \dots, 254 \} \newline -(1 + \mathrm{N}) = & (1 + \sum_{i=1}^{23} b_{23-i} 2^{-i}) \subset [1, 2 - 2^{-23}] -\end{aligned} -$$ - -![Calculation example of float under IEEE 754 standard](number_encoding.assets/ieee_754_float.png) - -Looking at the figure above, given a sample data $\mathrm{S} = 0$ , $\mathrm{E} = 124$ , $\mathrm{N} = 2^{-2} + 2^{-3} = 0.375$ , we have: - -$$ -\text { val } = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875 -$$ - -Now we can answer the original question:**The representation of `float` contains exponent bits, resulting in a much larger range of values than `int`**. Based on the above calculations, the largest positive number that can be represented by `float` is $2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}$ , and switching the sign bit will give us the smallest negative number. - -**Although floating-point numbers `float` extend the range of values, they have the side effect of sacrificing precision**. The integer type `int` dedicates all 32 bits to representing numbers, which are evenly distributed; whereas, the larger the value of a floating-point number `float`, the greater the difference between two neighboring numbers tends to be, due to the exponent bit. - -As shown in the table below, the exponent bits $E = 0$ and $E = 255$ have special meanings and **are used to denote zero, infinity, $\mathrm{NaN}$ and so on**. - -

    Table   Meaning of Index Bits

    - -| Exponential E | Score $\mathrm{N} = 0$ | Score $\mathrm{N} \ne 0$ | Calculation formula | -| ------------------ | ---------------------- | ------------------------ | ---------------------------------------------------------------------- | -| $0$ | $\pm 0$ | Subregular number | $(-1)^{\mathrm{S}} \times 2^{-126} \times (0.\mathrm{N})$ | -| $1, 2, \dots, 254$ | Regular number | Regular number | $(-1)^{\mathrm{S}} \times 2^{(\mathrm{E} -127)} \times (1.\mathrm{N})$ | -| $255$ | $\pm \infty$ | $\mathrm{NaN}$ | | - -It is worth stating that the subregular numbers significantly improve the precision of floating point numbers. The smallest positive regular number is $2^{-126}$ and the smallest positive subregular number is $2^{-126} \times 2^{-23}$. - -Double-precision `double` also uses a similar representation to `float`, which will not be repeated here. diff --git a/docs-en/chapter_data_structure/summary.md b/docs-en/chapter_data_structure/summary.md deleted file mode 100644 index a9631d6365..0000000000 --- a/docs-en/chapter_data_structure/summary.md +++ /dev/null @@ -1,33 +0,0 @@ -# Summary - -### Highlights - -- Data structures can be categorized in terms of both logical and physical structures. Logical structure describes the logical relationships between data elements, while physical structure describes how data is stored in computer memory. -- Common logical structures include linear, tree and net. Usually we categorize data structures into linear (arrays, linked lists, stacks, queues) and non-linear (trees, graphs, heaps) based on their logical structure. Implementations of hash tables may contain both linear and nonlinear structures. -- When a program runs, data is stored in computer memory. Each memory space has a corresponding memory address through which the program accesses the data. -- Physical structures are mainly categorized into continuous space storage (arrays) and distributed space storage (linked lists). All data structures are implemented as arrays, linked lists, or a combination of both. -- The basic data types in computers include integers `byte`, `short`, `int`, `long`, floats `float`, `double`, characters `char`, and booleans `boolean`. The range of their values depends on the amount of space occupied and the representation. -- True form, 1's complement code and 2's complement code are the three methods of coding numbers in a computer and they are interconvertible. The highest bit of the true form of an integer is the sign bit and the remaining bits are the value of the number. -- Integers are stored in computers as 2's complement code. Under the 2's complement code representation, computers can treat addition of positive and negative numbers equally, do not need to design separate special hardware circuits for subtraction operations, and do not have the problem of positive and negative zero ambiguity. -- Floating point numbers are encoded with 1 sign bit, 8 exponent bits, and 23 fractional bits. Because of the exponent bits, floating-point numbers have a much wider range of values than integers, at the expense of precision. -- ASCII is the earliest English character set, with a length of 1 byte and a total of 127 characters; GBK is the most commonly used Chinese character set, with a total of more than 20,000 Chinese characters; Unicode is committed to providing a complete character set standard, including characters of various languages in the world, so as to solve the problem of garbled characters caused by inconsistent character encoding methods. -- UTF-8 is the most popular Unicode encoding method, which is very versatile. It is a variable-length encoding method with good scalability, which effectively improves the efficiency of storage space usage.UTF-16 and UTF-32 are equal-length encoding methods, and UTF-16 occupies less space than UTF-8 when encoding Chinese. When encoding Chinese, UTF-16 takes less space than UTF-8. Programming languages like Java and C# use UTF-16 encoding by default. - -### Q & A - -!!! question "Why does a hash table contain both a linear data structure and a nonlinear data structure?" - - Hash table underlying the array, and in order to solve hash collision, we may use the "chained address" (subsequent hash table chapter will talk about): each bucket in the array to point to a chained list, when the length of the chained list exceeds a certain threshold, and may be transformed into a tree (usually a red-black tree). - From a storage point of view, the underlying level of a hash table is an array, where each bucket slot bit may contain a value, a linked list or a tree. Thus, hash tables may contain both linear (arrays, linked lists) and nonlinear (trees) data structures. - -!!! question "Is the length of the `char` type 1 byte?" - - The length of the `char` type is determined by the encoding method used by the programming language. For example, Java, JS, TS, C# all use UTF-16 encoding (save Unicode code point), so the length of char type is 2 bytes. - -!!! Question "Is it ambiguous that data structures implemented based on arrays are also called "static data structures"? Because the stack can also perform operations such as stack out and stack in, which are "dynamic"." - - Stacks do allow dynamic data manipulation, but the data structure is still "static" (immutable in length). Although data structures based on arrays can dynamically add or remove elements, their capacity is fixed. If the amount of data exceeds the pre-allocated size, a new, larger array needs to be created and the contents of the old array copied into the new one. - -!!! question "Why are stacks (queues) "static data structures" when their size is not specified at the time of construction?" - - In high-level programming languages, we do not have to manually specify the initial size of the stack (queue), this is done automatically from within the class. For example, Java's ArrayList usually has an initial capacity of 10. In addition, the expansion operation is also automatic. 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structure with a time complexity of $O(n)$ . -- **Adaptive Search**: it can achieve a time complexity of $O(\log n)$ or even $O(1)$ by exploiting the characteristic data organization or a priori information. - -In fact, **searching algorithms with a time complexity of $O(\log n)$ are usually implemented based on divide and conquer strategies**, such as binary lookup and trees. - -- Each step of the binary search breaks the problem (searching for the target element in the array) into a smaller problem (searching for the target element in one half of the array), and the process continues until the array is empty or the target element is found. -- Trees are representative of divide and conquer relationships, and in data structures such as binary search trees, AVL trees, and heaps, the time complexity of various operations is $O(\log n)$ . - -The divide and conquer strategy for binary search is shown below. - -- **Problems can be decomposed**: binary search recursively decomposes the original problem (searching in an array) into sub-problems (searching in one half of the array), this is done by comparing the middle element with the target element. -- **Subproblems are independent**: in binary search, only one subproblem is processed in each round and it is not affected by another subproblem. -- **Solutions of subproblems need not be merged**: binary search aims to find a specific element, so there is no need to merge the solutions of subproblems. When the subproblem is solved, the original problem is solved at the same time. - -Partitioning can improve search efficiency, essentially because brute-force search can only eliminate one option per round, **while partitioned search can eliminate half of the options per round**. - -### Realize The Dichotomy Based On The Divide And Conquer - -In the previous sections, binary search was implemented based on recursion (iteration). Now we implement it based on divide and conquer (recursion). - -!!! question - - Given an ordered array `nums` of length $n$ and all elements in the array are unique, find the element `target`. - -From the divide and conquer perspective, we denote the subproblem corresponding to the search interval $[i, j]$ as $f(i, j)$ . - -Starting from the original problem $f(0, n-1)$, perform a binary search by the following steps. - -1. Calculate the midpoint $m$ of the search interval $[i, j]$ and exclude half of the search interval based on it. -2. The recursion solves the subproblem that halves in size, possibly $f(i, m-1)$ or $f(m+1, j)$ . -3. Loop through steps `1.` and `2.` until `target` is found or the interval is empty. - -The figure below shows the process of dividing and conquering a binary search for the element $6$ in an array. - -![The divide and conquer process for binary search](binary_search_recur.assets/binary_search_recur.png) - -In the implementation code, we declare a recursion function `dfs()` to solve the problem $f(i, j)$ . - -```src -[file]{binary_search_recur}-[class]{}-[func]{binary_search} -``` diff --git a/docs-en/chapter_divide_and_conquer/build_binary_tree_problem.assets/build_tree_division_pointers.png b/docs-en/chapter_divide_and_conquer/build_binary_tree_problem.assets/build_tree_division_pointers.png deleted file mode 100644 index 52b769e2fbf64fdf38565f3f40020ccdad84cca9..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 17892 zcmd42XH*nH7dF^EfgDDXAUTLgRzyG$21UskBqxcIa}EO{Nis-Aq69@iGLi;Raug(o 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    - -| | index of root node in `preorder` | index interval of subtree in `inorder` | -| ------------- | -------------------------------- | -------------------------------------- | -| Current Tree | $i$ | $[l, r]$ | -| left subtree | $i + 1$ | $[l, m-1]$ | -| right subtree | $i + 1 + (m - l)$ | $[m+1, r]$ | - -Please note that $(m-l)$ in the index of the root node of the right-child node means "the number of nodes in the left subtree", and it is recommended to understand it with the figure below. - -![Index interval representation of root node and left and right subtrees](build_binary_tree_problem.assets/build_tree_division_pointers.png) - -### Code Implementation - -To improve the efficiency of the query $m$, we resort to a hash table `hmap` to store the mapping of the elements of the array `inorder` to the indexes. - -```src -[file]{build_tree}-[class]{}-[func]{build_tree} -``` - -The figure below illustrates the recursion process of building a binary tree, where nodes are built in a downward "recursion" process and edges (i.e., references) are built in an upward "recursion" process. - -=== "<1>" - ![Recursion process for building a binary tree](build_binary_tree_problem.assets/built_tree_step1.png) - -=== "<2>" - ![built_tree_step2](build_binary_tree_problem.assets/built_tree_step2.png) - -=== "<3>" - ![built_tree_step3](build_binary_tree_problem.assets/built_tree_step3.png) - -=== "<4>" - ![built_tree_step4](build_binary_tree_problem.assets/built_tree_step4.png) - -=== "<5>" - ![built_tree_step5](build_binary_tree_problem.assets/built_tree_step5.png) - -=== "<6>" - ![built_tree_step6](build_binary_tree_problem.assets/built_tree_step6.png) - -=== "<7>" - ![built_tree_step7](build_binary_tree_problem.assets/built_tree_step7.png) - -=== "<8>" - ![built_tree_step8](build_binary_tree_problem.assets/built_tree_step8.png) - -=== "<9>" - ![built_tree_step9](build_binary_tree_problem.assets/built_tree_step9.png) - -The result of the division of the preorder traversal `preorder` and the middle order traversal `inorder` within each recursion function is shown in the figure below. - -![Division results in each recursion function](build_binary_tree_problem.assets/built_tree_overall.png) - -Let the number of nodes in the tree be $n$ , initializing each node (executing a recursion function `dfs()` ) uses $O(1)$ time. **Therefore the overall time complexity is $O(n)$** . - -A hash table stores the mapping of `inorder` elements to indexes with a space complexity of $O(n)$ . 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zhpi_#!CNfD#ATmI_*U_9?~s9?%9L7lPKN|B)3DUWkIEleMrj@SLqOfe5&g{d~ttCHBdlS9)J#pUNP;3&^97*I>OU4-l`aobS7O@a$b+ z@ZQT#l})eTyZeomUQPGU;r%jcy_-WT&qsve}0qQvUtP> zD0w_mth}4}q*0#slbWDx8AKG6Jta7tUGjgn*2Ze>^4HH8Ht^iF**%w)!JvlWlvn27 TF8jM33_#%N>gTe~DWM4fbOV~E diff --git a/docs-en/chapter_divide_and_conquer/divide_and_conquer.md b/docs-en/chapter_divide_and_conquer/divide_and_conquer.md deleted file mode 100644 index 10b661f499..0000000000 --- a/docs-en/chapter_divide_and_conquer/divide_and_conquer.md +++ /dev/null @@ -1,91 +0,0 @@ -# Divide And Conquer Algorithms - -"Divide and conquer" is a very important and common algorithmic strategy. Divide and conquer is usually implemented based on recursion and consists of two steps: "divide" and "conquer". - -1. **Division (dividing stage)**: recursionally decomposes the original problem into two or more subproblems until it terminates when it reaches the smallest subproblem. -2. The **cure (merging phase)**: Starting from the smallest subproblem with a known solution, the solutions of the subproblems are merged from the bottom to the top to construct a solution to the original problem. - -As shown in the figure below, "merge sort" is one of the typical applications of divide and conquer strategy. - -1. **Divide**: recursively divide the original array (original problem) into two subarrays (subproblems) until only one element of the subarray remains (smallest subproblem). -2. **cure**: merge ordered subarrays (solutions to sub-problems) from bottom to top to obtain an ordered original array (solutions to the original problem). - -![Divide and conquer strategy for merge sort](divide_and_conquer.assets/divide_and_conquer_merge_sort.png) - -## How To Determine The Divide And Conquer Problem - -Whether a problem is suitable for solution using divide and conquer can usually be judged by referring to the following bases. - -1. **Problems can be decomposed**: the original problem can be decomposed into smaller, similar sub-problems, as well as being able to be divided recursively in the same way. -2. **Subproblems are independent**: there is no overlap between subproblems, no dependence on each other, and can be solved independently. -3. **The solutions of subproblems can be combined**: the solution of the original problem is obtained by combining the solutions of the subproblems. - -Clearly, merge sort is satisfying the above three bases of judgment. - -1. The **problem can be decomposed by**: recursively dividing the array (original problem) into two subarrays (subproblems). -2. **Subproblems are independent**: each subarray can be sorted independently (subproblems can be solved independently). -3. **Solutions of subproblems can be merged**: two ordered subarrays (solutions of subproblems) can be merged into one ordered array (solutions of the original problem). - -## Enhancing Efficiency Through Divide And Conquer - -Not only does divide and conquer effectively solve algorithmic problems, **it often leads to an increase in algorithmic efficiency**. In sorting algorithms, quick sort, merge sort, and heap sort are faster compared to selection, bubble, and insertion sort because they apply a partition strategy. - -So, we can't help but ask:**Why does dividing and conquering improve algorithm efficiency, and what is its underlying logic**? In other words, why are the steps of decomposing a large problem into multiple subproblems, solving the subproblems, and combining the solutions of the subproblems into the solution of the original problem more efficient than solving the original problem directly? This question can be discussed in terms of the number of operations and parallel computation. - -### Operation Number Optimization - -Take "bubble sort" for example, it takes $O(n^2)$ time to process an array of length $n$. Assuming we divide the array into two subarrays from the midpoint as shown in the figure below, it takes $O(n)$ time to divide, $O((n / 2)^2)$ time to sort each subarray, and $O(n)$ time to merge the two subarrays, with an overall time complexity of: - -$$ -O(n + (\frac{n}{2})^2 \times 2 + n) = O(\frac{n^2}{2} + 2n) -$$ - -![Bubble sort before and after dividing the array](divide_and_conquer.assets/divide_and_conquer_bubble_sort.png) - -Next, we compute the following inequality, whose left and right edges are the total number of operations before and after division, respectively: - -$$ -\begin{aligned} -n^2 & > \frac{n^2}{2} + 2n \newline -n^2 - \frac{n^2}{2} - 2n & > 0 \newline -n(n - 4) & > 0 -\end{aligned} -$$ - -**This means that when $n > 4$, the number of operations after division is smaller and the sorting should be more efficient**. Note that the time complexity after division is still of squared order $O(n^2)$ , just that the constant term in the complexity becomes smaller. - -Thinking further, **what if we keep dividing the subarray into two more subarrays from the midpoint until there is only one element left in the subarray**? This idea is actually a "merge sort" with a time complexity of $O(n \log n)$ . - -On second thought, **what if we set a few more division points and divide the original array into $k$ subarrays evenly**? This situation is very similar to "bucket sort", which is ideal for sorting large amounts of data, and has a theoretical time complexity of $O(n + k)$ . - -### Parallel Computing Optimization - -We know that the subproblems generated by partitioning are independent of each other, **and thus can usually be solved in parallel**. In other words, divide and conquer not only reduces the time complexity of the algorithm, **but also facilitates the parallel optimization of the operating system**. - -Parallel optimization is particularly effective in multi-core or multi-processor environments, as the system can process multiple sub-problems at the same time, making fuller use of computational resources and thus significantly reducing overall runtime. - -For example, in the "bucket sort" shown in the figure below, we will distribute the huge amount of data evenly into each bucket, then the sorting task of all the buckets can be distributed to each computing unit, and then merge the results after completion. - -![Parallel computing for bucket sorting](divide_and_conquer.assets/divide_and_conquer_parallel_computing.png) - -## Common Applications Of Divide And Conquer - -On the one hand, divide and conquer can be used to solve many classical algorithmic problems. - -- **Finding Nearest Point Pairs**: the algorithm first divides the set of points into two parts, then finds the nearest point pairs in each of the two parts, and finally finds the nearest point pairs that span the two parts. -- **Large integer multiplication**: e.g. the Karatsuba algorithm, which is a solution to large integer multiplication by breaking it down into multiplication and addition of several smaller integers. -- **Matrix multiplication**: for example, the Strassen algorithm, which is a solution to multiply a large matrix into multiple multiplications and additions of smaller matrices. -- **Hanota problem**: The Hanota problem can be considered as a typical divide and conquer strategy, solved by recursion. -- **Solving Inverse Pairs**: In a sequence, if the preceding number is greater than the following number, then these two numbers form an inverse pair. The problem of solving inverse order pairs can be solved by the idea of divide and conquer with the help of merge sort. - -On the other hand, divide and conquer is widely used in the design of algorithms and data structures. - -- **Binary Search**: binary search is to divide an ordered array into two parts from its midpoint index, then decide which half of the interval to exclude based on the result of comparing the target value with the value of the middle element, and then perform the same binary operation on the remaining interval. -- **Merge sort**: introduced at the beginning of the article and will not be repeated. -- **Quick Sort**: Quick sort is the process of picking a base value and then dividing the array into two subarrays, one with elements smaller than the base value and the other with elements larger than the base value, and then performing the same division operation on both parts until there is only one element left in the subarray. -- **Bucket Sort**: The basic idea of bucket sort is to spread the data into multiple buckets, then sort the elements in each bucket, and finally remove the elements from each bucket in turn to get an ordered array. -- **Trees**: e.g., binary search trees, AVL trees, red-black trees, B-trees, B+-trees, etc., whose operations of finding, inserting, and deleting can be considered as applications of divide and conquer. -- **Heap**: A heap is a special kind of complete binary tree whose various operations, such as insertion, deletion, and heapifying, actually imply the idea of partitioning. -- **Hash Tables**: although hash tables come do not directly apply divide and conquer, certain hash collision resolution strategies apply divide and conquer indirectly, e.g., long linked lists in chained addresses are transformed into red-black trees to improve query efficiency. - -It can be seen that **divide and conquer is a "silent" algorithmic idea**, implicit in various algorithms and data structures. diff --git 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For the Hanota problem, however, we use a different solution strategy. - -!!! question - - Given three columns, denoted `A`, `B` and `C` . In the starting state, column `A` is covered with $n$ disks, which are arranged from top to bottom in the order of smallest to largest. Our task is to move these $n$ disks to the column `C` and keep their original order. The following rules need to be observed while moving the disks. - - 1. Discs can only be taken out of the top of one post and put in from the top of another. - 2. Only one disk can be moved at a time. - 3. The smaller disk must be located above the larger disk at all times. - -![Hanota problem example](hanota_problem.assets/hanota_example.png) - -**We denote a Hanota problem of size $i$ as $f(i)$** . For example $f(3)$ represents the Hanota problem of moving $3$ disks from `A` to `C` . - -### Consider The Basics - -As shown in the figure below, for the problem $f(1)$ , i.e., when there is only one disk, we can just move it directly from `A` to `C` . - -=== "<1>" - ![Solution for problem of size 1](hanota_problem.assets/hanota_f1_step1.png) - -=== "<2>" - ![hanota_f1_step2](hanota_problem.assets/hanota_f1_step2.png) - -As shown in the figure below, for the problem $f(2)$ , i.e., when there are two discs, **Since it is necessary to satisfy at all times that the smaller disc is on top of the larger disc, the movement of is accomplished with the help of `B`** . - -1. First, move the top disk from `A` to `B`. -2. Then move the large disk from `A` to `C` . -3. Finally, move the disk from `B` to `C`. - -=== "<1>" - ![Solution for problem of size 2](hanota_problem.assets/hanota_f2_step1.png) - -=== "<2>" - ![hanota_f2_step2](hanota_problem.assets/hanota_f2_step2.png) - -=== "<3>" - ![hanota_f2_step3](hanota_problem.assets/hanota_f2_step3.png) - -=== "<4>" - ![hanota_f2_step4](hanota_problem.assets/hanota_f2_step4.png) - -The solution $f(2)$ can be summarized as follows:**Move two discs from `A` to `C`** with the help of `B` . Where `C` is called the target column and `B` is called the buffer column. - -### Sub-Problem Decomposition - -For the problem $f(3)$ , i.e., when there are three discs, the situation becomes slightly more complicated. - -Since the solutions of $f(1)$ and $f(2)$ are known, we can think in terms of dividing and conquer, **Consider the two disks at the top of `A` as a whole**. Perform the steps shown in the figure below. Thus the three disks are smoothly moved from `A` to `C`. - -1. With `B` as the target column and `C` as the buffer column, move the two disks from `A` to `B`. -2. Move one of the remaining disks in `A` directly from `A` to `C`. -3. With `C` as the target column and `A` as the buffer column, move the two disks from `B` to `C`. - -=== "<1>" - ![Solution for problem of size 3](hanota_problem.assets/hanota_f3_step1.png) - -=== "<2>" - ![hanota_f3_step2](hanota_problem.assets/hanota_f3_step2.png) - -=== "<3>" - ![hanota_f3_step3](hanota_problem.assets/hanota_f3_step3.png) - -=== "<4>" - ![hanota_f3_step4](hanota_problem.assets/hanota_f3_step4.png) - -Essentially, **we divide the problem $f(3)$ into two subproblems $f(2)$ and subproblem $f(1)$** . After solving these three subproblems in order, the original problem is then solved. This shows that the subproblems are independent and that the solutions can be combined. - -At this point, we can summarize the solution strategy for the Hanota problem shown in the figure below: divide and conquer the original problem $f(n)$ into two subproblems $f(n-1)$ and one subproblem $f(1)$ and solve these three subproblems in the following order. - -1. Move $n-1$ discs from `A` to `B` with the help of `C`. -2. Move the remaining $1$ discs directly from `A` to `C`. -3. Move $n-1$ discs from `B` to `C` with the help of `A`. - -For both subproblems $f(n-1)$ , **can be divided recursionally in the same way until the smallest subproblem $f(1)$ is reached**. And the solution of $f(1)$ is known with only one move operation. - -![Divide and conquer strategy for the hanota problem](hanota_problem.assets/hanota_divide_and_conquer.png) - -### Code Implementation - -In the code, we declare a recursion function `dfs(i, src, buf, tar)` that moves the $i$ discs at the top of the column `src` to the target column `tar` with the help of the buffer column `buf` . - -```src -[file]{hanota}-[class]{}-[func]{solve_hanota} -``` - -As shown in the figure below, the Hanota problem forms a recursion tree of height $n$, where each node represents a sub-problem, corresponding to an open `dfs()` function, **and thus has a time complexity of $O(2^n)$ and a space complexity of $O(n)$**. - -![Recursion tree for the hanota problem](hanota_problem.assets/hanota_recursive_tree.png) - -!!! quote - - The hanota problem stems from an ancient legendary story. In an ancient Indian temple, the monks had three tall diamond pillars and $64$ gold disks of various sizes. The monks were constantly moving the original disks, and they believed that the world would end the moment the last disk was placed correctly. - - However, even if the monks moved once every second, it would take a total of about $2^{64} \approx 1.84×10^{19}$ seconds, a contract of $585$ billion years, which is far more than the current estimate of the age of the universe. So, if this legend is true, we should not need to worry about the end of the world. diff --git a/docs-en/chapter_divide_and_conquer/index.md b/docs-en/chapter_divide_and_conquer/index.md deleted file mode 100644 index a208715f38..0000000000 --- a/docs-en/chapter_divide_and_conquer/index.md +++ /dev/null @@ -1,13 +0,0 @@ -# Divide And Conquer - -

    - -!!! abstract - - The puzzle is broken down layer by layer, each time making it easier. - - Divide and conquer reveals an important truth: start simple and nothing is complicated anymore. diff --git a/docs-en/chapter_divide_and_conquer/summary.md b/docs-en/chapter_divide_and_conquer/summary.md deleted file mode 100644 index ab9aae9348..0000000000 --- a/docs-en/chapter_divide_and_conquer/summary.md +++ /dev/null @@ -1,11 +0,0 @@ -# Summary - -- Partitioning algorithm is a common algorithm design strategy that consists of two phases, divide (divide) and conquer (merge), and is usually implemented based on recursion. -- The basis for determining whether a problem is a divide and conquer algorithm problem includes whether the problem can be divided and conquer, whether the subproblems are independent, and whether the subproblems can be merged. -- Merge sort is a typical application of the partition strategy, which recursively divides and conquers an array into two subarrays of equal length until only one element remains and starts merging them level by level, thus completing the sort. -- Introducing a divide and conquer strategy often leads to an increase in algorithm efficiency. On the one hand, the divide and conquer strategy reduces the number of operations; on the other hand, the partition facilitates the parallel optimization of the system. -- Divide and conquer is both a solution to many algorithmic problems and is widely used everywhere in data structure and algorithm design. -- Compared to brute-force search, adaptive search is more efficient. Searching algorithms with a time complexity of $O(\log n)$ are usually implemented based on a divide and conquer strategy. -- Binary search is another typical application of the divide and conquer strategy that does not include the step of combining the solutions of the subproblems. We can implement binary search by recursion division and conquer. -- In the constructed binary tree problem, the constructed tree (original problem) can be divided into constructed left subtrees and right subtrees (subproblems), which can be realized by dividing the index intervals of the preorder traversal and the middle order traversal. -- In the Hanota problem, a problem of size $n$ can be divided into two subproblems of size $n-1$ and one subproblem of size $1$. 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-In the previous section, we learned how dynamic programming solves problems by subproblem decomposition. In fact, subproblem decomposition is a generalized algorithmic idea with a different focus in divide and conquer, dynamic programming, and backtracking algorithm. - -- The backtracking algorithm recursively divides and conquers the original problem into a number of mutually independent subproblems up to the smallest subproblem and merges the solutions of the subproblems in backtracking to finally obtain a solution to the original problem. -- Dynamic programming also performs a recursion decomposition of the problem, but the main difference with the divide and conquer algorithm is that the subproblems in dynamic programming are interdependent, and many overlapping subproblems appear during the decomposition process. -- The backtracking algorithm exhausts all possible solutions in try and backtrack and avoids unnecessary search branches by pruning. The solution to the original problem consists of a series of decision steps, and we can view the subsequence preceding each decision step as a subproblem. - -In fact, dynamic programming is often used to solve optimization problems, which not only contain overlapping subproblems, but also have two other major properties: optimal substructure, and no posteriority. - -## Optimal Substructure - -We modify the stair-climbing problem slightly to make it more suitable for demonstrating the optimal substructure concept. - -!!! question "Minimal cost of stair climbing" - - Given a staircase where you can go up either $1$ or $2$ steps per step, each step of the staircase is labeled with a non-negative integer indicating the cost you need to pay at that step. Given an array of non-negative integers $cost$ , where $cost[i]$ denotes the cost you need to pay at the $i$th step, and $cost[0]$ is the ground level starting point. Please calculate the minimum cost to reach the top? - -As shown in the figure below, if the costs of the $1$, $2$, and $3$ steps are $1$, $10$, and $1$, respectively, then the minimum cost of climbing from the ground to the $3$ step is $2$. - -![Minimum cost to climb to 3rd order](dp_problem_features.assets/min_cost_cs_example.png) - -Let $dp[i]$ be the cumulative cost of climbing to the $i$th step. Since the $i$th step can only come from the $i - 1$th step or the $i - 2$th step, $dp[i]$ can only be equal to either $dp[i - 1] + cost[i]$ or $dp[i - 2] + cost[i]$. To minimize the cost, we should choose the smaller of the two: - -$$ -dp[i] = \min(dp[i-1], dp[i-2]) + cost[i] -$$ - -This leads to the meaning of optimal substructure:**the optimal solution of the original problem is constructed from the optimal solution of the subproblem**. - -This problem clearly has an optimal substructure: we pick the better of the two subproblem optimal solutions $dp[i-1]$ and $dp[i-2]$ and use it to construct an optimal solution to the original problem $dp[i]$. - -So, is there an optimal substructure for the stair-climbing problem in the previous section? Its goal is to solve for the number of solutions, which may seem like a counting problem, but if it is asked in a different way, "Solve for the maximum number of solutions." We unexpectedly find that **although the questions are equivalent before and after the modification, the optimal substructure surfaces**: the number of maximal solutions of order $n$ is equal to the sum of the number of maximal solutions of order $n-1$ and order $n-2$. So, the optimal substructure is interpreted in a flexible way and will have different meanings in different problems. - -Based on the state-trasition equation, and the initial states $dp[1] = cost[1]$ and $dp[2] = cost[2]$ , we get the dynamic programming code. - -```src -[file]{min_cost_climbing_stairs_dp}-[class]{}-[func]{min_cost_climbing_stairs_dp} -``` - -The figure below shows the dynamic programming process of the above code. - -![Dynamic programming process for minimizing the cost of stair climbing](dp_problem_features.assets/min_cost_cs_dp.png) - -This problem can also be space optimized by compressing one dimension to zero dimension, which makes the space complexity decrease from $O(n)$ to $O(1)$ . - -```src -[file]{min_cost_climbing_stairs_dp}-[class]{}-[func]{min_cost_climbing_stairs_dp_comp} -``` - -## No After-Effects - -The absence of posteriority is one of the important properties of dynamic programming that enables effective problem solving and is defined as: **Given a definite state, its future development is related only to the current state and not to all the states that the current state has experienced in the past**. - -Taking the stair climbing problem as an example, given the state $i$, it will develop the states $i+1$ and $i+2$, which correspond to jumping $1$ and $2$ steps respectively. When making these two choices, we do not need to consider the states before state $i$, they have no effect on the future of state $i$. - -However, if we add a constraint to the stair climbing problem, the situation is different. - -!!! question "climbing stairs with constraints" - - Given a staircase with a total of $n$ steps, you can go up either $1$ steps or $2$ steps per step, **but you can't jump $1$ steps in two consecutive rounds**. How many scenarios are there to get to the top of the building. - -For example, in the figure below, there are only $2$ feasible solutions left to climb up to the $3$ stage, among which the solution with three consecutive jumps of $1$ stage does not satisfy the constraints, and is therefore discarded. - -![Number of scenarios with constraints to climb to 3rd degree](dp_problem_features.assets/climbing_stairs_constraint_example.png) - -In that problem, if the previous round came up by jumping $1$ steps, then the next round must jump $2$ steps. **This means that the choice of next step cannot be independently determined by the current state (current stair step), but is also related to the previous state (previous round stair step)**. - -It is easy to find that this problem does not satisfy the no-laterality, and the state-trasition equation $dp[i] = dp[i-1] + dp[i-2]$ also fails because $dp[i-1]$ stands for the current round of jumping $1$ steps, but it contains many "previous round of jumping $1$ steps up" scenarios. In order to satisfy the constraint, we cannot directly count $dp[i-1]$ into $dp[i]$. - -To do so, we need to extend the state definition:**The state $[i, j]$ denotes that it is at the $i$th order and has jumped the $j$th order in the previous round**, where $j \in \{1, 2\}$ . This state definition effectively distinguishes whether the last round has jumped $1$ or $2$ steps, and we can use it to determine where the current state comes from. - -- When $1$ step was jumped in the previous round, the previous round can only choose to jump $2$ step, i.e. $dp[i, 1]$ can only be transferred from $dp[i-1, 2]$. -- When $2$ step is jumped in the previous round, the previous round can choose to jump $1$ step or $2$ step, i.e. $dp[i, 2]$ can be transferred from $dp[i-2, 1]$ or $dp[i-2, 2]$. - -As shown in the figure below, under this definition, $dp[i, j]$ denotes the number of programs corresponding to the state $[i, j]$. At this time, the state-transition equation is: - -$$ -\begin{cases} -dp[i, 1] = dp[i-1, 2] \\ -dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] -\end{cases} -$$ - -![Consider recursive relationships under constraints](dp_problem_features.assets/climbing_stairs_constraint_state_transfer.png) - -In the end, it is enough to return $dp[n, 1] + dp[n, 2]$, the sum of which represents the total number of scenarios that have climbed to the $n$th step. - -```src -[file]{climbing_stairs_constraint_dp}-[class]{}-[func]{climbing_stairs_constraint_dp} -``` - -In the above case, by considering only one more previous state, we can still extend the definition of the state to make the problem satisfy no posteriority again. However, some problems have very serious "consequences". - -!!! question "Stair Climbing and Obstacle Generation" - - Given a staircase with a total of $n$ steps, you can go up either $1$ steps or $2$ steps per step. **states that when climbing to the $i$th step, the system will automatically put obstacles on the $2i$th step, and all rounds after that are not allowed to jump to the $2i$th step**. For example, if the first two rounds are jumped to the $2$ and $3$ steps, you are not allowed to jump to the $4$ and $6$ steps after that. Ask how many options there are to climb to the top of the building. - -In this problem, the next jump depends on all past states, since each jump puts obstacles on higher rungs of the ladder and affects future jumps. Dynamic programming is often difficult to solve for this type of problem. - -In fact, many complex combinatorial optimization problems (e.g., the traveler's problem) do not satisfy no posteriority. 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zGVXpp6AGx3E(z?`TXrcwULBLSq3v$g%X?yN=y@O5uFrnfXu}W&qDy+nvXyqID|o=1 zmw3#yl9*q$0q@>bbmxbeAJ~WcTdapnC$(>mbzJm}(@B^q+NP=M;lQ~?rfh4^*~+=f zrVd*}g~yBL@5cbP=+orxZ0^DS_{y?rf>C7pL%=K~y?at+U$DI>*SnIF5doy+E0!JS d06-T>!" - ![edit dynamic programming process for distance](edit_distance_problem.assets/edit_distance_dp_step1.png) - -=== "<2>" - ![edit_distance_dp_step2](edit_distance_problem.assets/edit_distance_dp_step2.png) - -=== "<3>" - ![edit_distance_dp_step3](edit_distance_problem.assets/edit_distance_dp_step3.png) - -=== "<4>" - ![edit_distance_dp_step4](edit_distance_problem.assets/edit_distance_dp_step4.png) - -=== "<5>" - ![edit_distance_dp_step5](edit_distance_problem.assets/edit_distance_dp_step5.png) - -=== "<6>" - ![edit_distance_dp_step6](edit_distance_problem.assets/edit_distance_dp_step6.png) - -=== "<7>" - ![edit_distance_dp_step7](edit_distance_problem.assets/edit_distance_dp_step7.png) - -=== "<8>" - ![edit_distance_dp_step8](edit_distance_problem.assets/edit_distance_dp_step8.png) - -=== "<9>" - ![edit_distance_dp_step9](edit_distance_problem.assets/edit_distance_dp_step9.png) - -=== "<10>" - ![edit_distance_dp_step10](edit_distance_problem.assets/edit_distance_dp_step10.png) - -=== "<11>" - ![edit_distance_dp_step11](edit_distance_problem.assets/edit_distance_dp_step11.png) - -=== "<12>" - ![edit_distance_dp_step12](edit_distance_problem.assets/edit_distance_dp_step12.png) - -=== "<13>" - ![edit_distance_dp_step13](edit_distance_problem.assets/edit_distance_dp_step13.png) - -=== "<14>" - ![edit_distance_dp_step14](edit_distance_problem.assets/edit_distance_dp_step14.png) - -=== "<15>" - ![edit_distance_dp_step15](edit_distance_problem.assets/edit_distance_dp_step15.png) - -### Space Optimization - -Since $dp[i,j]$ is transferred from the upper $dp[i-1, j]$, the left $dp[i, j-1]$, and the upper-left state $dp[i-1, j-1]$, and the orthogonal traversal loses the upper-left $dp[i-1, j-1]$ , and the inverted traversal can not build $dp[i, j-1]$ in advance, both traversal orders are not desirable. - -To do this, we can use a variable `leftup` to temporarily store the top-left solution $dp[i-1, j-1]$ so that only the left and top solutions need to be considered. The situation is the same as the complete knapsack problem, and can be traversed using positive order. - -```src -[file]{edit_distance}-[class]{}-[func]{edit_distance_dp_comp} -``` diff --git a/docs-en/chapter_dynamic_programming/index.md b/docs-en/chapter_dynamic_programming/index.md deleted file mode 100644 index 4064c34549..0000000000 --- a/docs-en/chapter_dynamic_programming/index.md +++ /dev/null @@ -1,13 +0,0 @@ -# Dynamic Programming - -

    - -!!! abstract - - Streams merge into rivers and rivers merge into the sea. - - Dynamic programming brings together solutions to small problems into answers to large ones, leading us step by step to the solution of the problem. diff --git a/docs-en/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_dfs_memo_tree.png b/docs-en/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_dfs_memo_tree.png deleted file mode 100644 index bccc8424cde2ac2529a61cf23d09b6e9f086d0c8..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 28461 zcmb4qbyQVf@b9_z(hXA5ARygHNeD;?2m;b64HD8Iap?{T=|&m^K|u1NbR*r}ozlF+ z_x;}c`#sjWi*xoqd*(BY;0UzUmqSG{`>b& zL_{PkEbQ`k+vU~e-rnAUxmjy#>&?wgO?CD5_V&WULUeSrl$2C=clY%4bbel5M@L6S zdPe`md|hpAWo2bjQc_CE&$YERUtiyXf`Z=O-o}Q8#Kc5#aq(aE_5S|;t#hrnXJ>PB za}f~{73Jj%9MV}?S)nZh>nSNIm&eD$!^0sVAqNKsU%q^?va)JxYdbqT zi;s`@^zUY zv;FBe2sHmvL>FMT~rZrBPw6wJS^ZlZt 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lq%+jlFGL4=IJrKz94?5)l05sm0uEfYv4MsDYdzBK{{kQ44N3q2 diff --git a/docs-en/chapter_dynamic_programming/knapsack_problem.md b/docs-en/chapter_dynamic_programming/knapsack_problem.md deleted file mode 100644 index 7277e457f3..0000000000 --- a/docs-en/chapter_dynamic_programming/knapsack_problem.md +++ /dev/null @@ -1,168 +0,0 @@ -# 0-1 Knapsack Problem - -The knapsack problem is a great introductory topic to dynamic programming and is the most common form of the problem in dynamic programming. It has many variants, such as the 0-1 knapsack problem, the complete knapsack problem, and the multiple knapsack problem. - -In this section, we start by solving the most common 0-1 knapsack problem. - -!!! question - - Given $n$ items, the weight of the first $i$ item is $wgt[i-1]$, the value is $val[i-1]$ , and a backpack with capacity $cap$. Each item can be selected only once, and the maximum value of the item that can be put into the backpack without exceeding its capacity is asked. - -Observing the figure below, since the item number $i$ is counted from $1$ and the array index is counted from $0$, the item $i$ corresponds to the weight $wgt[i-1]$ and value $val[i-1]$ . - -![0-1 Example data for backpacks](knapsack_problem.assets/knapsack_example.png) - -We can view the 0-1 knapsack problem as a process consisting of $n$ rounds of decisions, with two decisions for each object: not to put it in and to put it in, and thus the problem is satisfying the decision tree model. - -The goal of the problem is to solve for the "maximum value given the capacity of the knapsack", so with a high probability it is a dynamic programming problem. - -**Step 1: Think about the decisions made in each round and define the state that leads to the $dp$ table** - -For each item, if it is not put into the backpack, the backpack capacity remains unchanged; if it is put into the backpack, the backpack capacity decreases. This leads to the state definition: the current item number $i$ and the remaining backpack capacity $c$ , denoted as $[i, c]$ . - -**The subproblem corresponding to the state $[i, c]$ is**: The maximum value of the previous $i$ items in the knapsack with remaining capacity $c$, denoted as $dp[i, c]$ . - -The solution to be solved is $dp[n, cap]$ , so a two-dimensional $dp$ table of size $(n+1) \times (cap+1)$ is needed. - -**Step 2: Find the optimal substructure, and thus derive the state-transition equation** - -When we make a decision for item $i$, what remains is the decision for the first $i-1$ items, which can be categorized into the following two cases. - -- **Do not put in items $i$** : The backpack capacity remains unchanged and the state changes to $[i-1, c]$ . -- **Put in item $i$** : Backpack capacity decreases $wgt[i-1]$ , value increases $val[i-1]$ , state changes to $[i-1, c-wgt[i-1]]$ . - -The above analysis reveals to us the optimal substructure of the problem: **The maximum value $dp[i, c]$ equals the one** that has the greater value of the two alternatives of leaving items out $i$ and putting items in $i$. This leads to the state-trasition equation: - -$$ -dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) -$$ - -Note that if the current item weight $wgt[i - 1]$ exceeds the remaining backpack capacity $c$, you can only choose not to put it into the backpack. - -**Step 3: Determine boundary conditions and state transfer order** - -The maximum value when there are no items or no remaining backpack capacity is $0$ , i.e. the first column $dp[i, 0]$ and the first row $dp[0, c]$ are both equal to $0$ . - -The current state $[i, c]$ is transferred from the top state $[i-1, c]$ and the top-left state $[i-1, c-wgt[i-1]]$, so it is sufficient to traverse the entire $dp$ table in positive order with a two-level loop. - -Based on the above analysis, we next implement the brute-force search, memorized search, and dynamic programming solutions in order. - -### Method One: Brute Force Search - -The search code contains the following elements. - -- **recursion parameter**: state $[i, c]$ . -- **Return value**: solution to the subproblem $dp[i, c]$ . -- **Termination condition**: when the item number crosses the boundary $i = 0$ or the remaining capacity of the backpack is $0$, the recursion is terminated and the value $0$ is returned. -- **Pruning**: if the weight of the current item exceeds the remaining capacity of the backpack, it can only be left out of the backpack. - -```src -[file]{knapsack}-[class]{}-[func]{knapsack_dfs} -``` - -As shown in the figure below, the time complexity is $O(2^n)$ since each item generates two search branches, unselected and selected. - -Observing the recursion tree, it is easy to find that there are overlapping subproblems in it, such as $dp[1, 10]$ and so on. And when there are more items and the knapsack capacity is large, especially when there are more items of the same weight, the number of overlapping subproblems will increase dramatically. - -![0-1 Backpacking's brute-force search recursion tree](knapsack_problem.assets/knapsack_dfs.png) - -### Method 2: Memorized Search - -To ensure that the overlapping subproblems are computed only once, we record the solutions of the subproblems with the help of a memoized list `mem` , where `mem[i][c]` corresponds to $dp[i, c]$ . - -With the introduction of memoization, **the time complexity depends on the number of subproblems**, which is $O(n \times cap)$ . - -```src -[file]{knapsack}-[class]{}-[func]{knapsack_dfs_mem} -``` - -The figure below shows the search branches that were clipped in the memorized recursion. - -![0-1 Memorized search recursion tree for backpacks](knapsack_problem.assets/knapsack_dfs_mem.png) - -### Method 3: Dynamic Programming - -Dynamic programming is essentially the process of populating the $dp$ table in a state transfer, as shown in the code below. - -```src -[file]{knapsack}-[class]{}-[func]{knapsack_dp} -``` - -As shown in the figure below, both time complexity and space complexity are determined by the size of the array `dp` , i.e. $O(n \times cap)$ . - -=== "<1>" - ![0-1 Dynamic programming process for backpacks](knapsack_problem.assets/knapsack_dp_step1.png) - -=== "<2>" - ![knapsack_dp_step2](knapsack_problem.assets/knapsack_dp_step2.png) - -=== "<3>" - ![knapsack_dp_step3](knapsack_problem.assets/knapsack_dp_step3.png) - -=== "<4>" - ![knapsack_dp_step4](knapsack_problem.assets/knapsack_dp_step4.png) - -=== "<5>" - ![knapsack_dp_step5](knapsack_problem.assets/knapsack_dp_step5.png) - -=== "<6>" - ![knapsack_dp_step6](knapsack_problem.assets/knapsack_dp_step6.png) - -=== "<7>" - ![knapsack_dp_step7](knapsack_problem.assets/knapsack_dp_step7.png) - -=== "<8>" - ![knapsack_dp_step8](knapsack_problem.assets/knapsack_dp_step8.png) - -=== "<9>" - ![knapsack_dp_step9](knapsack_problem.assets/knapsack_dp_step9.png) - -=== "<10>" - ![knapsack_dp_step10](knapsack_problem.assets/knapsack_dp_step10.png) - -=== "<11>" - ![knapsack_dp_step11](knapsack_problem.assets/knapsack_dp_step11.png) - -=== "<12>" - ![knapsack_dp_step12](knapsack_problem.assets/knapsack_dp_step12.png) - -=== "<13>" - ![knapsack_dp_step13](knapsack_problem.assets/knapsack_dp_step13.png) - -=== "<14>" - ![knapsack_dp_step14](knapsack_problem.assets/knapsack_dp_step14.png) - -### Space Optimization - -Since each state is only related to the state of its previous row, we can use two arrays to scroll forward, reducing the space complexity from $O(n^2)$ to $O(n)$ . - -Thinking further, can we achieve spatial optimization using only an array? Observation shows that each state is transferred from the lattice directly above or to the upper left. Assuming there is only one array, when starting to traverse the $i$th row, the array stores still the state of the $i-1$th row. - -- If we take positive order traversal, then when traversing to $dp[i, j]$, the top left $dp[i-1, 1]$ ~ $dp[i-1, j-1]$ values may have been overwritten, and then we cannot get the correct state transfer result. -- If an inverted traversal is taken, the overwrite problem does not occur and the state transfer can be performed correctly. - -The figure below shows the conversion from row $i = 1$ to row $i = 2$ under a single array. Consider the difference between an orthogonal traversal and an inverted traversal. - -=== "<1>" - ![0-1 Dynamic programming process after spatial optimization of backpacks](knapsack_problem.assets/knapsack_dp_comp_step1.png) - -=== "<2>" - ![knapsack_dp_comp_step2](knapsack_problem.assets/knapsack_dp_comp_step2.png) - -=== "<3>" - ![knapsack_dp_comp_step3](knapsack_problem.assets/knapsack_dp_comp_step3.png) - -=== "<4>" - ![knapsack_dp_comp_step4](knapsack_problem.assets/knapsack_dp_comp_step4.png) - -=== "<5>" - ![knapsack_dp_comp_step5](knapsack_problem.assets/knapsack_dp_comp_step5.png) - -=== "<6>" - ![knapsack_dp_comp_step6](knapsack_problem.assets/knapsack_dp_comp_step6.png) - -In the code implementation, we simply remove the first dimension $i$ of the array `dp` and change the inner loop to an inverted traversal. - -```src -[file]{knapsack}-[class]{}-[func]{knapsack_dp_comp} -``` diff --git a/docs-en/chapter_dynamic_programming/summary.md b/docs-en/chapter_dynamic_programming/summary.md deleted file mode 100644 index feaf457088..0000000000 --- a/docs-en/chapter_dynamic_programming/summary.md +++ /dev/null @@ -1,23 +0,0 @@ -# Summary - -- Dynamic programming decomposes the problem and achieves efficient computational efficiency by storing solutions to subproblems to circumvent repeated computations. -- All dynamic programming problems can be solved using backtracking (brute force search) without taking time into account, but there are a large number of overlapping subproblems in the recursion tree, which is extremely inefficient. By introducing a memorized list, the solutions of all computed subproblems can be stored, thus ensuring that overlapping subproblems are computed only once. -- Memorized recursion is a top-to-bottom recursion, while its counterpart, dynamic programming, is a bottom-to-top recursion, which is like "filling in a table". Since the current state only depends on some local states, we can eliminate one dimension of the $dp$ table, thus reducing the space complexity. -- Subproblem decomposition is a generalized algorithmic idea with different properties in divide and conquer, dynamic programming, and backtracking algorithm. -- Three main properties of dynamic programming problems: overlapping subproblems, optimal substructure, and no posteriority. -- An original problem has an optimal substructure if its optimal solution can be constructed from the optimal solution of the subproblem. -- A posteriority-free approach means that for a state, its future development is related only to that state, independent of all past states it has experienced. Many combinatorial optimization problems are not posteriority free and cannot be solved quickly using dynamic programming. - -**Knapsack problem** - -- The knapsack problem is the most typical dynamic programming topic with variants of the problem such as 0-1 knapsack, complete knapsack, and multiple knapsack. -- 0-1 The state of the backpack is defined as the maximum value of the first $i$ items in the backpack with remaining capacity $c$. Based on the two decisions of not putting into the backpack and putting into the backpack, the optimal substructure can be obtained and the state-trasition equation is constructed. In spatial optimization, since each state depends on the states directly above and above left, it is necessary to traverse the list in inverse order to avoid the state above left being covered. -- A full backpack has an unlimited number of selections per item, so the state transfer for selecting items to put into it is different from that of a 0-1 backpack. Since the state depends on the states directly above and directly to the left, it should be traversed in positive order in spatial optimization. -- The change exchange problem is a variant of the knapsack problem. It changes from finding the "maximum" value to finding the "minimum" number of coins, so $\max()$ in the state-transition equation should be changed to $\min()$. From "not exceeding" the backpack capacity to "exactly" the target amount, so $amt + 1$ is used to denote an invalid solution that "fails to reach the target amount". -- The Change Exchange II problem was changed from "Minimum number of coins" to "Number of combinations of coins", and the state-transition equation was changed from $\min()$ to a summation operator. - -**Edit distance problem** - -- The edit distance (Levenshtein distance) is used to measure the similarity between two strings, which is defined as the minimum number of editing steps from one string to another, and the editing operations include add, delete, and replace. -- The state of the edit distance problem is defined as the minimum number of edit steps required to change the first $i$ characters of $s$ to the first $j$ characters of $t$. When $s[i] \ne t[j]$ has three decisions: add, delete, and replace, they all have corresponding remaining subproblems. Accordingly, the optimal substructure can be found with the construction of state-trasition equation. And when $s[i] = t[j]$, there is no need to edit the current character. -- In edit distance, the state depends on the states directly above, directly to the left, and above left of it, so that neither positive-order nor inverted-order traversal after spatial optimization can correctly transfer the state. 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Knapsack Problem - -In this section, we start by solving another common knapsack problem: complete knapsacks, and then learn about a special case of it: change exchange. - -## Totally Backpacked - -!!! question - - Given $n$ items, the weight of the $i$th item is $wgt[i-1]$, the value is $val[i-1]$ , and a backpack with capacity $cap$. **Each item can be selected repeatedly**. Please return the maximum value of an item that can be put into the backpack without exceeding its capacity. - -![Example data for the complete knapsack problem](unbounded_knapsack_problem.assets/unbounded_knapsack_example.png) - -### Ideas For Dynamic Programming - -The full knapsack is very similar to the 0-1 knapsack problem, **differing only in not limiting the number of times an item can be selected**. - -- In 0-1 backpacks, there is only one item per backpack, so when you put an item $i$ into your backpack, you can only choose from the first $i-1$ items. -- In a Full Backpack, there are an infinite number of each item, so after placing an item $i$ into the backpack, **can still select from the previous $i$ items**. - -Under the full backpacking provision, changes in the state $[i, c]$ are categorized into two cases. - -- **Do not put items $i$** : Same as 0-1 backpack, transfer to $[i-1, c]$ . -- **Put in item $i$** : Transfer to $[i, c-wgt[i-1]]$ unlike 0-1 backpack. - -Thus the state-trasition equation becomes: - -$$ -dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) -$$ - -### Code Implementation - -Comparing the code of the two questions, one of the state transfers changes from $i-1$ to $i$, and the rest are identical. - -```src -[file]{unbounded_knapsack}-[class]{}-[func]{unbounded_knapsack_dp} -``` - -### Space Optimization - -Since the current state is transferred from the state on the left and above, **the spatial optimization should take a positive-order traversal of each row in the $dp$ table**. - -This traversal order is the opposite of 0-1 backpack. Please take the help of the figure below to understand the difference. - -=== "<1>" - ![Dynamic programming process after spatial optimization of a complete knapsack](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step1.png) - -=== "<2>" - ![unbounded_knapsack_dp_comp_step2](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step2.png) - -=== "<3>" - ![unbounded_knapsack_dp_comp_step3](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step3.png) - -=== "<4>" - ![unbounded_knapsack_dp_comp_step4](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step4.png) - -=== "<5>" - ![unbounded_knapsack_dp_comp_step5](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step5.png) - -=== "<6>" - ![unbounded_knapsack_dp_comp_step6](unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step6.png) - -The code is relatively simple to implement, simply removing the first dimension of the array `dp`. - -```src -[file]{unbounded_knapsack}-[class]{}-[func]{unbounded_knapsack_dp_comp} -``` - -## Change Exchange Problems - -The knapsack problem is representative of a large class of dynamic programming problems that have many variants, such as the change exchange problem. - -!!! question - - Given $n$ coins, the face value of the first $i$ coin is $coins[i - 1]$ and the target amount is $amt$ , **each coin can be selected repeatedly**, ask the minimum number of coins that can make the target amount. Return $-1$ if the target amount cannot be rounded up. - -![Example data for the coin exchange problem](unbounded_knapsack_problem.assets/coin_change_example.png) - -### Ideas For Dynamic Programming - -**Change exchange can be seen as a special case of full backpacking**. The two have the following connections and differences. - -- The two questions can be interchanged, with "item" corresponding to "coin", "item weight" corresponding to "coin denomination "Item" corresponds to "Coin", "Item Weight" corresponds to "Coin Face Value", and "Backpack Capacity" corresponds to "Target Amount". -- The optimization goals are reversed; the knapsack problem is to maximize the value of items and the change exchange problem is to minimize the number of coins. -- The knapsack problem is to find a solution that "does not exceed" the capacity of the knapsack, and the change exchange is to find a solution that "just happens" to be the target amount. - -**Step 1: Think about the decisions made in each round and define the state that leads to the $dp$ table** - -The subproblem corresponding to the state $[i, a]$ is: **The minimum number of coins that the first $i$ coins can come up with the amount $a$**, denoted as $dp[i, a]$ . - -The dimensions of the 2D $dp$ table are $(n+1) \times (amt+1)$ . - -**Step 2: Find the optimal substructure, and hence derive the state-transition equation** - -There are two differences between this problem and the state-trasition equation for a complete backpack, as follows. - -- This question requires the minimum value, so you need to change the operator $\max()$ to $\min()$. -- The subject of optimization is the number of coins, not the value of the item, so just execute $+1$ when you select the coins. - -$$ -dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1) -$$ - -**Step 3: Determine boundary conditions and state transfer order** - -When the target amount is $0$, the minimum number of coins to make it is $0$, i.e. all $dp[i, 0]$ in the first column are equal to $0$. - -When there are no coins, **cannot come up with any $> 0$ target amount**, i.e., it is an invalid solution. In order for the $\min()$ function in the state-transition equation to recognize and filter invalid solutions, we consider using $+ \infty$ to represent them, i.e., making all $dp[0, a]$ in the first row equal to $+ \infty$ . - -### Code Implementation - -Most programming languages do not provide $+ \infty$ variables, and the maximum value of the integer `int` can be used instead. This, in turn, can lead to large number overruns: the $+ 1$ operation in the state-transition equation can overflow. - -For this reason, we use the number $amt + 1$ to denote the null solution, since the maximum number of coins that make up $amt$ is $amt$. - -Before the final return, determine if $dp[n, amt]$ is equal to $amt + 1$, if it is, then return $-1$, which means the target amount cannot be reached. - -```src -[file]{coin_change}-[class]{}-[func]{coin_change_dp} -``` - -The figure below shows the dynamic programming process for change exchange, very similar to a full backpack. - -=== "<1>" - ![Dynamic programming process for the change exchange problem](unbounded_knapsack_problem.assets/coin_change_dp_step1.png) - -=== "<2>" - ![coin_change_dp_step2](unbounded_knapsack_problem.assets/coin_change_dp_step2.png) - -=== "<3>" - ![coin_change_dp_step3](unbounded_knapsack_problem.assets/coin_change_dp_step3.png) - -=== "<4>" - ![coin_change_dp_step4](unbounded_knapsack_problem.assets/coin_change_dp_step4.png) - -=== "<5>" - ![coin_change_dp_step5](unbounded_knapsack_problem.assets/coin_change_dp_step5.png) - -=== "<6>" - ![coin_change_dp_step6](unbounded_knapsack_problem.assets/coin_change_dp_step6.png) - -=== "<7>" - ![coin_change_dp_step7](unbounded_knapsack_problem.assets/coin_change_dp_step7.png) - -=== "<8>" - ![coin_change_dp_step8](unbounded_knapsack_problem.assets/coin_change_dp_step8.png) - -=== "<9>" - ![coin_change_dp_step9](unbounded_knapsack_problem.assets/coin_change_dp_step9.png) - -=== "<10>" - ![coin_change_dp_step10](unbounded_knapsack_problem.assets/coin_change_dp_step10.png) - -=== "<11>" - ![coin_change_dp_step11](unbounded_knapsack_problem.assets/coin_change_dp_step11.png) - -=== "<12>" - ![coin_change_dp_step12](unbounded_knapsack_problem.assets/coin_change_dp_step12.png) - -=== "<13>" - ![coin_change_dp_step13](unbounded_knapsack_problem.assets/coin_change_dp_step13.png) - -=== "<14>" - ![coin_change_dp_step14](unbounded_knapsack_problem.assets/coin_change_dp_step14.png) - -=== "<15>" - ![coin_change_dp_step15](unbounded_knapsack_problem.assets/coin_change_dp_step15.png) - -### Space Optimization - -The space-optimized handling of change redemption is consistent with a full backpack. - -```src -[file]{coin_change}-[class]{}-[func]{coin_change_dp_comp} -``` - -## Change Exchange Problem II - -!!! question - - Given $n$ coins, the face value of the first $i$ coin is $coins[i - 1]$ , and the target amount is $amt$ , and each coin can be selected repeatedly, **Please return the number of coin combinations** that are in the process of rounding up to the target amount. - -![Example data for Coin Exchange Problem II](unbounded_knapsack_problem.assets/coin_change_ii_example.png) - -### Ideas For Dynamic Programming - -Compared to the previous question, the goal of this question is the number of combinations, so the sub-problem becomes: **the number of combinations** of the first $i$ coins that can come up with the amount $a$. And the $dp$ table is still a 2D matrix of size $(n+1) \times (amt + 1)$. - -The number of combinations of the current state is equal to the sum of the number of combinations of the two decisions of not choosing the current coin and choosing the current coin. The state-transition equation is: - -$$ -dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] -$$ - -When the target amount is $0$, no coins need to be selected to get the target amount, so all $dp[i, 0]$ in the first row should be initialized to $1$ . When there are no coins, it is impossible to get any $>0$ target amount, so all $dp[0, a]$ in the first row are equal to $0$. - -### Code Implementation - -```src -[file]{coin_change_ii}-[class]{}-[func]{coin_change_ii_dp} -``` - -### Space Optimization - -Spatial optimization is handled in the same way, by removing the coin dimension. - -```src -[file]{coin_change_ii}-[class]{}-[func]{coin_change_ii_dp_comp} -``` diff --git a/docs-en/chapter_graph/graph.assets/adjacency_list.png b/docs-en/chapter_graph/graph.assets/adjacency_list.png deleted file mode 100644 index 2c91ed0f05d4226bddf1d6cb3d53fb5cf4315f91..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 20549 zcmc$_XHXPD8!kFCyGzblNehy5k_@g0k_AzM#1&BxBq&G@gNT6S9EDZMNt6rQ?R6O!eFS%=sHm8ooxOVX>W?2kI%hkaoSb@ldlwfMZES4n>+27W4>mS7ii?Z2wY7(r zh7wy7BAZ`6Zw$%H%X4>kx3#t1-Q9I_b2BzJj){r!^z{7n>C?OZaz8)6*RNmSxN&1- zWTd~pe|>#@XJ==Ab527;V|jUbY;0_3XsBttDXt|pr21LyNKH#iOJ-)Ks;Vjs!`;8S zYierly|#Vz>Q!K1;GH{njEs!Z($ah@eA2$8hQ1En+}|8u8SkC%tsMIB;>8Qx*PK6x ze>}^cKFo7?@ZdpsQ`o@5Kngzj*|TS5Wo6yn-SP49i`$Dy$w^rqnNw?%6B83rQBe&e z4aI#$k&%(b-9_PzVS(=h^1AY@vMqahdh&bnDhDcEi(FDllEV_iLdruvbbPoyd+{{@ z+<>*!uio|^Uz<>JS(Tu{+*ErOn527?Z?w?%wK;hlxDjOF`r^Z{k28?kbSt-3+_9~E z6(xYqRC_X2^T>i2_%h1l^h)=x)cFumt=5s{J^-uL=_%IL^<@N+NgZe(+Uj))xq`DD zJ0^JyZ=V}JXu+QCdqmEQ^R0fG=?zG!>;<-3*A?s+#U54)>Ojl~GC?f0`a6rK^~9F} zjO4R%gU!pq6S5gIY`O(UKaMHi%`JgAfsXBqRlyl}m(}lbZ})qT@2rj-+^cy>3j|gf zbRAo>%KN=Tv3*Zn^FKjXP@cfjvHMiXT+ZQW!9Y5xY7MhKxB}!g zK%>g0>|j8{2ENE|RGMv0(dtzC1rieYJ|x?(x9a`;hiXHTDSSX##}x#;+`EBwp9$Wf zt8njM=WH=VEw|hFy=Z2;C0&M#mYNp14{N@q_gS(n6&W;nFMSMKQ#pwPe>-|Fu?t9sk0{( z%FT$wiobFE7y=8o1El=g{$rFOKmO@2S{BUFbWX^H1tmCe1dv72V77VSXd<_R17sm+ zNs5O5JOoDe+)*)}d@!)$M(kFzU5Q|;hY*FBCMed2SPK9LLjkCe2}Bx-LL@kI0zx$j z8g`_`07Kj`P^^@uf7{S@W2#byf*54ElB+f4Ta`a#SZzkTW@JNplkM59i}B`cmGVPFU9sYAAL-%- zstoHE<=DHPu$^$G5nTU2{0~k(^=AIS8)LUW<&qeEW5znSJm=Hm{y@Bt>+V6`BIP`1 zfFQ+-^{IL`5Gh-{nX;SfR*)^@d8CkvHeG%$*Gx`M3|Bxvh2% z4@Hb?*@H4}{QP@I^NF;I&D@j5Z~!XWy1a+u-OkT>tlOOXAU{4crbm6z$>lcE?ZSMy zh8^4TTOR+tJloR5Pdq4ti%eGX9tGR_xy6_4JqC30HE4T6Im(_WK;br?8;f!eC1U0d z`8}_Ba2z59qEKrCr191QOgbj&w`wCjxDdYcpM#_0>e>>6nzjsa_tUgq5$*X{HO%~7 z(9ZT6E||z|?hr+oxcBfhiC%C;FtNZ+KvbB}m* z?{&&_Uw`|v-&rIYefY@s^)=DpWOp&|hQzB@A8&6o5#ty_ImU!t3jg-_VC(>1LANZO zJdqfqBJnh(yclPu5uI?s0?9X|Tm|lNZWs1d+to7x^OtT+{sV|9gL^N=fQ^9Ws}l#O z$Oj&=&&x@0k0-tsezTtNS`OpWL1hly{OChZugNpX7N1S#oZ*~;7*d#yk zM@w+(9@UgtXXFp%}k7YfDWsjc7Ds(ASNL3|+do=Y?kz(Hc8W9mUd>^^; 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zF@9dQpuM7?wi!xm(-&Y3$m8$lcndm>ubkT9cu>735hgrw8U_M=Z}Oet!6e7?sma6#}Z)33475$mhn-;H7~zFet~umtweML@}g z{E3L2yZRq46RvN3*q&k(6uHNouVtUkY`-aS{i@+(9!No{8b42d^h#4n>0ZtQETNOu TIvVl2HAG!mTd7RJ;?4g7$?D0a diff --git a/docs-en/chapter_graph/graph.md b/docs-en/chapter_graph/graph.md deleted file mode 100644 index 4979d14185..0000000000 --- a/docs-en/chapter_graph/graph.md +++ /dev/null @@ -1,83 +0,0 @@ -# Graph - -"A graph is a non-linear data structure consisting of a vertex and an edge. We can abstractly represent a graph $G$ as a set of vertices $V$ and a set of edges $E$. The following example shows a graph with 5 vertices and 7 edges. - -$$ -\begin{aligned} -V & = \{ 1, 2, 3, 4, 5 \} \newline -E & = \{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \} \newline -G & = \{ V, E \} \newline -\end{aligned} -$$ - -If we consider the vertices as nodes and the edges as references (pointers) connecting the individual nodes, we can think of a graph as a data structure expanded from a linked list. As shown in the figure below, **compared to linear relationships (linked lists) and divide and conquer relationships (trees), network relationships (graphs) have more degrees of freedom and thus are more complex**. - -![Relationship between linked lists, trees, and graphs](graph.assets/linkedlist_tree_graph.png) - -## Common Types Of Graphs And Terminology - -According to whether the edges are directed or not, it can be categorized into "undirected graph" and "directed graph" as shown in the figure below. - -- In an undirected graph, edges represent "bi-directional" connections between two vertices, such as "friend relationships" in WeChat or QQ. -- In a directed graph, edges are directional, i.e. $A \rightarrow B$ and $A \leftarrow B$ edges are independent of each other, e.g., the relationship between "following" and "being followed" on Weibo or Jitterbug. - -![Directed vs. undirected graphs](graph.assets/directed_graph.png) - -According to whether all the vertices are connected or not, it can be categorized into "connected graph" and "disconnected graph" as shown in the figure below. - -- For a connected graph, starting from a vertex, you can reach any of the remaining vertices. -- For a disconnected graph, at least one vertex is unreachable from some vertex. - -![Connected vs. disconnected graphs](graph.assets/connected_graph.png) - -We can also add "weight" variables to the edges to get a "weighted graph" as shown in the figure below. For example, in handheld games such as Honor of Kings, the system will calculate the "intimacy" between players based on the common playing time, and this kind of intimacy network can be represented by a weighted graph. - -![Weighted vs. unweighted graphs](graph.assets/weighted_graph.png) - -The graph data structure contains the following common terms. - -- "adjacency adjacency": two vertices are said to be "adjacency" when there are edges connecting them. In the figure above, the adjacency of vertex 1 is vertex 2, 3 and 5. -- "path": The sequence of edges passing from vertex A to vertex B is called a "path" from A to B. In the figure above, the edge sequence 1-5-2-4 is a path from vertex 1 to vertex 4. In the figure above, the edge sequence 1-5-2-4 is a path from vertex 1 to vertex 4. -- "degree": the number of edges a vertex has. For a directed graph, "in-degree" means how many edges are directed to the vertex, and "out-degree" means how many edges are directed away from the vertex. - -## Representation Of Graphs - -Common representations of graphs include "adjacency matrix" and "adjacency list". The following is an example of an undirected graph. - -### Adjacency Matrix - -Let the number of vertices of a graph be $n$, the "adjacency matrix" uses a matrix of size $n \times n$ to represent the graph, each row (column) represents a vertex, and the elements of the matrix represent edges, and $1$ or $0$ denotes the existence of an edge between two vertices. - -As shown in the figure below, let the adjacency matrix be $M$ and the list of vertices be $V$, then the matrix element $M[i, j] = 1$ means that there exists an edge between the vertex $V[i]$ to the vertex $V[j]$ and vice versa $M[i, j] = 0$ means that there is no edge between the two vertices. - -![Adjacency matrix representation of the graph](graph.assets/adjacency_matrix.png) - -The adjacency matrix has the following properties. - -- A vertex cannot be connected to itself, so the main diagonal elements of the adjacency matrix are meaningless. -- For undirected graphs, the edges in both directions are equivalent, when the adjacency matrix is symmetric about the main diagonal. -- By replacing the elements of the adjacency matrix from $1$ and $0$ to weights, the weighted graph can be represented. - -When using an adjacency matrix to represent a graph, we can directly access the matrix elements to get the edges, so the addition, deletion, and checking operations are very efficient with a time complexity of $O(1)$ . However, the space complexity of the matrix is $O(n^2)$ , which is more memory intensive. - -### Adjacency List - -"The adjacency list represents a graph using $n$ linked lists, where the nodes of the linked lists represent the vertices. The $i$th linked list corresponds to vertex $i$, which stores all the adjacency vertices (i.e., vertices connected to that vertex) of that vertex. The figure below shows an example of a graph stored using an adjacency list. - -![Adjacency list representation of the graph](graph.assets/adjacency_list.png) - -An adjacency list stores only the edges that actually exist, and the total number of edges is usually much less than $n^2$, so it is more space efficient. However, you need to find edges in an adjacency list by traversing the linked list, so it is not as time efficient as an adjacency matrix. - -Observing the figure above, **the structure of the adjacency list is very similar to the "separate chaining" of addresses in hash tables, so we can use a similar method to optimize the efficiency**. For example, when the linked list is long, it can be converted into an avl tree or a red-black tree to optimize the time efficiency from $O(n)$ to $O(\log n)$; or it can be converted into a hash table to reduce the time complexity to $O(1)$. - -## Graphs Common Applications - -As shown in the table below, many realistic systems can be modeled by graphs, and the corresponding problems can be approximated as graph computation problems. - -

      Table   Common Real-Life Graphs

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      Table   Adjacency matrix vs. adjacency list

      - -| | Adjacency matrix | Adjacency list (linked list) | Adjacency table (hash table) | -| --------------------------- | ---------------- | ---------------------------- | ---------------------------- | -| determine whether adjacency | $O(1)$ | $O(m)$ | $O(1)$ | -| Add Edge | $O(1)$ | $O(1)$ $O(1)$ $O(1)$ | -| Delete Edge | $O(1)$ | $O(m)$ | $O(1)$ | -| Add vertex | $O(n)$ | $O(1)$ | $O(1)$ | -| Delete vertex | $O(n^2)$ | $O(n + m)$ | $O(n)$ | -| Memory Space Occupancy | $O(n^2)$ | $O(n + m)$ | $O(n + m)$ | - -Observing the above table, it seems that the adjacency list (hash table) is optimal in terms of time and space efficiency. But in fact, the efficiency of operating edges in the adjacency matrix is higher, only one array access or assignment operation is needed. 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zs83-61c1Nbbr=T6Fd20$7|WNdxE;WhHu|E)_>0AeGWl1^8as-axs6(TV!E>(v$a`& z3r;&G=u?;g0pO3zh*J5dW5HNJWV8clZIi=!5n1Yru9t31z2Hfh8OhdSjSVeQpAuK^ zv6AHC`43pLS$_-W(PjQVw3}0ZW&+$FdCwq#0RK=<y8db5=xj zP?b9*M0snGhD4W~X|YEXy{B1g&&l^z3C)_#`dcu|7h85tbIPI(yNCdWKQRGr?q2{Y zWB|}pmUZ)pV>T=6%LV^>^Ms2(mfkbV#LA_W-rW8crrc7W`sq7CHZp*j3?Prc9l+6B z-Y;(vK!E>*pJaCNitc{vy}&V-0n9el1bAu!*lka|PcQ%i{5R}~1kd`N_X4W`o~j9O z^e~kHte*mR3g7l7fyd+9Jp+LMg=wG4nEgKLXMi$*slEuJ1K2GLr#-RXj~D>Jue=)! zZ(jvI)fYib1^_T%rg^;ze7XkEP*;It(LWRr00zxe1&q~IVB;FVQ(3DtHrd6s- zAZ2kjMl&Xj19crZcjD@$Fm(h00ER^*E{>gCT>}orRUCtjajptDu@qBbIB?*=F^EI5 zlu6aUv#|+Grj%j?d{D>&^0;ZB1pqJ*P5ojn`&T>=17OK2rV-4tR29&z6DXNDaNxi( zfJ5icRsBn|Szxf56^yC^+Kxydk3|II$z&ov007);v5J}a5Oa|~$FBM}3M|%eWdR?g z=3j!hVbfhY{?z?g%>@(|an7HE&Fw1Y-)n?@68*_<#TMUnv& z-uHI+d1Hvo`g#5+>+Ao$?W8sn7)vS5E9h$i0I;~$#P?{Pwm9==N_+UQ4OPx`*4lfV zN(2B{UWrrhwVsq&b%d`rNtd$al!)jA0QeI`lu|a" - ![Breadth-first traversal steps for graphs](graph_traversal.assets/graph_bfs_step1.png) - -=== "<2>" - ![graph_bfs_step2](graph_traversal.assets/graph_bfs_step2.png) - -=== "<3>" - ![graph_bfs_step3](graph_traversal.assets/graph_bfs_step3.png) - -=== "<4>" - ![graph_bfs_step4](graph_traversal.assets/graph_bfs_step4.png) - -=== "<5>" - ![graph_bfs_step5](graph_traversal.assets/graph_bfs_step5.png) - -=== "<6>" - ![graph_bfs_step6](graph_traversal.assets/graph_bfs_step6.png) - -=== "<7>" - ![graph_bfs_step7](graph_traversal.assets/graph_bfs_step7.png) - -=== "<8>" - ![graph_bfs_step8](graph_traversal.assets/graph_bfs_step8.png) - -=== "<9>" - ![graph_bfs_step9](graph_traversal.assets/graph_bfs_step9.png) - -=== "<10>" - ![graph_bfs_step10](graph_traversal.assets/graph_bfs_step10.png) - -=== "<11>" - ![graph_bfs_step11](graph_traversal.assets/graph_bfs_step11.png) - -!!! question "Is the sequence of breadth-first traversals unique?" - - No, it isn't. Breadth-first traversal only requires traversing in "near to far" order, **while the traversal order of multiple vertices of the same distance is allowed to be arbitrarily disrupted**. Take the figure above as an example, the access order of vertices $1$, $3$ can be swapped, and the access order of vertices $2$, $4$, $6$ can be swapped arbitrarily. - -### Complexity Analysis - -**Time complexity:** All vertices are queued in and out once, using $O(|V|)$ time; during traversal of the adjacency vertices, since it is an undirected graph, all the edges are accessed $2$ times, using $O(2|E|)$ time; and overall using $O(|V| + |E|)$ time. - -**Space complexity:** The maximum number of vertices in a list `res` , hash table `visited` , and queue `que` is $|V|$ , using $O(|V|)$ space. - -## Depth-First Traversal - -**Depth-first traversal is a type of traversal that prioritizes going to the end and turning back when there is no way out**. As shown in the figure below, starting from the top-left vertex, you visit one of the current vertex's adjacency vertices until you return at the end of your walk, then continue to the end and return, and so on until all vertex traversal is complete. - -![depth-first traversal of graph](graph_traversal.assets/graph_dfs.png) - -### Algorithm Implementation - -This algorithmic paradigm of "go to the end and back" is usually implemented based on recursion. Similar to breadth-first traversal, depth-first traversal requires a hash table `visited` to keep track of the vertices that have already been visited, in order to avoid revisiting vertices. - -```src -[file]{graph_dfs}-[class]{}-[func]{graph_dfs} -``` - -The algorithm flow for depth-first traversal is shown in the figure below. - -- The **straight dashed line represents a downward recursion**, indicating that a new recursion method is opened to access the new vertex. -- The **curved dashed line represents upward backtracking**, indicating that this recursion method has returned, back to the point where it was opened. - -For a better solution, it is recommended to combine the graphs with the code to simulate the entire DFS process in your head (or draw it down with a pen), including when each recursion method opens and when it returns. - -=== "<1>" - ![Depth-first traversal steps for graphs](graph_traversal.assets/graph_dfs_step1.png) - -=== "<2>" - ![graph_dfs_step2](graph_traversal.assets/graph_dfs_step2.png) - -=== "<3>" - ![graph_dfs_step3](graph_traversal.assets/graph_dfs_step3.png) - -=== "<4>" - ![graph_dfs_step4](graph_traversal.assets/graph_dfs_step4.png) - -=== "<5>" - ![graph_dfs_step5](graph_traversal.assets/graph_dfs_step5.png) - -=== "<6>" - ![graph_dfs_step6](graph_traversal.assets/graph_dfs_step6.png) - -=== "<7>" - ![graph_dfs_step7](graph_traversal.assets/graph_dfs_step7.png) - -=== "<8>" - ![graph_dfs_step8](graph_traversal.assets/graph_dfs_step8.png) - -=== "<9>" - ![graph_dfs_step9](graph_traversal.assets/graph_dfs_step9.png) - -=== "<10>" - ![graph_dfs_step10](graph_traversal.assets/graph_dfs_step10.png) - -=== "<11>" - ![graph_dfs_step11](graph_traversal.assets/graph_dfs_step11.png) - -!!! question "Is the sequence of depth-first traversal unique?" - - Similar to breadth-first traversal, the order of the depth-first traversal sequence is not unique. Given a vertex, it is possible to explore in either direction first, i.e., the order of the adjacency vertices can be arbitrarily disrupted, and it is all depth-first traversal. - - Taking tree traversal as an example, "root $\rightarrow$ left $\rightarrow$ right" and "left $\rightarrow$ root $\rightarrow$ right", "left $\rightarrow$ right $\rightarrow$ root" corresponds to preorder, middle-order, and postorder traversals, respectively, and they demonstrate three different traversal priorities, however, all three are depth-first traversals. - -### Complexity Analysis - -**Time complexity:** All vertices are visited $1$ times, using $O(|V|)$ time; all edges are visited $2$ times, using $O(2|E|)$ time; and overall $O(|V| + |E|)$ time. - -**Space complexity:** lists `res` , hash tables `visited` The maximum number of vertices is $|V|$ , and the maximum depth of recursion is $|V|$ , so $O(|V|)$ space is used. diff --git a/docs-en/chapter_graph/index.md b/docs-en/chapter_graph/index.md deleted file mode 100644 index 0dd7ffb9f4..0000000000 --- a/docs-en/chapter_graph/index.md +++ /dev/null @@ -1,13 +0,0 @@ -# Graph - -

      - -!!! abstract - - In the journey of life, we are like each node, connected by countless invisible edges. - - Every time we meet and leave each other, we leave a unique mark on this huge network graph. diff --git a/docs-en/chapter_graph/summary.md b/docs-en/chapter_graph/summary.md deleted file mode 100644 index 035b3c9133..0000000000 --- a/docs-en/chapter_graph/summary.md +++ /dev/null @@ -1,30 +0,0 @@ -# Summary - -### Highlights - -- A graph consists of vertices and edges and can be represented as a set consisting of a set of vertices and a set of edges. -- Compared to linear relationships (linked lists) and divide and conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex. -- The edges of a directed graph are directional, any vertex in the connected graph is reachable, and each edge of a weighted graph contains weight variables. -- An adjacency matrix uses a matrix to represent a graph, where each row (column) represents a vertex and the matrix elements represent edges, using $1$ or $0$ to indicate that there is an edge or no edge between two vertices. Adjacency matrix is very efficient in addition, deletion, and checking operations, but it takes up more space. -- An adjacency table uses multiple linked lists to represent a graph, with the $i$th linked list corresponding to vertex $i$, which stores all the neighboring vertices of that vertex. Adjacency lists are more space-efficient compared to adjacency matrices, but are less time-efficient due to the need to traverse the linked lists to find edges. -- When the linked list in the adjacency list is too long, it can be converted to a red-black tree or hash table to improve query efficiency. -- From the point of view of algorithmic thinking, the adjacency matrix embodies "space for time", and the adjacency list embodies "time for space". -- Graphs can be used to model all kinds of real-world systems, such as social networks and subway lines. -- Trees are a special case of graphs, and tree traversal is a special case of graph traversal. -- Breadth-first traversal of a graph is a search that expands from near to far, level by level, usually implemented with the help of a queue. -- Depth-first traversal of a graph is a search that prioritizes going to the bottom and backtracking when there is no way out, often based on recursion. - -### Q & A - -!!! question "Is a path defined as a sequence of vertices or a sequence of edges?" - - Wikipedia has inconsistent definitions in different languages: the English version says "a path is a sequence of edges", while the Chinese version says "a path is a sequence of vertices". Here is the original English version: In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. - In this paper, a path is considered to be a sequence of edges rather than a sequence of vertices. This is because there may be multiple edges connected between two vertices, when each edge corresponds to a path. - -!!! Question "Are there points in a disconnected graph that cannot be traversed?" - - In a disconnected graph, at least one vertex is unreachable from some vertex. Traversing a disconnected graph requires setting up multiple starting points to traverse to all connected components of the graph. - -!!! question "In the adjacency list, is there a requirement for the vertex order of "all vertices connected to this vertex"?" - - It can be any order. 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a/docs-en/chapter_greedy/fractional_knapsack_problem.md +++ /dev/null @@ -1,50 +0,0 @@ -# Fractional Knapsack Problem - -!!! question - - Given $n$ items, the weight of the $i$th item is $wgt[i-1]$, the value is $val[i-1]$ , and a backpack with capacity $cap$. Each item can be selected only once, **but a portion of the item can be selected, and the value is calculated in proportion to the weight of the selection**. Please return the maximum value of the items in the backpack without exceeding the capacity of the backpack. - -![Example data for the fractional knapsack problem](fractional_knapsack_problem.assets/fractional_knapsack_example.png) - -Fractional backpacks are very similar to 0-1 backpacks in general, in that the state contains the current item $i$ and the capacity $c$ , and the goal is to find the maximum value without exceeding the capacity of the backpack. - -The difference is that this question allows to select only a part of the item. As shown in the figure below, **we can slice the item arbitrarily and calculate the value of the item in proportion to its weight**. - -1. For an item $i$ , its value per unit weight is $val[i-1] / wgt[i-1]$ , abbreviated as unit value. -2. Assuming that a portion of the item is put in $i$ and the weight is $w$, the value added to the backpack is $w \times val[i-1] / wgt[i-1]$ . - -![Value of item per unit weight](fractional_knapsack_problem.assets/fractional_knapsack_unit_value.png) - -### Greedy Strategy Determination - -Maximizing the total value of the items in the backpack, **essentially means maximizing the value of the items per unit weight**. This leads to the greedy strategy shown in the figure below. - -1. Sort items from highest to lowest unit value. -2. Iterate over all items, **greedily choosing the item with the highest unit value each round**. -3. If the remaining backpack capacity is insufficient, simply use a portion of the current item to fill the backpack. - -![Greedy strategy for fractional knapsacks](fractional_knapsack_problem.assets/fractional_knapsack_greedy_strategy.png) - -### Code Implementation - -We create an item class `Item` in order to sort items by unit value. The loop makes a greedy selection, jumps out when the backpack is full and returns the solution. - -```src -[file]{fractional_knapsack}-[class]{}-[func]{fractional_knapsack} -``` - -In the worst case, the entire list of items needs to be traversed, **hence the time complexity is $O(n)$** , where $n$ is the number of items. - -Since a list of `Item` objects is initialized, **the space complexity is $O(n)$**. - -### Proof Of Correctness - -The counterfactual method is used. Assume that item $x$ is the item with the highest unit value, and use an algorithm to find the maximum value as `res` , but the solution does not include item $x$ . - -Now take any item of unit weight from the backpack and replace it with an item of unit weight $x$ . Since item $x$ has the highest unit value, the total value after replacement must be greater than `res` . **This contradicts the fact that `res` is the optimal solution and shows that the optimal solution must include the item $x$** . - -For the other items in this solution, we can also construct the above contradiction. In sum, **the item with a greater unit value is always the better choice**, which shows that the greedy strategy is effective. - -As shown in the figure below, the fractional knapsack problem can be transformed into "find the maximum enclosing area for a finite interval of the horizontal axis" by considering the weight of the item and the unit value of the item as the horizontal and vertical axes, respectively, of a 2D graph. 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      oW=K~x{d}p4OyC@>wrr&!0Yvul)LJjQi@?g zso@qIO&R5`R)jW5BqvT^I^VN@-6a~|+#%6L1&1_MUHJ#z6=;fKlL3*796z`Vap=SZ zcM1b`YI$`;B-W}=OnGg}?_mNktCpEpto+)hH32xLTCG>z4S zOo^FE#LXo;5+j*(xk(>iMp-&@5^ZWG?J@YC)r?PzaoBpla>rX1D~3fC`XHH@PA3 z>Qu2Ct93D7M3D?vxJ~Aw1b{e2yMV@ojqR5kumT{?7h&MPVgb04k^$@qW9_DyhatLN zLeNPWG8lkiM-wtZiufYzTq80V2E&Ypj@=9DkcGsGVJyGcNsxdt?K_C47&ajbS=2g) z$OB-u>8u!>%z`4j=F){(GHr7N60oA>M+|5KyIp=#t{LeI>J;c!Rw^Kx{&xzz)@AL@ zm5lAAS9(Ah85!_8@AN0o=d``^NjK%C^N}1%20;Mnz9vSiPXOX_Ci=>Y?+uSS?#XLo z0N!V&mVpFgF(-xm{@#_;+7YisFA!1@6F{6ltEtCddJ(5pgHlqhpFEe@#&B#X;0w2t zJ$9L5?0)~U<)5^Tpozyo89M^F<(9v;MX_IHVOjWoc*X$H#h42L+9cFBlT$FkUe##^ zZJ!K}+$+BFIEQtm0st>5cVDWXZQn_uOuy^SmpV!M(xfO=6as-zp}v@?mU1XAUDRNz zR$I86FG=qrcoWVIU^wCqUj`j^?AqQL#a=f%@mSUaC(^OEP4OQ(!Sw@sy|csa`K$HM z8lwqi#2;u7vyv$fco^Sg2Y{?JHK9C8VeQjj#gZMTcOI52vXJ_u_2-N;*qL5WU;uD> z^(pU5Qi_Qe59p3Y%ZW(oT|7Ur`8wcq;Q*ag)b3o~`qZtH2V*S7f4=C#tnJW?93q=F zQ&OsC`1ef(`FRAkfOh2Q zZ0gawN!q0+x6CNj9~b}tg7JssgyO9qyT`6C#-F^5zFhWx^kv0L0~|nQdK>pGR*@vx z3Ej{TW4qNO6x~SzY*KO3TohiWUb8&??!Yiqc#B+^7QkZVnDmNoC9XXJfb7R$+)Z;q UVdv)Z^X*$rSyQP*!8G{40X(#&X#fBK diff --git a/docs-en/chapter_greedy/max_product_cutting_problem.md b/docs-en/chapter_greedy/max_product_cutting_problem.md deleted file mode 100644 index 5fb9a632e8..0000000000 --- a/docs-en/chapter_greedy/max_product_cutting_problem.md +++ /dev/null @@ -1,85 +0,0 @@ -# Maximum Cut Product Problem - -!!! question - - Given a positive integer $n$ , cut it into the sum of at least two positive integers and find the maximum product of all the integers after the cut. - -![Problem definition for the maximum cut product](max_product_cutting_problem.assets/max_product_cutting_definition.png) - -Suppose we slice $n$ into $m$ integer factors, where the $i$th factor is denoted as $n_i$ , i.e. - -$$ -n = \sum_{i=1}^{m}n_i -$$ - -The goal of this problem is to find the maximum product of all integer factors of - -$$ -\max(\prod_{i=1}^{m}n_i) -$$ - -What we need to think about is: how big should the number of cuts $m$ be, and how much should each $n_i$ be? - -### Greedy Strategy Determination - -As a rule of thumb, the product of two integers is often larger than their sum. Suppose a factor $2$ is divided from $n$ , then their product is $2(n-2)$ . We compare this product with $n$: - -$$ -\begin{aligned} -2(n-2) & \geq n \newline -2n - n - 4 & \geq 0 \newline -n & \geq 4 -\end{aligned} -$$ - -As shown in the figure below, when $n \geq 4$, the product will be larger after cutting out a $2$, **This means that any integer greater than or equal to $4$ should be cut**. - -**Greedy Strategy I**: If the slicing scheme contains a factor of $\geq 4$, then it should be continued to be sliced. Only the factors $1$, $2$, and $3$ should appear in the final slicing scheme. - -![Cutting leads to larger product](max_product_cutting_problem.assets/max_product_cutting_greedy_infer1.png) - -Next think about which factor is optimal. Among the three factors $1$, $2$, and $3$, it is clear that $1$ is the worst because $1 \times (n-1) < n$ holds, i.e., slicing out $1$ will instead result in a smaller product. - -As shown in the figure below, when $n = 6$, there are $3 \times 3 > 2 \times 2 \times 2$ . **This means that slicing out $3$ is better than slicing out $2$**. - -**Greedy Strategy II**: In the cut scheme, at most two $2$ should exist. This is because three $2$ can always be replaced by two $3$ , resulting in a larger product. - -![Optimal Cutting Factor](max_product_cutting_problem.assets/max_product_cutting_greedy_infer2.png) - -Summarizing the above, the following greedy strategy can be introduced. - -1. Input an integer $n$ , from which the factors $3$ are continuously sliced until the remainders are $0$, $1$, $2$. -2. When the remainder is $0$, it means that $n$ is a multiple of $3$, so nothing is done. -3. When the remainder is $2$, do not continue the division and keep it. -4. When the remainder is $1$, the last $3$ should be replaced by $2$ since $2 \times 2 > 1 \times 3$. - -### Code Implementation - -As shown in the figure below, instead of slicing the integers by loops, we can use downward division to get the number of $3$ $a$ and modulo operation to get the remainder $b$, which we have: - -$$ -n = 3 a + b -$$ - -Note that for the boundary case of $n \leq 3$, a $1$ must be split and the product is $1 \times (n - 1)$ . - -```src -[file]{max_product_cutting}-[class]{}-[func]{max_product_cutting} -``` - -![Calculation of Maximum Cutting Product](max_product_cutting_problem.assets/max_product_cutting_greedy_calculation.png) - -**The time complexity depends on the programming language's implementation of the power operation**. In Python, for example, there are three commonly used power calculation functions. - -- Both the operator `**` and the function `pow()` have a time complexity of $O(\log a)$. -- The function `math.pow()` internally calls the C library's `pow()` function, which performs floating-point power-taking with a time complexity of $O(1)$. - -The variables $a$ and $b$ use extra space of constant size, **so the space complexity is $O(1)$** . - -### Proof Of Correctness - -Using the counterfactual, analyze only the case of $n \geq 3$. - -1. **All factors $\leq 3$** : Assuming that there is a factor $\geq 4$ of $x$ in the optimal cutoff scheme, it must be possible to continue dividing it into $2(x-2)$ and thus obtain a larger product. This contradicts the assumption. -2. The **-slicing scheme does not contain $1$** : Assuming that a factor $1$ exists in the optimal slicing scheme, it must be possible to merge it into another factor to obtain a larger product. This contradicts the assumption. -3. **The cut scheme contains at most two $2$** : Assuming that the optimal cut scheme contains three $2$, then it must be possible to replace it with two $3$ with a larger product. This contradicts the assumption. diff --git a/docs-en/chapter_greedy/summary.md b/docs-en/chapter_greedy/summary.md deleted file mode 100644 index 94f4fb5759..0000000000 --- a/docs-en/chapter_greedy/summary.md +++ /dev/null @@ -1,12 +0,0 @@ -# Summary - -- Greedy algorithms are commonly used to solve optimization problems based on the principle of making locally optimal decisions at each decision stage in the expectation of obtaining a globally optimal solution. -- The greedy algorithm iteratively makes one greedy choice after another, transforming the problem into a smaller subproblem each round until the problem is solved. -- The greedy algorithm is not only simple to implement, but also highly efficient in solving problems. Compared to dynamic programming, greedy algorithms usually have lower time complexity. -- In the change exchange problem, the greedy algorithm is guaranteed to find an optimal solution for some coin combinations; for others it is not, and the greedy algorithm may find a very poor solution. -- Problems suitable for solution with greedy algorithms have two main properties: the greedy choice property and the optimal substructure. The greedy selection property represents the effectiveness of the greedy strategy. -- For some complex problems, proof of the nature of greedy choice is not simple. It is relatively easier to falsify, for example, the change exchange problem. -- Solving greedy problems is divided into three main steps: problem analysis, greedy strategy determination, and correctness proof. Among them, greedy strategy determination is the core step, and correctness proof is often the hard part. -- The fractional knapsack problem is based on the 0-1 knapsack and allows to select a fraction of the items, so it can be solved using a greedy algorithm. The correctness of the greedy strategy can be proved using the inverse method. -- The maximum capacity problem can be solved using the exhaustive enumeration method with a time complexity of $O(n^2)$ . The time complexity can be optimized to $O(n)$ by designing a greedy strategy that moves the short board inward each round. -- In the Maximum Tangent Product Problem, we successively reason about two greedy strategies: $\geq 4$ integers should all continue to be tangent, and the optimal tangent factor is $3$ . 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zZO|!0vM>|d7f$!yz~dMOaziqqQUju3G?@k@4@3^Mr}U6Ti2`fopqPj*T(B@55*xxw zjWrd+C^ko^o_j7tigFS|T8>ca5w=GmRNN4xK!V5dGZnn5e;>`H13C9E%`w@A_yR$I z6VfQBr}Xr2b8X2Ph)gi#E#%T;pmUw75(Gk=AJ}p!e5(AzfAi+8AUKxMgyiM+Fqz?_ z0WPL*(_M%DXgdDQsKwwD7@+&qXY}#Gavfu9Z zOB=E%2+u7Z#gmXll_9)8M3yM=_pahFydS8gdwrnP!$Yy=Zw0S*6`p}Q{b|LhKm Table   Common hash algorithms

      - -| | MD5 | SHA-1 | SHA-2 | SHA-3 | -| -------------- | -------------------------------------------------- | -------------------------- | --------------------------------------------------------------- | -------------------------------------- | -| Launched | 1992 | 1995 | 2002 | 2008 | -| output length | 128 bits | 160 bits | 256 / 512 bits | 224/256/384/512 bits | -| hash collision | more | more | less | less | less | less | -| Security Level | Low, Successfully Attacked | Low, Successfully Attacked | High | High | -| Applications | Deprecated, still used for data integrity checking | Deprecated | Cryptocurrency transaction validation, digital signatures, etc. | Can be used as a replacement for SHA-2 | - -## Hash Values For Data Structures - -As we know, the `key` of a hash table can be a data type such as integer, decimal, or string. Programming languages usually provide built-in hash algorithms for these data types, which are used to compute bucket indexes in hash tables. In Python, for example, we can call the `hash()` function to compute hash values for various data types. - -- The hash values of integers and boolean quantities are themselves. -- The calculation of hash values for floating point numbers and strings is more complex, so if you are interested, please learn on your own. -- A hash of a tuple is hashed for each of its elements, and then those hashes are combined to get a single hash. -- An object's hash is generated based on its memory address. Generating hashes based on content can be achieved by overriding the object's hash method. - -!!! tip - - Note that the definitions and methods of the built-in hash function for calculating hash values differ from one programming language to another. - -=== "Python" - - ```python title="built_in_hash.py" - num = 3 - hash_num = hash(num) - # The hash value of integer 3 is 3 - - bol = True - hash_bol = hash(bol) - # The hash value of the boolean True is 1 - - dec = 3.14159 - hash_dec = hash(dec) - # The hash value of the decimal 3.14159 is 326484311674566659 - - str = "Hello Algo" - hash_str = hash(str) - # String Hello The algorithm's hash value is 4617003410720528961 - - tup = (12836, "small ha") - hash_tup = hash(tup) - # The hash value of the tuple (12836, 'smallha') is 1029005403108185979 - - obj = ListNode(0) - hash_obj = hash(obj) - # Node object hash value 274267521 - ``` - -=== "C++" - - ```cpp title="built_in_hash.cpp" - int num = 3; - size_t hashNum = hash()(num); - // The integer 3 hash is 3 - - bool bol = true; - size_t hashBol = hash()(bol); - // Boolean 1 has a hash value of 1 - - double dec = 3.14159; - size_t hashDec = hash()(dec); - // The hash value of the decimal 3.14159 is 4614256650576692846 - - string str = "Hello Algo"; - size_t hashStr = hash()(str); - // The hash value of the Hello Algo is 15466937326284535026. - - // In C++, the built-in std:hash() provides hash calculations for basic data types only. - // Hash calculations for arrays and objects are self-implemented. - ``` - -=== "Java" - - ```java title="built_in_hash.java" - int num = 3; - int hashNum = Integer.hashCode(num); - // The integer 3 hash is 3 - - boolean bol = true; - int hashBol = Boolean.hashCode(bol); - // The hash value of the boolean true is 1231 - - double dec = 3.14159; - int hashDec = Double.hashCode(dec); - // The hash value for the decimal 3.14159 is -1340954729 - - String str = "Hello Algo"; - int hashStr = str.hashCode(); - // The string Hello Algo has a hash value of -727081396. - - Object[] arr = { 12836, "littleha" }; - int hashTup = Arrays.hashCode(arr); - // The array [12836, small hash] has a hash value of 1151158. - - ListNode obj = new ListNode(0); - int hashObj = obj.hashCode(); - // The hash value of the node object utils.ListNode@7dc5e7b4 is 2110121908 - ``` - -=== "C#" - - ```csharp title="built_in_hash.cs" - int num = 3; - int hashNum = num.GetHashCode(); - // 整数 3 的哈希值为 3; - - bool bol = true; - int hashBol = bol.GetHashCode(); - // the hash value of the boolean true is 1. - - double dec = 3.14159; - int hashDec = dec.GetHashCode(); - // The hash value of the decimal 3.14159 is -1340954729. - - string str = "Hello Algo"; - int hashStr = str.GetHashCode(); - // The string Hello Algo has a hash value of -586107568. - - object[] arr = { 12836, "littleha" }; - int hashTup = arr.GetHashCode(); - // The array [12836, small hash] has a hash value of 42931033. - - ListNode obj = new(0); - int hashObj = obj.GetHashCode(); - // The hash value of node object 0 is 39053774; // The hash value of node object 0 is 39053774. - ``` - -=== "Go" - - ```go title="built_in_hash.go" - // Go does not provide a built-in hash code function - ``` - -=== "Swift" - - ```swift title="built_in_hash.swift" - let num = 3 - let hashNum = num.hashValue - // integer 3 hash value 9047044699613009734 - - let bol = true - let hashBol = bol.hashValue - // The hash value of the boolean true is -4431640247352757451 - - let dec = 3.14159 - let hashDec = dec.hashValue - // The hash value of the decimal 3.14159 is -2465384235396674631 - - let str = "Hello Algo" - let hashStr = str.hashValue - // The algorithm for the string Hello has a hash value of -7850626797806988787. - - let arr = [AnyHashable(12836), AnyHashable("small ha")] - let hashTup = arr.hashValue - // The array [AnyHashable(12836), AnyHashable("small hash")] has a hash of -2308633508154532996 - - let obj = ListNode(x: 0) - let hashObj = obj.hashValue - // The hash of the node object utils.ListNode is -2434780518035996159 - ``` - -=== "JS" - - ```javascript title="built_in_hash.js" - // JavaScript does not provide a built-in hash code function - ``` - -=== "TS" - - ```typescript title="built_in_hash.ts" - // TypeScript does not provide a built-in hash code function. - ``` - -=== "Dart" - - ```dart title="built_in_hash.dart" - int num = 3; - int hashNum = num.hashCode; - // The hash of integer 3 is 34803. - - bool bol = true; - int hashBol = bol.hashCode; - // The hash value of the boolean true is 1231 - - double dec = 3.14159; - int hashDec = dec.hashCode; - // The hash value of the decimal 3.14159 is 2570631074981783 - - String str = "Hello Algo"; - int hashStr = str.hashCode; - // The algorithm for the string Hello has a hash value of 468167534. - - List arr = [12836, "Little Hah"];; - int hashArr = arr.hashCode; - // The array [12836, small hash] has a hash value of 976512528. - - ListNode obj = new ListNode(0); - int hashObj = obj.hashCode; - // The hash value of the node object Instance of 'ListNode' is 1033450432. - ``` - -=== "Rust" - - ```rust title="built_in_hash.rs" - use std::collections::hash_map::DefaultHasher; - use std::hash::{Hash, Hasher}; - - let num = 3; - let mut num_hasher = DefaultHasher::new(); - num.hash(&mut num_hasher); - let hash_num = num_hasher.finish(); - // integer 3 hash 568126464209439262 - - let bol = true; - let mut bol_hasher = DefaultHasher::new(); - bol.hash(&mut bol_hasher); - let hash_bol = bol_hasher.finish(); - // The hash value of the boolean true is 4952851536318644461 - - let dec: f32 = 3.14159; - let mut dec_hasher = DefaultHasher::new(); - dec.to_bits().hash(&mut dec_hasher); - let hash_dec = dec_hasher.finish(); - println!("The hash of decimal {} is {}", dec, hash_dec); - // The hash value of the decimal 3.14159 is 2566941990314602357 - - let str = "Hello Algo"; - let mut str_hasher = DefaultHasher::new(); - str.hash(&mut str_hasher); - let hash_str = str_hasher.finish(); - // The hash value of the Hello Algo is 16092673739211250988. - - let arr = (&12836, &"small-ha"); - let mut tup_hasher = DefaultHasher::new(); - arr.hash(&mut tup_hasher); - let hash_tup = tup_hasher.finish(); - // The hash value of the tuple (12836, "smallha") is 1885128010422702749 - - let node = ListNode::new(42); - let mut hasher = DefaultHasher::new(); - node.borrow().val.hash(&mut hasher); - let hash = hasher.finish(); - // The node object RefCell { value: ListNode { val: 42, next: None } } has a hash value of 15387811073369036852 - ``` - -=== "C" - - ```c title="built_in_hash.c" - // C does not provide a built-in hash code function - ``` - -=== "Zig" - - ```zig title="built_in_hash.zig" - - ``` - -In many programming languages, **only immutable objects can be used as the `key`** of a hash table. If we use a list (dynamic array) as a `key`, when the contents of the list change, its hash value changes as well, and we can't look up the original `value` in the hash table. - -Although the member variables of a custom object (such as a linked list node) are mutable, it is hashable. **This is because the hash value of an object is usually generated based on the memory address**. Even if the content of the object changes, its memory address remains the same and the hash value remains the same. - -If you're careful, you may notice that the output hash value is different when you run the program in different consoles. **This is because the Python interpreter adds a random salt value to the string hash function at each startup**. 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Specifically, if we input a key `key` into a hash table, we can get the corresponding value `value` within $O(1)$ time. - -As shown in the figure below, given $n$ students, each student has two pieces of data, "name" and "student number". If we want to realize the query function of "input a student number, return the corresponding name", we can use the hash table shown in the figure below to realize it. - -![Abstract representation of hash table](hash_map.assets/hash_table_lookup.png) - -In addition to hash tables, arrays and linked lists can also realize the query function, and their efficiency comparison is shown in the table below. - -- **Add element**: simply add the element to the end of the array (linked list), using $O(1)$ time. -- **Query element**: since arrays (linked lists) are disordered, you need to iterate through all the elements in them, using $O(n)$ time. -- **Delete Element**: you need to query for the element and then delete it from the array (linked list), using $O(n)$ time. - -

      Table   Element Query Efficiency Comparison

      - -| | arrays | linked lists | hash tables | -| -------------- | ------ | -------------------- | ----------- | -| Find Elements | $O(n)$ | $O(n)$ | $O(1)$ | -| Add Element | $O(1)$ | $O(1)$ $O(1)$ $O(1)$ | -| Delete Element | $O(n)$ | $O(n)$ | $O(1)$ | - -Observe that the time complexity of all additions, deletions, checks, **and modifications in a hash table by is $O(1)$** , which is very efficient. - -## Common Operations On Hash Tables - -Common operations on hash tables include: initialization, query operations, adding key-value pairs and deleting key-value pairs. - -=== "Python" - - ```python title="hash_map.py" - # Initialize the hash table - hmap: dict = {} - - # Add operations - # Add key-value pairs (key, value) to the hash table - hmap[12836] = "little ha" - hmap[15937] = "Xiaoluo" - hmap[16750] = "small count" - hmap[13276] = "Little Law" - hmap[10583] = "Duckling" - - # Query operations - # Enter the key into the hash table to get the value value - name: str = hmap[15937] - - # Delete operations - # Remove key-value pairs from hash table (key, value) - hmap.pop(10583) - ``` - -=== "C++" - - ```cpp title="hash_map.cpp" - /* Initialize hash table */ - unordered_map map; - - /* Add operation */ - // Add the key-value pair (key, value) to the hash table. - map[12836] = "Little Hah"; - map[15937] = "Xiaoluo". - map[16750] = "Small count"; - map[13276] = "Minor Law". - map[10583] = "Duckling"; - - /* Query operations */ - // Enter the key into the hash table, and get the value - string name = map[15937]; - - /* Delete operation */ - // Remove the key-value pair (key, value) from the hash table. - map.erase(10583); - ``` - -=== "Java" - - ```java title="hash_map.java" - /* Initialize hash table */ - Map map = new HashMap<>(); - - /* Add operation */ - // Add the key-value pair (key, value) to the hash table. - map.put(12836, "smallha"); - map.put(15937, "Xiaoluo");; - map.put(16750, "Small count"); - map.put(13276, "Little Law"); - map.put(10583, "Duckling"). - - /* Query operations */ - // Enter the key into the hash table, and get the value - String name = map.get(15937); - - /* Delete operation */ - // Remove the key-value pair (key, value) from the hash table. - map.remove(10583); - ``` - -=== "C#" - - ```csharp title="hash_map.cs" - /* Initialize hash table */ - Dictionary map = new() { - /* Add operation */ - // Add the key-value pair (key, value) to the hash table. - { 12836, "little ha" }. - { 15937, "Koro" }. - { 16750, "small calculations" }. - { 13276, "Minor law" }. - { 10583, "Duckling" } - }; - - /* Query operations */ - // Enter the key into the hash table, and get the value - string name = map[15937]; - - /* Delete operation */ - // Remove the key-value pair (key, value) from the hash table. - map.Remove(10583); - ``` - -=== "Go" - - ```go title="hash_map_test.go" - /* Initialize hash table */ - hmap := make(map[int]string) - - /* Add operation */ - // Add the key-value pair (key, value) to the hash table. - hmap[12836] = "little ha" - hmap[15937] = "Xiaoluo" - hmap[16750] = "small count" - hmap[13276] = "Little Law" - hmap[10583] = "Duckling" - - /* Query operations */ - // Enter the key into the hash table, and get the value - name := hmap[15937] - - /* Delete operation */ - // Remove the key-value pair (key, value) from the hash table. - delete(hmap, 10583) - ``` - -=== "Swift" - - ```swift title="hash_map.swift" - /* Initialize hash table */ - var map: [Int: String] = [:] - - /* Add operation */ - // Add the key-value pair (key, value) to the hash table. - map[12836] = "Xiaoha" - map[15937] = "Xiaoluo" - map[16750] = "small count" - map[13276] = "Little Law" - map[10583] = "Duckling" - - /* Query operations */ - // Enter the key into the hash table, and get the value - let name = map[15937]! - - /* Delete operation */ - // Remove the key-value pair (key, value) from the hash table. - map.removeValue(forKey: 10583) - ``` - -=== "JS" - - ```javascript title="hash_map.js" - /* Initialize hash table */ - const map = new Map(); - /* Add operation */ - // Add the key-value pair (key, value) to the hash table. - map.set(12836, 'small-ha'); - map.set(15937, 'Xiaoluo'); - map.set(16750, 'Small count'); - map.set(13276, 'Small Law'); - map.set(10583, 'Duckling'); - - /* Query operations */ - // Enter the key into the hash table, and get the value - let name = map.get(15937); - - /* Delete operation */ - // Remove the key-value pair (key, value) from the hash table. - map.delete(10583); - ``` - -=== "TS" - - ```typescript title="hash_map.ts" - /* Initialize hash table */ - const map = new Map(); - /* Add operation */ - // Add the key-value pair (key, value) to the hash table. - map.set(12836, 'small-ha'); - map.set(15937, 'Xiaoluo'); - map.set(16750, 'Small count'); - map.set(13276, 'Small Law'); - map.set(10583, 'Duckling'); - console.info('\nAfter the addition is complete, the hash table is \nKey -> Value'); - console.info(map); - - /* Query operations */ - // Enter the key into the hash table, and get the value - let name = map.get(15937); - console.info('\n input student number 15937, query name ' + name); - - /* Delete operation */ - // Remove the key-value pair (key, value) from the hash table. - map.delete(10583); - console.info('After \n deletes 10583, hash table is \nKey -> Value'); - console.info(map); - ``` - -=== "Dart" - - ```dart title="hash_map.dart" - /* Initialize hash table */ - Map map = {}; - - /* Add operation */ - // Add the key-value pair (key, value) to the hash table. - map[12836] = "Little Hah"; - map[15937] = "Xiaoluo". - map[16750] = "Small count"; - map[13276] = "Minor Law". - map[10583] = "Duckling"; - - /* Query operations */ - // Enter the key into the hash table, and get the value - String name = map[15937]; - - /* Delete operation */ - // Remove the key-value pair (key, value) from the hash table. - map.remove(10583); - ``` - -=== "Rust" - - ```rust title="hash_map.rs" - use std::collections::HashMap; - - /* Initialize hash table */ - let mut map: HashMap = HashMap::new(); - - /* Add operation */ - // Add the key-value pair (key, value) to the hash table. - map.insert(12836, "Xiaoha".to_string()); - map.insert(15937, "Xiaoluo".to_string());; - map.insert(16750, "small count".to_string()); - map.insert(13279, "Small Law".to_string()); - map.insert(10583, "duckling".to_string()); - - /* Query operations */ - // Enter the key into the hash table to get the value value. - let _name: Option<&String> = map.get(&15937); - - /* Delete operation */ - // Remove the key-value pair (key, value) from the hash table. - let _removed_value: Option = map.remove(&10583); - ``` - -=== "C" - - ```c title="hash_map.c" - // C does not provide a built-in hash table - ``` - -=== "Zig" - - ```zig title="hash_map.zig" - - ``` - -Hash tables have three common traversals: traversing key-value pairs, traversing keys, and traversing values. - -=== "Python" - - ```python title="hash_map.py" - # Iterate over the hash table - # Iterate over key-value pairs key->value - for key, value in hmap.items(): - print(key, "->", value) - # Iterate over the key alone key - for key in hmap.keys(): - print(key) - # Individual traversal value value - for value in hmap.values(): - print(value) - ``` - -=== "C++" - - ```cpp title="hash_map.cpp" - /* Iterate over the hash table */ - // Iterate over key-value pairs key->value - for (auto kv: map) { - cout << kv.first << " -> " << kv.second << endl; - } - // Iterate over key->value using iteration. - for (auto iter = map.begin(); iter != map.end(); iter++) { - cout << iter->first << "->" << iter->second << endl; - } - ``` - -=== "Java" - - ```java title="hash_map.java" - /* Iterate over the hash table */ - // Iterate over key-value pairs key->value - for (Map.Entry kv: map.entrySet()) { - System.out.println(kv.getKey() + " -> " + kv.getValue()); - } - // Iterate over keys individually key - for (int key: map.keySet()) { - System.out.println(key); - } - // Iterate over values individually value - for (String val: map.values()) { - System.out.println(val); - } - ``` - -=== "C#" - - ```csharp title="hash_map.cs" - /* Iterate over the hash table */ - // Iterate over key-value pairs Key->Value - foreach (var kv in map) { - Console.WriteLine(kv.Key + " -> " + kv.Value); - } - // Iterate over keys individually key - foreach (int key in map.Keys) { - Console.WriteLine(key); - } - // Iterate over values individually value - foreach (string val in map.Values) { - Console.WriteLine(val); - } - ``` - -=== "Go" - - ```go title="hash_map_test.go" - /* Iterate over the hash table */ - // Iterate over key-value pairs key->value - for key, value := range hmap { - fmt.Println(key, "->", value) - } - // Iterate over keys individually key - for key := range hmap { - fmt.Println(key) - } - // Iterate over values individually value - for _, value := range hmap { - fmt.Println(value) - } - ``` - -=== "Swift" - - ```swift title="hash_map.swift" - /* Iterate over the hash table */ - // Iterate over key-value pairs Key->Value - for (key, value) in map { - print("\(key) -> \(value)") - } - // Iterate through the keys individually Key - for key in map.keys { - print(key) - } - // Iterate through the values individually Value - for value in map.values { - print(value) - } - ``` - -=== "JS" - - ```javascript title="hash_map.js" - /* Iterate over the hash table */ - console.info('\n traversing Key->Value pair Key->Value');; - for (const [k, v] of map.entries()) { - console.info(k + ' -> ' + v); - } - console.info('\n traversing Key Key alone');; - for (const k of map.keys()) { - console.info(k); - } - console.info('\n traversing the values individually Value');; - for (const v of map.values()) { - console.info(v); - } - ``` - -=== "TS" - - ```typescript title="hash_map.ts" - /* Iterate over the hash table */ - console.info('\n traversing Key->Value pair Key->Value');; - for (const [k, v] of map.entries()) { - console.info(k + ' -> ' + v); - } - console.info('\n traversing Key Key alone');; - for (const k of map.keys()) { - console.info(k); - } - console.info('\n traversing the values individually Value');; - for (const v of map.values()) { - console.info(v); - } - ``` - -=== "Dart" - - ```dart title="hash_map.dart" - /* Iterate over the hash table */ - // Iterate over key-value pairs Key->Value - map.forEach((key, value) { - print('$key -> $value'); - }); - - // Iterate through the keys individually Key - map.keys.forEach((key) { - print(key); - }); - - // Iterate through the values individually Value - map.values.forEach((value) { - print(value); - }); - ``` - -=== "Rust" - - ```rust title="hash_map.rs" - /* Iterate over the hash table */ - // Iterate over key-value pairs Key->Value - for (key, value) in &map { - println!("{key} -> {value}"); - } - - // Iterate through the keys individually Key - for key in map.keys() { - println!("{key}"); - } - - // Iterate through the values individually Value - for value in map.values() { - println!("{value}"); - } - ``` - -=== "C" - - ```c title="hash_map.c" - // C does not provide a built-in hash table - ``` - -=== "Zig" - - ```zig title="hash_map.zig" - - ``` - -## Simple Implementation Of Hash Tables - -Let's first consider the simplest case, **implementing a hash table using only an array**. In a hash table, we call each empty space in the array a "bucket bucket", and each bucket can store a key-value pair. Therefore, the query operation is to find the bucket corresponding to `key` and get `value` in the bucket. - -So, how to locate the corresponding bucket based on `key`? This is achieved by the hash function. The function of hash function is to map a larger input space to a smaller output space. In a hash table, the input space is all the `keys` and the output space is all the buckets (array indexes). In other words, by inputting a `key` , **we can use the hash function to get the storage location of the key-value pair corresponding to that `key` in the array**. - -Input a `key` , the hash function is computed in the following two steps. - -1. The hash value is calculated by some hash algorithm `hash()`. -2. The hash value is modulo the number of buckets (array length) `capacity` to get the index of the array `index` corresponding to the `key`. - -```shell -index = hash(key) % capacity -``` - -We can then use `index` to access the corresponding bucket in the hash table to get `value` . - -Given the length of the array `capacity = 100`, the hash algorithm `hash(key) = key`, it is easy to obtain the hash function as `key % 100`. The figure below shows how the hash function works, using `key` school number and `value` name as examples. - -![How hash functions work](hash_map.assets/hash_function.png) - -The following code implements a simple hash table. In it, we encapsulate `key` and `value` into a class `Pair` to represent key-value pairs. - -```src -[file]{array_hash_map}-[class]{array_hash_map}-[func]{} -``` - -## Hash Collision And Capacity Expansion - -Essentially, the hash function maps the input space consisting of all the `keys` to the output space consisting of all the indexes of the array, and the input space is often much larger than the output space. Therefore, **theoretically there must be "multiple inputs correspond to the same output" case**. - -For the hash function in the above example, when the last two digits of the input `key` are the same, the output of the hash function is also the same. For example, when querying the two students with student numbers 12836 and 20336, we get: - -```shell -12836 % 100 = 36 -20336 % 100 = 36 -``` - -As shown in the figure below, two school numbers point to the same name, which is obviously not correct. We call this case of multiple inputs corresponding to the same output "hash collision". - -![Example of hash collision](hash_map.assets/hash_collision.png) - -It is easy to think that the larger the hash table capacity $n$ is, the lower the probability that multiple `key` are assigned to the same bucket, and the fewer hash collisions. Therefore, **we can reduce hash collision by expanding the hash table**. - -As shown in the figure below, the key-value pairs `(136, A)` and `(236, D)` were in conflict before expansion, and the conflict disappears after expansion. - -![hash table expansion](hash_map.assets/hash_table_reshash.png) - -Similar to array expansion, hash table expansion requires migrating all key-value pairs from the original hash table to the new hash table, which is very time-consuming. And because the hash table capacity `capacity` change, we need to recalculate all the key-value pairs through the hash function to store the location, which further increases the computational overhead of the expansion process. For this reason, programming languages usually reserve a large enough hash table capacity to prevent frequent expansion. - -"Load factor" is an important concept of hash table, which is defined as the number of elements divided by the number of buckets, used to measure the severity of hash collision, **is also often used as a trigger for the expansion of the hash table capacity**. For example, in Java, when the load factor exceeds $0.75$, the system will expand the hash table capacity to $2$ times its original size. diff --git a/docs-en/chapter_hashing/index.md b/docs-en/chapter_hashing/index.md deleted file mode 100644 index 7bdced46f3..0000000000 --- a/docs-en/chapter_hashing/index.md +++ /dev/null @@ -1,13 +0,0 @@ -# Hash Table - -
      - -![hash table](../assets/covers/chapter_hashing.jpg){ width="600" } - -
      - -!!! abstract - - In the computer world, hash tables are like an intelligent librarian. - - He knows how to calculate the claim number so he can find the target book quickly. diff --git a/docs-en/chapter_hashing/summary.md b/docs-en/chapter_hashing/summary.md deleted file mode 100644 index fb621c0cea..0000000000 --- a/docs-en/chapter_hashing/summary.md +++ /dev/null @@ -1,47 +0,0 @@ -# Summary - -### Highlights - -- Input `key` , hash table can look up `value` in $O(1)$ time, which is very efficient. -- Common hash table operations include querying, adding key-value pairs, deleting key-value pairs, and traversing the hash table. -- The hash function maps `key` to an array index to access the corresponding bucket and get `value`. -- Two different `key` may be after the hash function to get the same array index, resulting in query results, this phenomenon is known as hash collision. -- The larger the hash table capacity, the lower the probability of hash collision. Therefore, hash collision can be mitigated by expanding the hash table. Similar to array expansion, hash table expansion operation has a high overhead. -- The load factor is defined as the number of elements in the hash table divided by the number of buckets, reflecting the severity of hash collision, and is often used as a condition to trigger the expansion of the hash table. -- Separate chaining addresses store all conflicting elements in the same chained list by converting individual elements into a linked list. However, a linked list that is too long reduces query efficiency, which can be further improved by converting the linked list into a red-black tree. -- Open addressing handles hash collisions by probing multiple times. Linear probing uses a fixed step size, the disadvantage is that it can not delete elements, and is prone to aggregation. Multiple hashing uses multiple hash functions for probing, compared to linear probing is less likely to produce aggregation, but multiple hash functions increase the amount of computation. -- Different programming languages take different implementations of hash tables. For example, Java's `HashMap` uses separate chaining of addresses, while Python's `Dict` uses open addressing. -- In hash table, we want the hash algorithm to be deterministic, efficient and uniformly distributed. In cryptography, the hashing algorithm should also be collision resistant and avalanche effect. -- Hashing algorithms usually use large prime numbers as moduli to maximize the even distribution of hash values and reduce hash collisions. -- Common hash algorithms include MD5, SHA-1, SHA-2, and SHA3, etc. MD5 is commonly used to verify file integrity, and SHA-2 is commonly used in security applications and protocols. -- Programming languages usually provide built-in hashing algorithms for data types to compute bucket indexes in hash tables. Typically, only immutable objects are hashable. - -### Q & A - -!!! question "Why is the time complexity of hash tables not $O(n)$ ?" - - When hash collision is more severe, the time complexity of the hash table degrades to $O(n)$ . When the hash function is well designed, the capacity is set reasonably, and hash collision is more even, the time complexity is $O(1)$ . When we use a programming language's built-in hash table, we usually consider the time complexity to be $O(1)$ . - -!!! question "Why not use the hash function $f(x) = x$? That way there is no conflict" - - Under the $f(x) = x$ hash function, each element corresponds to a unique bucket index, which is equivalent to an array. However, the input space is usually much larger than the output space (the length of the array), so the final step of a hash function is often to model the length of the array. In other words, the goal of hash tables is to map a larger state space to a smaller one and provide $O(1)$ query efficiency. - -!!! Question "The underlying implementations of hash tables are arrays, linked lists, and binary trees, but why can they be more efficient than them?" - - First, hash tables become more time efficient, but less space efficient. A significant portion of the hash table's memory is unused, the - - Second, it just becomes more time efficient in specific usage scenarios. If a function can be implemented using arrays or linked lists with the same time complexity, it is usually faster than a hash table. This is because hash functions require overhead to compute and have a larger constant term for time complexity. - - Finally, the time complexity of hash tables may deteriorate. For example, in separate chaining, we take to perform a lookup operation in a linked list or a red-black tree and still run the risk of degrading to $O(n)$ time. - -!!! Question "Does multiple hashing have the flaw of not being able to delete elements directly? For marking a deleted space, can this space be used again?" - - Multiple hashes are a type of open addressing, and all open addressing methods have the drawback of not being able to delete elements directly; they need to be deleted by marking them. Marked as deleted space can be used again. When a new element is inserted into the hash table, and the location marked as deleted is found through the hash function, the location can be used by the new element. This keeps the probe sequence of the hash table unchanged and ensures the space utilization of the hash table. - -!!! question "Why does hash collision occur when looking up elements in linear probing?" - - Find the corresponding bucket and key-value pairs through the hash function, found that the `key` does not match, which means that there is a hash collision. Therefore, the linear probing method will be based on a pre-set step down the search, until the correct key-value pairs found or can not be found to jump out until. - -!!! question "Why does hash table expansion mitigate hash collisions?" - - The last step of the hash function is often the length of the array $n$ to take the remainder, so that the output value falls into the array index range; after the expansion, the length of the array $n$ changed, and the index corresponding to the `key` may also change. 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z+L#A?%{g)JRD#7NCQ_YuV0CkcYED2yrDpD!RM048N^n%HIwj}2faI%;v%!vm?`=M3 zi&U=P>2i2Q>O0$ZhG)R|c3m&Otb^)-vb2VwwFRs|M<*Ph=E*;BN^i39j^WOKhnMG34nLPirCgm>{heCj{uIL)+6cQ<6{{ Table   Efficiency of heap operations

      , greater> minHeap; - // Initialize the max heap. - priority_queue, less> maxHeap; - - /* Element into heap */ - maxHeap.push(1); - maxHeap.push(3); - maxHeap.push(2); - maxHeap.push(5); - maxHeap.push(4); - - /* Get the top element of the heap */ - int peek = maxHeap.top(); // 5 - - /* Top-of-heap element out of heap */ - // The elements out of the heap will form a sequence from largest to smallest - maxHeap.pop(); // 5 - maxHeap.pop(); // 4 - maxHeap.pop(); // 3 - maxHeap.pop(); // 2 - maxHeap.pop(); // 1 - - /* Get heap size */ - int size = maxHeap.size(); - - /* Determine if the heap is empty */ - bool isEmpty = maxHeap.empty(); - - /* Input list and build heap */ - vector input{1, 3, 2, 5, 4}; - priority_queue, greater> minHeap(input.begin(), input.end()); - ``` - -=== "Java" - - ```java title="heap.java" - /* Initialize heap */ - // Initialize the min heap. - Queue minHeap = new PriorityQueue<>(); - // Initialize the max heap (just modify the Comparator with a lambda expression). - Queue maxHeap = new PriorityQueue<>((a, b) -> b - a); - - /* Element into heap */ - maxHeap.offer(1); - maxHeap.offer(3); - maxHeap.offer(2); - maxHeap.offer(5); - maxHeap.offer(4); - - /* Get the top element of the heap */ - int peek = maxHeap.peek(); // 5 - - /* Top-of-heap element out of heap */ - // The elements out of the heap will form a sequence from largest to smallest - peek = maxHeap.poll(); // 5 - peek = maxHeap.poll(); // 4 - peek = maxHeap.poll(); // 3 - peek = maxHeap.poll(); // 2 - peek = maxHeap.poll(); // 1 - - /* Get heap size */ - int size = maxHeap.size(); - - /* Determine if the heap is empty */ - boolean isEmpty = maxHeap.isEmpty(); - - /* Input list and build heap */ - minHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4)); - ``` - -=== "C#" - - ```csharp title="heap.cs" - /* Initialize heap */ - // Initialize the min heap. - PriorityQueue minHeap = new(); - // Initialize the max heap (just modify the Comparator with a lambda expression). - PriorityQueue maxHeap = new(Comparer.Create((x, y) => y - x)); - - /* Element into heap */ - maxHeap.Enqueue(1, 1); - maxHeap.Enqueue(3, 3); - maxHeap.Enqueue(2, 2); - maxHeap.Enqueue(5, 5); - maxHeap.Enqueue(4, 4); - - /* Get the top element of the heap */ - int peek = maxHeap.Peek();//5 - - /* Top-of-heap element out of heap */ - // The elements out of the heap will form a sequence from largest to smallest - peek = maxHeap.Dequeue(); // 5 - peek = maxHeap.Dequeue(); // 4 - peek = maxHeap.Dequeue(); // 3 - peek = maxHeap.Dequeue(); // 2 - peek = maxHeap.Dequeue(); // 1 - - /* Get heap size */ - int size = maxHeap.Count; - - /* Determine if the heap is empty */ - bool isEmpty = maxHeap.Count == 0; - - /* Input list and build heap */ - minHeap = new PriorityQueue(new List<(int, int)> { (1, 1), (3, 3), (2, 2), (5, 5), (4, 4), }); - ``` - -=== "Go" - - ```go title="heap.go" - // Integer max heaps can be built in Go by implementing heap. - // Implementing heap.Interface requires also implementing sort. - type intHeap []any - - // Push heap.Interface's method to push elements into the heap. - func (h *intHeap) Push(x any) { - // Push and Pop use the pointer receiver as an argument. - // because they not only adjust the contents of the slice, but also modify the length of the slice. - *h = append(*h, x.(int)) - } - - // Pop heap.Interface's method to pop the top-of-heap element. - func (h *intHeap) Pop() any { - // The heap elements are stored at the end. - last := (*h)[len(*h)-1] - *h = (*h)[:len(*h)-1] - return last - } - - // Methods of the Len sort. - func (h *intHeap) Len() int { - return len(*h) - } - - // Methods of Less sort. - func (h *intHeap) Less(i, j int) bool { - // If implementing a min heap, it needs to be adjusted to less than sign - return (*h)[i].(int) > (*h)[j].(int) - } - - // Methods of the Swap sort. - func (h *intHeap) Swap(i, j int) { - (*h)[i], (*h)[j] = (*h)[j], (*h)[i] - } - - // Top gets the top element of the heap. - func (h *intHeap) Top() any { - return (*h)[0] - } - - /* Driver Code */ - func TestHeap(t *testing.T) { - /* Initialize heap */ - // Initialize the max heap. - maxHeap := &intHeap{} - heap.Init(maxHeap) - /* Element into heap */ - // Call the heap. - heap.Push(maxHeap, 1) - heap.Push(maxHeap, 3) - heap.Push(maxHeap, 2) - heap.Push(maxHeap, 4) - heap.Push(maxHeap, 5) - - /* Get the top element of the heap */ - top := maxHeap.Top() - fmt.Printf("The top element of the heap is %d\n", top) - - /* Top-of-heap element out of heap */ - // Call the heap. - heap.Pop(maxHeap) // 5 - heap.Pop(maxHeap) // 4 - heap.Pop(maxHeap) // 3 - heap.Pop(maxHeap) // 2 - heap.Pop(maxHeap) // 1 - - /* Get heap size */ - size := len(*maxHeap) - fmt.Printf("The number of heap elements is %d\n", size) - - /* Determine if the heap is empty */ - isEmpty := len(*maxHeap) == 0 - fmt.Printf("Is heap empty %t\n", isEmpty) - } - ``` - -=== "Swift" - - ```swift title="heap.swift" - // Swift does not provide a built-in Heap class. - ``` - -=== "JS" - - ```javascript title="heap.js" - // JavaScript does not provide a built-in Heap class. - ``` - -=== "TS" - - ```typescript title="heap.ts" - // TypeScript does not provide a built-in Heap class. - ``` - -=== "Dart" - - ```dart title="heap.dart" - // Dart does not provide a built-in Heap class - ``` - -=== "Rust" - - ```rust title="heap.rs" - use std::collections::BinaryHeap; - use std::cmp::Reverse; - - /* Initialize heap */ - // Initialize the min heap. - let mut min_heap = BinaryHeap::>::new(); - // Initialize the max heap. - let mut max_heap = BinaryHeap::new(); - - /* Element into heap */ - max_heap.push(1); - max_heap.push(3); - max_heap.push(2); - max_heap.push(5); - max_heap.push(4); - - /* Get the top element of the heap */ - let peek = max_heap.peek().unwrap(); // 5 - - /* Top-of-heap element out of heap */ - // The elements out of the heap will form a sequence from largest to smallest - let peek = max_heap.pop().unwrap(); // 5 - let peek = max_heap.pop().unwrap(); // 4 - let peek = max_heap.pop().unwrap(); // 3 - let peek = max_heap.pop().unwrap(); // 2 - let peek = max_heap.pop().unwrap(); // 1 - - /* Get heap size */ - let size = max_heap.len(); - - /* Determine if the heap is empty */ - let is_empty = max_heap.is_empty(); - - /* Input list and build heap */ - let min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]); - ``` - -=== "C" - - ```c title="heap.c" - // C does not provide a built-in Heap class - ``` - -=== "Zig" - - ```zig title="heap.zig" - - ``` - -## Heap Implementation - -The following implementation is a max heap. To convert it to a min heap, simply invert all size logic judgments (e.g., replace $\geq$ with $\leq$). Interested readers can implement it by themselves. - -### Heap Storage And Representation - -We learned in the binary tree chapter that complete binary trees are well suited to be represented by arrays. Since a heap is exactly a complete binary tree, **we will use an array to store the heap**. - -When an array is used to represent a binary tree, the elements represent the node values and the indexes represent the position of the nodes in the binary tree. **The node pointers are realized by the index mapping formula**. - -As shown in the figure below, given an index $i$, its left-child node index is $2i + 1$, its right-child node index is $2i + 2$, and its parent node index is $(i - 1) / 2$ (rounded down). When the index is out of bounds, it means empty node or node does not exist. - -![Representation and storage of heap](heap.assets/representation_of_heap.png) - -We can encapsulate the index mapping formula into a function for easy subsequent use. - -```src -[file]{my_heap}-[class]{max_heap}-[func]{parent} -``` - -### Accessing The Top Element Of The Heap - -The top element of the heap is the root node of the binary tree, which is the first element of the list. - -```src -[file]{my_heap}-[class]{max_heap}-[func]{peek} -``` - -### Element Into The Heap - -Given the element `val` , we first add it to the bottom of the heap. After the addition, the conditions under which the heap was founded may have been broken since val may be larger than any other element in the heap. Therefore, **needs to repair each node on the path from the insertion node to the root node**, an operation known as 'heapify heapify'. - -**Consider performing heapify from bottom to top starting from the entry node**. As shown in the figure below, we compare the values of the insertion node with its parent node and swap them if the insertion node is larger. This operation is then continued by repairing each node in the heap from bottom to top until it ends when the root node is crossed or when a node is encountered that does not need to be swapped. - -=== "<1>" - ![Element into heap step](heap.assets/heap_push_step1.png) - -=== "<2>" - ![heap_push_step2](heap.assets/heap_push_step2.png) - -=== "<3>" - ![heap_push_step3](heap.assets/heap_push_step3.png) - -=== "<4>" - ![heap_push_step4](heap.assets/heap_push_step4.png) - -=== "<5>" - ![heap_push_step5](heap.assets/heap_push_step5.png) - -=== "<6>" - ![heap_push_step6](heap.assets/heap_push_step6.png) - -=== "<7>" - ![heap_push_step7](heap.assets/heap_push_step7.png) - -=== "<8>" - ![heap_push_step8](heap.assets/heap_push_step8.png) - -=== "<9>" - ![heap_push_step9](heap.assets/heap_push_step9.png) - -Let the total number of nodes be $n$ , then the height of the tree node is $O(\log n)$ . It can be seen that the maximum number of rounds of the heapify operation is $O(\log n)$ , **and the time complexity of the element into heap operation is $O(\log n)$** . - -```src -[file]{my_heap}-[class]{max_heap}-[func]{sift_up} -``` - -### The Top Element Of The Heap Is Out Of The Heap - -The top element of the heap is the root node of the binary tree, i.e. the first element of the list. If we directly remove the first element from the list, the indexes of all nodes in the binary tree will change, which will make subsequent repair using heapify difficult. In order to minimize the changes in the indexes of the elements, we use the following procedure. - -1. Swap the top element of the heap with the bottom element of the heap (i.e., swap the root node with the rightmost leaf node). -2. When the swap is complete, the bottom of the heap is removed from the list (note that since it has been swapped, it is the original top element of the heap that is actually removed). -3. Starting from the root node, **performs heapify from top to bottom**. - -As shown in the figure below, **"top-to-bottom heapify" operates in the opposite direction of "bottom-to-top heapify"**we compare the value of the root node with the values of its two child nodes, and exchange the biggest We compare the value of the root node with the values of its two children, and exchange the largest child node with the root node. We then loop through this operation until we cross a leaf node or encounter a node that does not need to be exchanged. - -=== "<1>" - ![Heap top element exit step](heap.assets/heap_pop_step1.png) - -=== "<2>" - ![heap_pop_step2](heap.assets/heap_pop_step2.png) - -=== "<3>" - ![heap_pop_step3](heap.assets/heap_pop_step3.png) - -=== "<4>" - ![heap_pop_step4](heap.assets/heap_pop_step4.png) - -=== "<5>" - ![heap_pop_step5](heap.assets/heap_pop_step5.png) - -=== "<6>" - ![heap_pop_step6](heap.assets/heap_pop_step6.png) - -=== "<7>" - ![heap_pop_step7](heap.assets/heap_pop_step7.png) - -=== "<8>" - ![heap_pop_step8](heap.assets/heap_pop_step8.png) - -=== "<9>" - ![heap_pop_step9](heap.assets/heap_pop_step9.png) - -=== "<10>" - ![heap_pop_step10](heap.assets/heap_pop_step10.png) - -Similar to the element-in operation, the top-of-heap element-out operation has a time complexity of $O(\log n)$ . - -```src -[file]{my_heap}-[class]{max_heap}-[func]{sift_down} -``` - -## Common Applications Of Heaps - -- **Priority Queue**: heaps are often used as the preferred data structure to implement priority queues, with time complexity of $O(\log n)$ for both in-queue and out-queue operations and $O(n)$ for queue-building operations, which are very efficient. -- **Heap sort**: given a set of data, we can build a heap with them and then keep performing element-out operations to get ordered data. However, we usually implement heap sorting using a more elegant approach, as detailed in the subsequent chapter on heap sorting. -- **Getting the largest $k$ elements**: this is a classic algorithmic problem and also a typical application, e.g., selecting the top 10 hot news as a microblogging hot search, selecting the top 10 selling products, etc. diff --git a/docs-en/chapter_heap/index.md b/docs-en/chapter_heap/index.md deleted file mode 100644 index 94b84e5bb0..0000000000 --- a/docs-en/chapter_heap/index.md +++ /dev/null @@ -1,13 +0,0 @@ -# Heap - -
      - -![heap](../assets/covers/chapter_heap.jpg){ width="600" } - -
      - -!!! abstract - - The heaps are like the peaks of a mountain; they are layered and undulating and have different shapes. - - Every mountain has its height and the highest peaks are always the first to come into view. diff --git a/docs-en/chapter_heap/summary.md b/docs-en/chapter_heap/summary.md deleted file mode 100644 index 606af8b40d..0000000000 --- a/docs-en/chapter_heap/summary.md +++ /dev/null @@ -1,17 +0,0 @@ -# Summary - -### Highlights - -- A heap is a complete binary tree that can be categorized into a max heap and a min heap based on the establishment conditions. The top element of the heap of a big (small) top heap is the largest (smallest). -- A priority queue is defined as a queue with an outgoing priority, usually implemented using a heap. -- The common operations of heap and their corresponding time complexity include: elements into the heap $O(\log n)$, elements out of the top of the heap $O(\log n)$ and access to the top of the heap $O(1)$, and so on. -- Complete binary trees are well suited for representation in arrays, so we usually use arrays to store heaps. -- The heapify operation is used to maintain the properties of the heap and is used in both in-heap and out-heap operations. -- The time complexity of inputting $n$ elements and building a heap can be optimized to $O(n)$ , which is very efficient. -- Top-K is a classical algorithmic problem that can be solved efficiently using a heap data structure with a time complexity of $O(n \log k)$ . - -### Q & A - -!!! Question "Is the "heap" of data structure the same concept as the "heap" of memory management?" - - The two are not the same concept, they just happen to both be called heaps. The heap in a computer system's memory is part of a dynamic memory allocation that a program can use to store data at runtime. Programs can request a certain amount of heap memory for storing complex structures such as objects and arrays. When this data is no longer needed, the program needs to free this memory to prevent memory leaks. 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that the pointers $i$ and $j$ are closer to `target`**. -- When `nums[m] == target`, it means that the element smaller than `target` is in the interval $[i, m - 1]$, so $j = m - 1$ is used to narrow the interval, **so that the pointer $j$ is closer to the element smaller than `target`**. - -When the loop finishes, $i$ points to the leftmost `target` and $j$ points to the first element that is smaller than `target`, **so index $i$ is the insertion point**. - -=== "<1>" - ![Steps for binary search for insertion points of duplicate elements](binary_search_insertion.assets/binary_search_insertion_step1.png) - -=== "<2>" - ![binary_search_insertion_step2](binary_search_insertion.assets/binary_search_insertion_step2.png) - -=== "<3>" - ![binary_search_insertion_step3](binary_search_insertion.assets/binary_search_insertion_step3.png) - -=== "<4>" - ![binary_search_insertion_step4](binary_search_insertion.assets/binary_search_insertion_step4.png) - -=== "<5>" - ![binary_search_insertion_step5](binary_search_insertion.assets/binary_search_insertion_step5.png) - -=== "<6>" - ![binary_search_insertion_step6](binary_search_insertion.assets/binary_search_insertion_step6.png) - -=== "<7>" - ![binary_search_insertion_step7](binary_search_insertion.assets/binary_search_insertion_step7.png) - -=== "<8>" - ![binary_search_insertion_step8](binary_search_insertion.assets/binary_search_insertion_step8.png) - -Observe the following code and determine that the branches `nums[m] > target` and `nums[m] == target` operate the same way, so both can be merged. - -Even so, we can still keep the judgment condition unfolded because its logic is clearer and more readable. - -```src -[file]{binary_search_insertion}-[class]{}-[func]{binary_search_insertion} -``` - -!!! tip - - The code in this section is written in the "double-closed interval" style. If you are interested, you can implement the "left-closed-right-open" method by yourself. - -In a nutshell, a binary search is nothing more than setting a target for pointers $i$ and $j$, which may be a specific element (e.g., `target`) or a range of elements (e.g., elements less than `target`). - -In a continuous loop of bisection, the pointers $i$ and $j$ are both approaching a predefined goal. Eventually, they either succeed in finding the answer, or they cross the edge and stop. diff --git a/docs-en/chapter_searching/index.md b/docs-en/chapter_searching/index.md deleted file mode 100644 index b9a0aae2c8..0000000000 --- a/docs-en/chapter_searching/index.md +++ /dev/null @@ -1,13 +0,0 @@ -# Search - -

      - -!!! abstract - - Searching is an adventure into the unknown, we may need to travel to every corner of the mysterious space, or we may be able to quickly pinpoint our target. - - On this journey of discovery, every quest may yield an unanticipated answer. diff --git a/docs-en/chapter_searching/replace_linear_by_hashing.assets/two_sum_brute_force.png b/docs-en/chapter_searching/replace_linear_by_hashing.assets/two_sum_brute_force.png deleted file mode 100644 index f4e24fc41058b3f650af687652dcb9eee4ed167c..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 15890 zcmcJ$1yohv_b<9na5x|mf^-}}1f@es(LXhqjj(~It z(jC$*opjcTsKA;4J3BiU7nju3)V$u@p{2o& zjt)INJyliJgoK3s{rx{%^Rcn9AP7Ev`gD7HJ1HsY)2B~fUS67-nsRb-i;Ih2zkYRh zcOMxU>FVmr%gg)r?c3E1ckOfTQhMR|9nJ2Hj0`K^fy%*(xy@N|{oasI zbBY%It-U+;0b`@1qx;AEsU2SuTH=pRkIH{+=zY?CpRw?3U^enc#NNT#(CA!A?c~_l zn1+2{<3vMhOKMVEQt$BY_~d+J+jQ+{?aabqOhfeW?6LlZe&dvHyY 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zYDjdBT)&aMCES9qEz(+?7u7e*ArP>8I0_Z$T3FT++t^BkEQzdA(dImuS3Rq82OhK%7RuXZ&4h(qBMSYMJM+*ECfW4+_`T%%bGQ+C2F>k1KXG@cst - -| | Linear search | Binary search | Tree search | Hash search | -| --------------------------- | ------------- | ------------------ | ------------------------ | ----------------------- | -| Find Elements | $O(n)$ | $O(\log n)$ | $O(\log n)$ | $O(1)$ | -| insert element | $O(1)$ | $O(n)$ | $O(\log n)$ | $O(1)$ $O(1)$ | -| delete element | $O(n)$ | $O(n)$ | $O(\log n)$ | $O(1)$ | -| Extra Space | $O(1)$ | $O(1)$ | $O(n)$ | $O(n)$ | -| Data Preprocessing | / | Sort $O(n \log n)$ | Build Tree $O(n \log n)$ | Build Hash Table $O(n)$ | -| Whether the data is ordered | Unordered | Ordered | Ordered | Unordered | - -The choice of searching algorithm also depends on the volume of data, search performance requirements, frequency of data queries and updates, etc. - -**Linear search** - -- It is more general and does not require any data preprocessing. If we only need to query the data once, the data preprocessing of the other three methods takes longer than the linear search. -- Suitable for smaller volumes of data, in which case time complexity has less impact on efficiency. -- Suitable for scenarios where data is updated more frequently, as this method does not require any additional maintenance of the data. - -**binary search** - -- It is suitable for the case of large data volume and has a stable efficiency performance with a worst-case time complexity of $O(\log n)$ . -- The amount of data cannot be too large because storing arrays requires contiguous memory space. -- Not suitable for high-frequency data addition and deletion scenarios because of the overhead of maintaining an ordered array. - -**hash lookup** - -- It is suitable for scenarios that require high query performance with an average time complexity of $O(1)$ . -- Not suitable for scenarios that require ordered data or range lookups, as hash tables cannot maintain the orderliness of the data. -- The high dependency on hash functions and hash collision handling strategies carries a high risk of performance degradation. -- Not suitable for excessive data volumes, as hash tables require extra space to minimize conflicts and thus provide good query performance. - -**Tree Finder** - -- Suitable for massive data because tree nodes are stored decentralized in memory. -- Ideal for scenarios where you need to maintain ordered data or range lookups. -- During the continuous addition and removal of nodes, the binary search tree may become skewed and the time complexity deteriorates to $O(n)$ . -- If an AVL tree or a red-black tree is used, the operations can be stabilized with $O(\log n)$ efficiency, but the operation of maintaining the tree balance will add extra overhead. diff --git a/docs-en/chapter_searching/summary.md b/docs-en/chapter_searching/summary.md deleted file mode 100644 index af130d5f0a..0000000000 --- a/docs-en/chapter_searching/summary.md +++ /dev/null @@ -1,8 +0,0 @@ -# Summary - -- Binary search relies on the orderliness of the data and is achieved by gradually reducing the search interval by half through a loop. It requires ordered input data and only works with arrays or data structures implemented based on arrays. -- Brute-force search locates data by traversing data structures. Linear search is applicable to arrays and linked lists, breadth-first search and depth-first search are applicable to graphs and trees. These algorithms are generalizable and do not require data preprocessing, but have high time complexity $O(n)$. -- Hash lookups, tree lookups and binary searches are efficient search methods that can quickly locate a target element in a particular data structure. These algorithms are efficient, with a time complexity of $O(\log n)$ or even $O(1)$, but usually require the use of additional data structures. -- In practice, we need to specifically analyze factors such as data volume, search performance requirements, data query and update frequency, so as to choose a suitable search method. -- Linear search is suitable for small or frequently updated data; binary search is suitable for large, sorted data; hash search is suitable for data that requires high query efficiency and does not require a range query; tree search is suitable for large, dynamic data that requires maintenance of order and support for range queries. -- Replacing linear lookups with hash lookups is a commonly used strategy to optimize runtime, reducing the time complexity from $O(n)$ to $O(1)$ . diff --git a/docs-en/chapter_sorting/bubble_sort.assets/bubble_operation_step1.png b/docs-en/chapter_sorting/bubble_sort.assets/bubble_operation_step1.png deleted file mode 100644 index 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The process is like a bubble rising from the bottom to the top, hence the name bubble sort. - -As shown in the figure below, the bubbling process can be simulated by using the element swapping operation: start traversing the array from the leftmost end to the right, compare the sizes of the neighboring elements in turn, and swap them if "left element > right element". After the traversal is complete, the largest element will be moved to the right end of the array. - -=== "<1>" - ![Simulate bubbling using element swapping operations](bubble_sort.assets/bubble_operation_step1.png) - -=== "<2>" - ![bubble_operation_step2](bubble_sort.assets/bubble_operation_step2.png) - -=== "<3>" - ![bubble_operation_step3](bubble_sort.assets/bubble_operation_step3.png) - -=== "<4>" - ![bubble_operation_step4](bubble_sort.assets/bubble_operation_step4.png) - -=== "<5>" - ![bubble_operation_step5](bubble_sort.assets/bubble_operation_step5.png) - -=== "<6>" - ![bubble_operation_step6](bubble_sort.assets/bubble_operation_step6.png) - -=== "<7>" - ![bubble_operation_step7](bubble_sort.assets/bubble_operation_step7.png) - -## Algorithmic Flow - -Let the length of the array be $n$ , the steps of bubble sort are shown in the figure below. - -1. First, perform a "bubble" on $n$ elements, **swapping the largest element of the array to the correct position**. -2. Next, perform a "bubble" on the remaining $n - 1$ elements, **swapping the second largest element to the correct position**. -3. And so on, after $n - 1$ rounds of "bubbling", **the first $n - 1$ larger elements are swapped to the correct position**. -4. The only remaining element must be the smallest element and does not need to be sorted, so the array is sorted. - -![bubble sort process](bubble_sort.assets/bubble_sort_overview.png) - -```src -[file]{bubble_sort}-[class]{}-[func]{bubble_sort} -``` - -## Efficiency Optimization - -We have found that if no swap operation is performed during a round of bubbling, the array has been sorted and the result can be returned directly. Therefore, you can add a flag `flag` to monitor this situation and return the result as soon as it occurs. - -After optimization, the worst and average time complexity of bubble sort is still $O(n^2)$; however, the best time complexity $O(n)$ can be achieved when the input array is completely ordered. - -```src -[file]{bubble_sort}-[class]{}-[func]{bubble_sort_with_flag} -``` - -## Algorithm Properties - -- **Time complexity is $O(n^2)$, Adaptive sorting**: each round of "bubbling" traverses the array with lengths $n - 1$, $n - 2$, $\dots$, $2$, $1$, and the total is $(n - 1) n / 2$. After introducing the `flag` optimization, the optimal time complexity can reach $O(n)$ . -- **Space complexity of $O(1)$, In-place sorting**: pointers $i$ and $j$ use extra space of constant size. -- **Stable ordering**: since equal elements are not exchanged when encountered in "bubbling". diff --git a/docs-en/chapter_sorting/bucket_sort.assets/bucket_sort_overview.png b/docs-en/chapter_sorting/bucket_sort.assets/bucket_sort_overview.png deleted file mode 100644 index 9d18db05e0eee3ce639f63d73ff484867333a672..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 45635 zcmc$_Ra6{Z6DT?}3?AHrLvSYq2s*d~cMSx02<{9H!QCx5A-D$_+#P~zaCeu(_n-H> z?!#T{c0aV0SMA!hrNfmJq|i{_qW}N^G#P1e6#xMK?G`kF1ow8i%V#lpdw5ooQffOF9{HcDuyS~4_zrMb{xqIH(*}1s5 z*x1-OI5;>xJ-xlX<>%+`=;(NOe7S#qx_f$geSEliJD(mte*8E;KM#RGW@cu(y1Md4 za{vAN_mJzAkdV;Z+dDBa5fv3xR8(Z&de-0Hzq-1bo}S*&by`|lIy^jFv^>|g)Rs`N z`}p{HbaYfzRaMh|0$YZ~{@iVCZ5(?W!baZqy 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The time complexity of these sorting algorithms cannot exceed $O(n \log n)$. Next, we will look at some non-comparison sorting algorithms, which can have time complexity up to linear order. - -"Bucket sort" is a typical application of divide and conquer strategy. It distributes data evenly among buckets by setting up some buckets with size order, each bucket corresponds to a data range; then, it performs sorting inside each bucket; finally, it merges all the data according to the order of the buckets. - -## Algorithmic Flow - -Consider an array of length $n$ with elements of floating point numbers in the range $[0, 1)$. The flow of bucket sort is shown in the figure below. - -1. Initialize $k$ buckets and assign $n$ elements to $k$ buckets. -2. Perform a separate sort for each bucket (in this article, we use the programming language's built-in sort function). -3. Merge the results in order of the buckets from smallest to largest. - -![Bucket sorting algorithm flow](bucket_sort.assets/bucket_sort_overview.png) - -```src -[file]{bucket_sort}-[class]{}-[func]{bucket_sort} -``` - -## Algorithm Properties - -Bucket sort is suitable for processing data with large volume. For example, the input data contains 1 million elements, and due to space constraints, the system memory cannot load all the data at once. At this point, you can divide the data into 1000 buckets, then sort each bucket separately, and finally merge the results. - -- **Time complexity $O(n + k)$** : Assuming that the elements are evenly distributed across the buckets, the number of elements in each bucket is $\frac{n}{k}$ . Assuming that sorting a single bucket uses $O(\frac{n}{k} \log\frac{n}{k})$ time, then sorting all buckets uses $O(n \log\frac{n}{k})$ time. **When the number of buckets $k$ is large, the time complexity tends to $O(n)$** . Merging the results requires traversing all buckets and elements, taking $O(n + k)$ time. -- **Adaptive Sorting**: in the worst case, all data is assigned to a bucket and sorting that bucket uses $O(n^2)$ time. -- **Space complexity $O(n + k)$, non-in-place sorting**: requires additional space with the help of $k$ buckets and a total of $n$ elements. -- The stability of a bucket sort depends on the stability of the sorting algorithm that sorts the elements in the bucket. - -## How To Achieve Equal Distribution - -The time complexity of bucket sort can theoretically reach $O(n)$ , **the key is to evenly distribute the elements into each bucket**, because the actual data is often not evenly distributed. For example, we want to evenly distribute all the products on Taobao into 10 buckets by price range, but the product prices are unevenly distributed, with very many below $100 and very few above $1000. If the price range is divided into 10 equal parts, the difference in the number of items in each bucket will be very large. - -To achieve an even distribution, we can start by setting an approximate dividing line and roughly dividing the data into 3 buckets. **Afterallocation, the buckets with more commodities are then continued to be divided into 3 buckets until the number of elements in all buckets is roughly equal**. - -As shown in the figure below, this method essentially creates a recursion tree, and the goal is to make the values of the leaf nodes as average as possible. Of course, it is not necessary to divide the data into 3 buckets per round, the specific division can be flexibly selected according to the data characteristics. - -![Recursion dividing bucket](bucket_sort.assets/scatter_in_buckets_recursively.png) - -If we know the probability distribution of the commodity price in advance, **then we can set the price cutoff for each bucket based on the data probability distribution**. It is worth noting that the data distribution does not necessarily need to be intentionally statistical, but can also be approximated using some probability model based on the data characteristics. - -As shown in the figure below, we assume that the price of the commodity follows a normal distribution so that the price interval can be set reasonably well so that the commodity can be distributed evenly across the buckets. - -![Sort buckets according to probability distribution](bucket_sort.assets/scatter_in_buckets_distribution.png) diff --git a/docs-en/chapter_sorting/counting_sort.assets/counting_sort_overview.png b/docs-en/chapter_sorting/counting_sort.assets/counting_sort_overview.png deleted file mode 100644 index 6afdf0ea1667264117a544e16e9342124d439216..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 32476 zcmc$^bx<5%@Gm;Mi@Uo!L4yUi;4UFZAS`adJ=o$NG`J-pKoWuzBq1#B?!LIY`y=1` zt6R6~z5m{;dUJMac4m6|boc2`pYAggt)-!ejX{9{006eKlAI0zAV2*D52Hbz4p(2e 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zXfDQ@xAnfYJ8=+J^6tzVoduqy`zRFObhF&Y?F|1E2>_&P4&YB10*HT$2>?F|0Ym@+ z00Be*0saw-I;2xo+qPmDzHht+0Ym@+{w-Th=u{PLcrgs$H(oQD0Pb!dKiWA6Ai(4e z0UQ{-7=~{v?fr81*#Jb4dLN6co!^Aff%1aQR;$v_6Q&9sil3=Ml9kJfzRlBmwj zVR5fK-02VbY||#CB?1Tl4v|7>L3=>vRC;Du4x%SLd8p zx^k7AiqWv$b6yuTaNC^Yx4AJ|6GmyBv(Yaeo`S3>V@{p^ki|CDEq$7J0P~Ch;7a)x zeZ_Lj<1xoN9fVJ(Tp_=#VU019lgIV2c>ENmT5|q4KOI2vodDW9ab0!@Ai#ej%91Wz za`k&Z3Ma+^48I>hy355i!3GfEzp>b}l={CtfZ-1Zkh0y?ByhFb-4Ou%FU&4%T%rdI zpneJL4*@)vWw%%2?#?CvfL{-wfA$jC(*ZOI0ARvQ9YAl`0UGlg9YE7W0RdprOhpIq zW^Nn;7&dwbhyVh>w3){pU=!X4FhBqS;6HH^z(5Bu3Cp4nAQA`w6J_=REXpuB25_qb z=v&wT0>HGX{^lKX(U_wNteiN|s@ERBZrfhfx>~;Drz>h6J`0ndIwPbC4_ZQl41t7k!0(|M1zx)0rGJ1662evBU&7yN; z0)-0zFgZfz&*>I7s(*IiQ_s@{+}<>cu5cVr6O#On8*aGa_AlQgPjj52TQoOY-9L{; z3R%EeG7rO2HL!3104AkS^((`0p3G$b3OeBh!CED7D;rprO|t+1n2ay&)Vm1;F$_gf zgD^&hnT;kAN>zqIp{j6uQ|8)cqx}@H=AGNa?cLo~WA3ksJkpzsgx3OVgsL7RJpcgy z+-qkVQP=)h=t>`Qm$krRW)%QnD`r;K{-md`vePZhjiv$sunC$Pv)tr4{dBq!%fJ8t i+mM0D{7e7" - ![counting sort steps](counting_sort.assets/counting_sort_step1.png) - -=== "<2>" - ![counting_sort_step2](counting_sort.assets/counting_sort_step2.png) - -=== "<3>" - ![counting_sort_step3](counting_sort.assets/counting_sort_step3.png) - -=== "<4>" - ![counting_sort_step4](counting_sort.assets/counting_sort_step4.png) - -=== "<5>" - ![counting_sort_step5](counting_sort.assets/counting_sort_step5.png) - -=== "<6>" - ![counting_sort_step6](counting_sort.assets/counting_sort_step6.png) - -=== "<7>" - ![counting_sort_step7](counting_sort.assets/counting_sort_step7.png) - -=== "<8>" - ![counting_sort_step8](counting_sort.assets/counting_sort_step8.png) - -The implementation code for counting sort is shown below. - -```src -[file]{counting_sort}-[class]{}-[func]{counting_sort} -``` - -## Algorithm Properties - -- **Time complexity $O(n + m)$** : Involves traversing `nums` and traversing `counter` , both using linear time. In general $n \gg m$ , the time complexity tends to $O(n)$ . -- **Space complexity $O(n + m)$, Non-in-place sorting**: with the help of arrays `res` and `counter` of lengths $n$ and $m$ respectively. -- **Stable Sort**: Since the order of filling `res` with elements is "right-to-left", traversing `nums` in reverse order avoids changing the relative positions of the equal elements, and thus achieves a stable sort. In fact, an orthogonal traversal of `nums` will also give the correct sorting result, but the result will be non-stable. - -## Limitations - -You might think that counting sorts are very clever, and that you can achieve an efficient sorting job just by counting quantities. However, the preconditions for using counting sort are relatively strict. - -**The counting sort only works with non-negative integers**. If you want to use it for other types of data, you need to make sure that they can be converted to non-negative integers and that the relative size relationships between the elements cannot be changed during the conversion. For example, for an array of integers containing negative numbers, you can just add a constant to all the numbers, convert all the numbers to positive numbers, and then convert them back when the sorting is complete. - -**Counting sort is suitable for cases where the data volume is large but the data range is small**. For example, in the above example $m$ cannot be too large or it will take up too much space. 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zj{m&wy8`?F6i#wY?3lD`*%AqWlA~hxQKAgNXWEHBM5uVN%n`#rz~%9>k8$-+ScBsLdQcSUf_wbA!xUMDs^|c$2xqg^g4|@IGC{bZcLtO5WE( zQ4k_<#=Hx>dh7ky6;oQKowP|X?b)zgEn0**n&aRBgUsa5SDhaBX>M&cZ}j!usdDKq z(y@V1@XRK^*y~7>=}>=Xt_2^UMY_-T^Vr24jZud16yn0UhiR+&k~b1Zuj{$}@^fmk z!t(Yl{ki-~^Vgv8et%&|Kgv&N4zC1&r#aW32Gi9yi!R<0+Urswt-^RNbH}CjEEcN| znweyGw`P%4s=gJEFbgz_I*>~!s%s3Ps1~oYU&zoIpC3@jeYy2B_4EMHPzOYHGo}Ti z?|)lfHEgkdDdJyt7V4lQG16=oGBM>((UPxlAfAbut0Wz0oU>{Cr zL(@CbpX2-9Hqva*?$mQYz+uFxfBG$JotH2G7=J-ky#~4*AUhN-P?1qi)a?LqWI|(_ zusQ}XZw!Fsh^shsgvG&}5Ws!I<-nd7cVA3oWJh?yRH^+ra?zX;A$rke781h@vjVUm zIRu=6BW*Hp0Q4{?m&rgRuz`dGUUPHq*NKrI1Hd}+};-!Qy#c;w}b-{I~yDK1YJUrH+8SGSXC;iXz zs|)EK7PvM+10oQph+bn(u=@S7IA51IZIgGjN21Cd;cD2d)>rd37q40=u`~$hzR}Kd zwx&GTNJ#c#Pl}2K`f5t=IH}+BfvVS{r2)`H8^n)8OF`14g||C(e)N2{pk# z>6@5MXwq2H($!QRdM!_k=Wf>jD&%SjmW9n`8^o+sRPKCwr?x>OJ@H@(H1uVORZ^)X yqx0}Je0t|^<()`wfQNg6gjV" - ![heap sort steps](heap_sort.assets/heap_sort_step1.png) - -=== "<2>" - ![heap_sort_step2](heap_sort.assets/heap_sort_step2.png) - -=== "<3>" - ![heap_sort_step3](heap_sort.assets/heap_sort_step3.png) - -=== "<4>" - ![heap_sort_step4](heap_sort.assets/heap_sort_step4.png) - -=== "<5>" - ![heap_sort_step5](heap_sort.assets/heap_sort_step5.png) - -=== "<6>" - ![heap_sort_step6](heap_sort.assets/heap_sort_step6.png) - -=== "<7>" - ![heap_sort_step7](heap_sort.assets/heap_sort_step7.png) - -=== "<8>" - ![heap_sort_step8](heap_sort.assets/heap_sort_step8.png) - -=== "<9>" - ![heap_sort_step9](heap_sort.assets/heap_sort_step9.png) - -=== "<10>" - ![heap_sort_step10](heap_sort.assets/heap_sort_step10.png) - -=== "<11>" - ![heap_sort_step11](heap_sort.assets/heap_sort_step11.png) - -=== "<12>" - ![heap_sort_step12](heap_sort.assets/heap_sort_step12.png) - -In the code implementation, we use the same top-to-bottom heap `sift_down()` function as in the heap chapter. It is worth noting that since the length of the heap decreases as the largest element is extracted, we need to add a length parameter $n$ to the `sift_down()` function to specify the current effective length of the heap. - -```src -[file]{heap_sort}-[class]{}-[func]{heap_sort} -``` - -## Algorithm Properties - -- **Time complexity $O(n \log n)$, non-adaptive sorting**: the heap building operation uses $O(n)$ time. The time complexity of extracting the largest element from the heap is $O(\log n)$ , with a total of $n - 1$ rounds. -- **Space complexity $O(1)$, in-place sorting**: several pointer variables use $O(1)$ space. Element swapping and heapify operations are performed on the original array. -- **Unstable ordering**: the relative positions of equal elements may change when exchanging the top element of the heap and the bottom element of the heap. diff --git a/docs-en/chapter_sorting/index.md b/docs-en/chapter_sorting/index.md deleted file mode 100644 index 8a28b1c49b..0000000000 --- a/docs-en/chapter_sorting/index.md +++ /dev/null @@ -1,13 +0,0 @@ -# Sort - -

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Insertion Sort - -The "insertion sort insertion sort" is a simple sorting algorithm that works very similar to the process of manually organizing a deck of cards. - -Specifically, we select a baseline element in the unsorted interval, compare the size of that element with the elements in the sorted interval to its left, one by one, and insert that element into the correct position. - -The figure below shows the process of inserting elements into an array. Let the base element be `base`, we need to move all the elements from the target index to `base` one place to the right, and then assign `base` to the target index. - -![Single insertion operation](insertion_sort.assets/insertion_operation.png) - -## Algorithmic Flow - -The overall flow of the insertion sort is shown in the figure below. - -1. In the initial state, the 1st element of the array is already sorted. -2. After selecting the 2nd element of the array as `base` , and inserting it in the correct position, **the first 2 elements of the array are sorted**. -3. After selecting the 3rd element as `base` , and inserting it in the correct position, **the first 3 elements of the array are sorted**. -4. And so on, in the last round, the last element is selected as `base` , and after inserting it in the correct position, **all elements are sorted**. - -![insertion sort process](insertion_sort.assets/insertion_sort_overview.png) - -```src -[file]{insertion_sort}-[class]{}-[func]{insertion_sort} -``` - -## Algorithm Properties - -- **Time complexity $O(n^2)$, Adaptive sorting**: in the worst case, each insertion operation needs to loop $n - 1$, $n-2$, $\dots$, $2$, $1$ respectively, and the summation yields $(n - 1) n / 2$ , so the time complexity is $O(n^2)$ . The insertion operation terminates early when encountering ordered data. When the input array is completely ordered, insertion sort reaches the optimal time complexity $O(n)$ . -- **Space complexity $O(1)$, In-place sorting**: pointers $i$ and $j$ use extra space of constant size. -- **Stable sort**: during insertion operation, we will insert the elements to the right of the equal elements without changing their order. - -## Insertion Sort Advantages - -The time complexity of insertion sort is $O(n^2)$ and the time complexity of the quick sort we are about to learn is $O(n \log n)$ . Even though the time complexity of insertion sort is higher compared to quick sort, **insertion sort is usually faster when the amount of data is small**. - -This conclusion is similar to the conclusion about the applicability of linear and binary search. Algorithms such as quick sorting algorithm $O(n \log n)$ are partition-based sorting algorithms, which tend to contain more unitary computational operations. While the values of $n^2$ and $n \log n$ are closer and complexity does not play a dominant role when the amount of data is small; the number of unit operations in each round plays a decisive factor. - -In fact, the built-in sorting functions of many programming languages (e.g., Java) use insertion sort, along the lines of a partition-based sorting algorithm, such as quick sort, for long arrays, or direct insertion sort for short arrays. - -Although the time complexity of bubble sort, selection sort and insertion sort is $O(n^2)$ , in practice, **insertion sort is used significantly more often than bubble sort and selection sort for the following main reasons**. - -- The bubble sort is implemented based on element swapping and involves 3 unit operations with the help of a temporary variable; the insertion sort is implemented based on element assignment and requires only 1 unit operation. 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a/docs-en/chapter_sorting/quick_sort.md +++ /dev/null @@ -1,98 +0,0 @@ -# Quick Sort - -"Quick sort" is a sorting algorithm based on the divide and conquer strategy, which is efficient and widely used. - -The core operation of quick sort is the "sentinel division", the goal is: selection of an element in the array as the "base number", all the elements smaller than the base number of elements moved to its left, and greater than the base number of elements moved to its right. Specifically, the process of sentinel division is shown in the figure below. - -1. Select the leftmost element of the array as the reference number, and initialize two pointers `i` and `j` to point to the two ends of the array respectively. -2. Set up a loop that uses `i` (`j`) in each round to find the first element that is larger (smaller) than the baseline number, respectively, and then swaps the two elements. -3. Loop through step `2.` until `i` and `j` meet and stop, and finally swap the datums to the dividing line of the two subarrays. - -=== "<1>" - ![Sentinel division step](quick_sort.assets/pivot_division_step1.png) - -=== "<2>" - ![pivot_division_step2](quick_sort.assets/pivot_division_step2.png) - -=== "<3>" - ![pivot_division_step3](quick_sort.assets/pivot_division_step3.png) - -=== "<4>" - ![pivot_division_step4](quick_sort.assets/pivot_division_step4.png) - -=== "<5>" - ![pivot_division_step5](quick_sort.assets/pivot_division_step5.png) - -=== "<6>" - ![pivot_division_step6](quick_sort.assets/pivot_division_step6.png) - -=== "<7>" - ![pivot_division_step7](quick_sort.assets/pivot_division_step7.png) - -=== "<8>" - ![pivot_division_step8](quick_sort.assets/pivot_division_step8.png) - -=== "<9>" - ![pivot_division_step9](quick_sort.assets/pivot_division_step9.png) - -After the sentinel division is done, the original array is divided into three parts: the left subarray, the base number, and the right subarray, and it satisfies "any element of the left subarray $\leq$ base number $\leq$ any element of the right subarray". Therefore, we only need to sort these two arrays next. - -!!! note "A divide and conquer strategy for quick sort" - - The essence of sentinel division is to simplify the problem of sorting a longer array into a problem of sorting two shorter arrays. - -```src -[file]{quick_sort}-[class]{quick_sort}-[func]{partition} -``` - -## Algorithmic Flow - -The overall process of quick sort is shown in the figure below. - -1. First, perform a "sentinel division" on the original array to get the unsorted left and right subarrays. -2. Then, the "sentinel division" is performed recursionally on the left and right subarrays. -3. The recursion continues until the length of the subarray reaches 1, thus completing the sorting of the entire array. - -![Quick sort process](quick_sort.assets/quick_sort_overview.png) - -```src -[file]{quick_sort}-[class]{quick_sort}-[func]{quick_sort} -``` - -## Algorithm Properties - -- **Time complexity $O(n \log n)$, Adaptive Sorting**: In the average case, the number of recursion levels for sentinel division is $\log n$ , the total number of loops in each level is $n$, and the overall time complexity used is $O(n \log n)$. In the worst case, each round of sentinel division operation divides the array of length $n$ into two subarrays of length $0$ and $n - 1$, at which point the number of recursion levels reaches $n$, the number of loops in each level is $n$, and the overall time used is $O(n^2)$. -- **Space complexity $O(n)$, in-place sorting**: reaches the worst recursion depth $n$ with the input array completely inverted, using $O(n)$ stack frame space. The sorting operation is performed on the original array without the help of additional arrays. -- **Unstable ordering**: in the last step of the sentinel division, the base numbers may be swapped to the right of equal elements. - -## Why Is Fast Platoon Fast? - -As you can see from the name, quick sort should have a certain advantage in terms of efficiency. Although the average time complexity of quick sort is the same as that of merge sort and heap sort, quick sort is usually more efficient for the following reasons. - -- **Low probability of worst-case**: although the worst-case time complexity of quick sort is $O(n^2)$, which is not as stable as merge sort, in the vast majority of cases, quick sort can run at a time complexity of $O(n \log n)$. -- **Efficient use of cache**: When performing a sentinel division operation, the system can load the entire subarray into the cache, thus accessing the elements more efficiently. Algorithms such as heap sort lack this feature as they require jumping to access elements. -- Low constant factor of **complexity**: Among the above three algorithms, quick sorting has the lowest total number of comparison, assignment, and swap operations. This is similar to the reason why Insertion Sort is faster than Bubble Sort. - -## Benchmark Optimization - -**Quick sort may be less time efficient with certain inputs**. As an extreme example, suppose the input array is completely inverted, and since we choose the leftmost element as the base number, then after the completion of the sentinel division, the base number is exchanged to the rightmost end of the array, resulting in a left subarray length of $n - 1$ and a right subarray length of $0$ . As this recursion continues, the length of the right subarray after each round of sentinel division is $0$, the divide and conquer strategy fails, and the quick sort degenerates into a "bubble sort". - -To try to avoid this situation, **we can optimize the selection strategy of the benchmark number in sentinel division**. For example, we can randomly pick an element as the benchmark number. However, if we are unlucky and pick an undesirable benchmark number every time, the efficiency is still unsatisfactory. - -Note that programming languages often generate "pseudo-random numbers". If we construct a specific test case for a sequence of pseudo-random numbers, the efficiency of the quick sort may still be degraded. - -For further improvement, we can select three candidate elements in the array (usually the first, last, and midpoint elements of the array), **and use the median of these three candidate elements as the base number**. In this way, the probability that the base number is "neither too small nor too large" will be greatly increased. Of course, we can also select more candidate elements to further improve the robustness of the algorithm. With this approach, the probability of time complexity degradation to $O(n^2)$ is greatly reduced. - -```src -[file]{quick_sort}-[class]{quick_sort_median}-[func]{partition} -``` - -## Tail Recursion Optimization - -**Under certain inputs, the quick sort may take up more space**. Taking a completely inverted input array as an example, since the length of the right child array after each round of sentinel division is $0$, the height of the recursion tree will reach $n - 1$, which will take up $O(n)$ size of stack frame space. - -To prevent the accumulation of stack frame space, we can compare the lengths of the two subarrays after each round of sentinel sorting is completed, **recursion only on the shorter subarray**. 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where the amount of data $n$ is large but the data range $m$ is small. Suppose we need to sort $n = 10^6$ school numbers, which is a $8$ bit number, which means the data range $m = 10^8$ is very large, and using counting sort would require allocating a lot of memory space, which can be avoided by radix sort. - -The core idea of "radix sort" is the same as counting sort, which is also realized by counting the number of digits. Based on this, radix sort utilizes the progressive relationship between each digit and sorts each digit in turn to get the final sort result. - -## Algorithmic Flow - -Taking the school number data as an example, assuming that the lowest digit of the number is the $1$th digit and the highest digit is the $8$th digit, the flow of radix sort is shown in the figure below. - -1. Initialization bits $k = 1$ . -2. Performs "counting sort" on the $k$ digit of the student number. When finished, the data is sorted from smallest to largest by the $k$ digit. -3. Increase $k$ by $1$ and return to step `2.` Continue iteration until it ends when all bits are sorted. - -![Radix sort algorithm flow](radix_sort.assets/radix_sort_overview.png) - -Let's dissect the code implementation. For a $d$ progressive number $x$, to get its first $k$ bit $x_k$, you can use the following formula: - -$$ -x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d -$$ - -where $\lfloor a \rfloor$ denotes rounding down the floating point number $a$ and $\bmod \: d$ denotes rounding up $d$. For school number data, $d = 10$ and $k \in [1, 8]$ . - -In addition, we need to make a small change to the counting sort code so that it can sort the numbers according to the $k$th bit of the number. - -```src -[file]{radix_sort}-[class]{}-[func]{radix_sort} -``` - -!!! Question "Why do you start sorting from the lowest position?" - - In consecutive sorting rounds, the later sorting round overwrites the result of the previous sorting round. For example, if the result of the first sorting round $a < b$ and the result of the second sorting round $a > b$, then the result of the second round will replace the result of the first round. Since the higher digits of a number have higher priority than the lower digits, we should sort the lower digits first before sorting the higher digits. - -## Algorithm Properties - -Compared to counting sort, radix sort is suitable for larger ranges of values, **provided that the data can be represented in a fixed bit format and the number of bits is not too large**. For example, floating point numbers are not suitable for radix sort because the number of bits $k$ is too large, which may result in a time complexity $O(nk) \gg O(n^2)$ . - -- **Time complexity $O(nk)$**: Given a data size of $n$, a data $d$ progression, and a maximum number of bits $k$, it takes $O(n + d)$ time to perform a counting sort on a particular bit, and $O((n + d)k)$ time to sort all the $k$ bits. 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step](selection_sort.assets/selection_sort_step1.png) - -=== "<2>" - ![selection_sort_step2](selection_sort.assets/selection_sort_step2.png) - -=== "<3>" - ![selection_sort_step3](selection_sort.assets/selection_sort_step3.png) - -=== "<4>" - ![selection_sort_step4](selection_sort.assets/selection_sort_step4.png) - -=== "<5>" - ![selection_sort_step5](selection_sort.assets/selection_sort_step5.png) - -=== "<6>" - ![selection_sort_step6](selection_sort.assets/selection_sort_step6.png) - -=== "<7>" - ![selection_sort_step7](selection_sort.assets/selection_sort_step7.png) - -=== "<8>" - ![selection_sort_step8](selection_sort.assets/selection_sort_step8.png) - -=== "<9>" - ![selection_sort_step9](selection_sort.assets/selection_sort_step9.png) - -=== "<10>" - ![selection_sort_step10](selection_sort.assets/selection_sort_step10.png) - -=== "<11>" - ![selection_sort_step11](selection_sort.assets/selection_sort_step11.png) - -In the code, we use $k$ to record the smallest element in the unsorted interval. - -```src -[file]{selection_sort}-[class]{}-[func]{selection_sort} -``` - -## Algorithm Properties - -- **Time complexity is $O(n^2)$, non-adaptive sorting**: the outer loop consists of a total of $n - 1$ rounds, the length of the unsorted interval in the first round is $n$ , and the length of the unsorted interval in the last round is $2$ , i.e., each round of the outer loop contains $n$, $n - 1$, $\dots$, $3$, $2$ rounds of inner loops, and the summation is $\frac{(n - 1)(n + 2)}{2}$ . -- **Space complexity $O(1)$, In-place sorting**: pointers $i$ and $j$ use extra space of constant size. -- **Unstable ordering**: as shown in the figure below, it is possible for an element `nums[i]` to be swapped to the right of an element equal to it, resulting in a change in the relative order of the two. - -![Example of selection sort non-instability](selection_sort.assets/selection_sort_instability.png) diff --git 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Non-comparative sorting does not use comparison operators, and the time complexity can be up to $O(n)$, but its generality is relatively poor. - -## Ideal Sorting Algorithm - -**runs fast, in situ, stable, positively adaptive, and generalizes well**. Obviously, no sorting algorithm that combines all of these properties has been found so far. 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Assuming that we "search from left to right" first, then if we can't find an element smaller than the base number, **we will jump out of the loop at `i == j`, and then `nums[j] == nums[i] > nums[left]`**. That is, at this point, the last step of the swap operation will swap an element larger than the baseline number to the leftmost end of the array, causing the sentinel division to fail. - - For example, given an array `[0, 0, 0, 0, 0, 1]`, if you first "find from left to right", the sentinel divides the array into `[1, 0, 0, 0, 0]`, which is incorrect. - - Thinking deeper, if we choose `nums[right]` as the base number, then it's the other way around: we have to "find from left to right" first. - -!!! question "Regarding tail recursion optimization, why is it that choosing a shorter array ensures that the recursion depth does not exceed $\log n$ ?" - - The recursion depth is the number of recursion methods that are currently not returned. Each round of sentinel division we divide the original array into two subarrays. After tail recursion optimization, the maximum length of the downward recursion subarray is half the length of the original array. Assuming the worst case, which is always half the length, the final depth of recursion is $\log n$ . - - Recalling the original quick sort, it is possible to recursion successively through arrays of large lengths, in the worst case $n$, $n - 1$, $\dots$, $2$, $1$, with a recursion depth of $n$. The tail recursion optimization avoids this. - -!!! question "Is the time complexity of quick sort $O(n^2)$ when all elements in the array are equal? How should I handle this degenerate case?" - - Yes. This situation can be considered by dividing the array into three parts by sentinel division: less than, equal to, and greater than the base number. Only the less than and greater than parts are recursed downwards. In this method, an array with all equal elements is input and only one round of sentinel division completes the sorting. - -!!! question "Why is the worst time complexity of bucket sort $O(n^2)$ ?" - - In the worst case, all elements are sorted into the same bucket. 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zM@L8F;^K00avB;MIyyR%lat@PdGqz_*Xru(;^N|rjEuInw*20_`1tt3!ot+l)UB - -| method name | description | time complexity | -| ----------- | ------------------------------------- | --------------- | -| pushFirst() | Add element to top of queue | $O(1)$ | -| pushLast() | Add element to end of queue | $O(1)$ | -| popFirst() | Delete the first element of the queue | $O(1)$ | -| popLast() | Delete end-of-queue element | $O(1)$ | -| peekFirst() | access the first element of the queue | $O(1)$ | -| peekLast() | access the end-of-queue element | $O(1)$ | - -Similarly, we can directly use the double-ended queue class already implemented in the programming language. - -=== "Python" - - ```python title="deque.py" - # Initialize the double-ended queue - deque: deque[int] = collections.deque() - - # Element in the queue - deque.append(2) # add to the end of the queue - deque.append(5) - deque.append(4) - deque.appendleft(3) # add to head of queue - deque.appendleft(1) - - # Access to elements - front: int = deque[0] # head element of the queue - rear: int = deque[-1] # end of queue element - - # Element out of line - pop_front: int = deque.popleft() # head element out of queue - pop_rear: int = deque.pop() # end-of-queue element out of queue - - # Get the length of the double-ended queue - size: int = len(deque) - - # Determine if the double-ended queue is empty - is_empty: bool = len(deque) == 0 - ``` - -=== "C++" - - ```cpp title="deque.cpp" - /* Initialize double-ended queue */ - deque deque; - - /* Element in queue */ - deque.push_back(2); // add to the end of the queue - deque.push_back(5); - deque.push_back(4); - deque.push_front(3); // add to the top of the queue - deque.push_front(1); - - /* Access to elements */ - int front = deque.front(); // head element of the queue - int back = deque.back(); // end of queue element - - /* Element out of line */ - deque.pop_front(); // the top element of the queue is out of the queue - deque.pop_back(); // the last element of the queue is out of the queue - - /* Get the length of the double-ended queue */ - int size = deque.size(); - - /* Determine if the double-ended queue is empty */ - bool empty = deque.empty(); - ``` - -=== "Java" - - ```java title="deque.java" - /* Initialize double-ended queue */ - Deque deque = new LinkedList<>(); - - /* Element in queue */ - deque.offerLast(2); // add to the end of the queue - deque.offerLast(5); - deque.offerLast(4); - deque.offerFirst(3); // add to the top of the queue - deque.offerFirst(1); - - /* Access to elements */ - int peekFirst = deque.peekFirst(); // first element of the queue - int peekLast = deque.peekLast(); // the last element of the queue - - /* Element out of line */ - int popFirst = deque.pollFirst(); // first element out of queue - int popLast = deque.pollLast(); // end of queue out of queue - - /* Get the length of the double-ended queue */ - int size = deque.size(); - - /* Determine if the double-ended queue is empty */ - boolean isEmpty = deque.isEmpty(); - ``` - -=== "C#" - - ```csharp title="deque.cs" - /* Initialize double-ended queue */ - // In C#, think of a linked list, a LinkedList, as a double-ended queue. - LinkedList deque = new(); - - /* Element in queue */ - deque.AddLast(2); // add to the end of the queue - deque.AddLast(5); - deque.AddLast(4); - deque.AddFirst(3); // add to the top of the queue - deque.AddFirst(1); - - /* Access to elements */ - int peekFirst = deque.First.Value; // first element of the queue - int peekLast = deque.Last.Value; // the last element of the queue - - /* Element out of line */ - deque.RemoveFirst(); // the first element of the queue is out of the queue - deque.RemoveLast(); // the end of the queue is out of the queue - - /* Get the length of the double-ended queue */ - int size = deque.Count; - - /* Determine if the double-ended queue is empty */ - bool isEmpty = deque.Count == 0; - ``` - -=== "Go" - - ```go title="deque_test.go" - /* Initialize double-ended queue */ - // Use lists as double-ended queues in Go. - deque := list.New() - - /* Element in queue */ - deque.PushBack(2) // add to the end of the queue - deque.PushBack(5) - deque.PushBack(4) - deque.PushFront(3) // add to the top of the queue - deque.PushFront(1) - - /* Access to elements */ - front := deque.Front() // head element of the queue - rear := deque.Back() // the end element of the queue - - /* Element out of line */ - deque.Remove(front) // head element out of queue - deque.Remove(rear) // the end of the queue is out of the queue - - /* Get the length of the double-ended queue */ - size := deque.Len() - - /* Determine if the double-ended queue is empty */ - isEmpty := deque.Len() == 0 - ``` - -=== "Swift" - - ```swift title="deque.swift" - /* Initialize double-ended queue */ - // Swift doesn't have a built-in double-ended queue class, so you can use Array as a double-ended queue. - var deque: [Int] = [] - - /* Element in queue */ - deque.append(2) // add to the end of the queue - deque.append(5) - deque.append(4) - deque.insert(3, at: 0) // add to top of queue - deque.insert(1, at: 0) - - /* Access to elements */ - let peekFirst = deque.first! // The first element of the queue. - let peekLast = deque.last! // The last element of the queue. - - /* Element out of line */ - // The complexity of popFirst is O(n) when simulating with Array. - let popFirst = deque.removeFirst() // the first element of the queue is out of the queue - let popLast = deque.removeLast() // the last element of the queue is out of the queue - - /* Get the length of the double-ended queue */ - let size = deque.count - - /* Determine if the double-ended queue is empty */ - let isEmpty = deque.isEmpty - ``` - -=== "JS" - - ```javascript title="deque.js" - /* Initialize double-ended queue */ - // JavaScript doesn't have a built-in double-ended queue, you can only use Array as a double-ended queue. - const deque = []; - - /* Element in queue */ - deque.push(2); - deque.push(5); - deque.push(4); - // Note that the unshift() method has a time complexity of O(n) because it is an array. - deque.unshift(3); - deque.unshift(1); - console.log("Double-ended queue dequeue = ", deque); - - /* Access to elements */ - const peekFirst = deque[0]; - console.log("Team head element peekFirst = " + peekFirst); - const peekLast = deque[deque.length - 1]; - console.log("Last element of queue peekLast = " + peekLast); - - /* Element out of line */ - // Note that the shift() method has a time complexity of O(n) because it's an array. - const popFront = deque.shift(); - console.log("First element out of the queue popFront = " + popFront + ", first out of the queue after deque = " + deque); - const popBack = deque.pop(); - console.log("End-of-queue out element popBack = " + popBack + ", end-of-queue out after deque = " + deque); - - /* Get the length of the double-ended queue */ - const size = deque.length; - console.log("Double-ended queue length size = " + size); - - /* Determine if the double-ended queue is empty */ - const isEmpty = size === 0; - console.log("Whether double-ended queue is empty = " + isEmpty); - ``` - -=== "TS" - - ```typescript title="deque.ts" - /* Initialize double-ended queue */ - // TypeScript doesn't have a built-in double-ended queue; you can only use Array as a double-ended queue. - const deque: number[] = []; - - /* Element in queue */ - deque.push(2); - deque.push(5); - deque.push(4); - // Note that the unshift() method has a time complexity of O(n) because it is an array. - deque.unshift(3); - deque.unshift(1); - console.log("Double-ended queue dequeue = ", deque); - - /* Access to elements */ - const peekFirst: number = deque[0]; - console.log("Team head element peekFirst = " + peekFirst); - const peekLast: number = deque[deque.length - 1]; - console.log("Last element of queue peekLast = " + peekLast); - - /* Element out of line */ - // Note that the shift() method has a time complexity of O(n) because it's an array. - const popFront: number = deque.shift() as number; - console.log("First element out of the queue popFront = " + popFront + ", first out of the queue after deque = " + deque); - const popBack: number = deque.pop() as number; - console.log("End-of-queue out element popBack = " + popBack + ", end-of-queue out after deque = " + deque); - - /* Get the length of the double-ended queue */ - const size: number = deque.length; - console.log("length of double-ended queue size = " + size); - - /* Determine if the double-ended queue is empty */ - const isEmpty: boolean = size === 0; - console.log("Whether double-ended queue is empty = " + isEmpty); - ``` - -=== "Dart" - - ```dart title="deque.dart" - /* Initialize double-ended queue */ - // In Dart, Queues are defined as double-ended queues. - Queue deque = Queue(); - - /* Element in queue */ - deque.addLast(2); // add to the end of the queue - deque.addLast(5); - deque.addLast(4); - deque.addFirst(3); // add to the top of the queue - deque.addFirst(1); - - /* Access to elements */ - int peekFirst = deque.first; // first element of the queue - int peekLast = deque.last; // the last element of the queue - - /* Element out of line */ - int popFirst = deque.removeFirst(); // first element out of queue - int popLast = deque.removeLast(); // the last element out of the queue - - /* Get the length of the double-ended queue */ - int size = deque.length; - - /* Determine if the double-ended queue is empty */ - bool isEmpty = deque.isEmpty;W - ``` - -=== "Rust" - - ```rust title="deque.rs" - /* Initialize double-ended queue */ - let mut deque: VecDeque = VecDeque::new(); - - /* Element in queue */ - deque.push_back(2); // add to the end of the queue - deque.push_back(5); - deque.push_back(4); - deque.push_front(3); // add to the top of the queue - deque.push_front(1); - - /* Access to elements */ - if let Some(front) = deque.front() { // the first element of the queue - } - if let Some(rear) = deque.back() { // end-of-queue element - } - - /* Element out of line */ - if let Some(pop_front) = deque.pop_front() { // the top element of the queue is out of the queue - } - if let Some(pop_rear) = deque.pop_back() { // end-of-queue element out of queue - } - - /* Get the length of the double-ended queue */ - let size = deque.len(); - - /* Determine if the double-ended queue is empty */ - let is_empty = deque.is_empty(); - ``` - -=== "C" - - ```c title="deque.c" - // C does not provide a built-in double-ended queue - ``` - -=== "Zig" - - ```zig title="deque.zig" - - ``` - -## Double-Ended Queue Implementation - -The implementation of a double-ended queue is similar to a queue, and you can choose either a linked list or an array as the underlying data structure. - -### Implementation Based On Bi-Directional Linked Lists - -Recall from the previous section that we used an ordinary unidirectional linked list to implement the queue because it makes it easy to delete the head node (corresponding to an out-of-queue operation) and add a new node after the tail node (corresponding to an in-queue operation). - -For a double-ended queue, both the head and the tail can perform in-queue and out-queue operations. In other words, a double-ended queue needs to realize another symmetric direction. For this reason, we use a "bidirectional linked list" as the underlying data structure of a double-ended queue. - -As shown in the figure below, we consider the head and tail nodes of a double-ended queue as the head and tail of the double-ended queue, and realize the function of adding and deleting nodes at both ends at the same time. - -=== "LinkedListDeque" - ![Implementing in-queue and out-queue operations for double-ended queues based on linked lists](deque.assets/linkedlist_deque.png) - -=== "pushLast()" - ![linkedlist_deque_push_last](deque.assets/linkedlist_deque_push_last.png) - -=== "pushFirst()" - ![linkedlist_deque_push_first](deque.assets/linkedlist_deque_push_first.png) - -=== "popLast()" - ![linkedlist_deque_pop_last](deque.assets/linkedlist_deque_pop_last.png) - -=== "popFirst()" - ![linkedlist_deque_pop_first](deque.assets/linkedlist_deque_pop_first.png) - -The implementation code is shown below. - -```src -[file]{linkedlist_deque}-[class]{linked_list_deque}-[func]{} -``` - -### An Array-Based Implementation - -As shown in the figure below, similar to implementing queues based on arrays, we can also implement double-ended queues using ring arrays. - -=== "ArrayDeque" - ![Implementing in-queue and out-queue operations for double-ended queues based on arrays](deque.assets/array_deque.png) - -=== "pushLast()" - ![array_deque_push_last](deque.assets/array_deque_push_last.png) - -=== "pushFirst()" - ![array_deque_push_first](deque.assets/array_deque_push_first.png) - -=== "popLast()" - ![array_deque_pop_last](deque.assets/array_deque_pop_last.png) - -=== "popFirst()" - ![array_deque_pop_first](deque.assets/array_deque_pop_first.png) - -On the basis of the queue implementation, only need to add the "queue head into the queue" and "queue tail out of the queue" method. - -```src -[file]{array_deque}-[class]{array_deque}-[func]{} -``` - -## Double-Ended Queue Application - -A double-ended queue combines the logic of both stacks and queues, **so it can realize all the application scenarios of both, while providing a higher degree of freedom**. - -As we know, the "undo" function of software is usually realized by using a stack: the system `push` each change operation to the stack, and then `pop` to realize the undo. However, software usually limits the number of undo steps (e.g. only $50$ steps are allowed to be saved) due to the limitation of system resources. When the length of the stack exceeds $50$, the software needs to perform a delete operation at the bottom of the stack (i.e., the head of the queue). **However, the stack is unable to fulfill this function, and then a double-ended queue needs to be used instead of the stack**. Note that the core logic of "undo" still follows the stack's first-in, first-out principle, but the double-ended queue is able to implement some additional logic more flexibly. diff --git a/docs-en/chapter_stack_and_queue/index.md b/docs-en/chapter_stack_and_queue/index.md deleted file mode 100644 index 7731ec39fd..0000000000 --- a/docs-en/chapter_stack_and_queue/index.md +++ /dev/null @@ -1,13 +0,0 @@ -# Stack And Queue - -
      - -![stack and queue](../assets/covers/chapter_stack_and_queue.jpg){ width="600" } - -
      - -!!! abstract - - Stacks are like stacking cats and queues are like cats queuing. - - The two represent the logical relationships of first-in-last-out and first-in-first-out, respectively. diff --git a/docs-en/chapter_stack_and_queue/queue.assets/array_queue.png b/docs-en/chapter_stack_and_queue/queue.assets/array_queue.png deleted file mode 100644 index 4e5dfd095cb216374e50c1d38979051fe3184dbe..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 25642 zcmcG#byQSQ_W*kC%rJBbNT-xasif3^f`F7tNh?UVNJ(5&S`kE$lmQixE|F#c5l~XP z2LT5r9#*5jXZ_uX;!*>%p@bBWN^zClUGLIwb!R9Cxx8vr80Bcg`{ zNf`Fu30o6@ZCy=0m6MZ`+1c4A)xn#ao5z3u?(gp(9UUDW9!5k&EG{liPfzde?(XdD z{QmuWZEa0fR<@y`fg$+&_V)J5O){C-@R5-bCnqN_clTeve!YA54mXY)8yj0*UdG{Y zot>QrhX>8g%`aZO`1b8vUteEsZ7m*;ud1q2Q&ZE|*KcWQ!D6wQ9T}#krc+Z>t*xy} zN=gO!`AJDh3Gwl1X=y1bDMLd;a&mGT8ygD?3pO@34h{|j0|TX{rPMTwJ`my4u^@TU1n(o11H4VNqUQ{^7%inC9Q@?d=K*3XzeKj~_o)QBjGC zihB0!*~G*|SXh{rmR9TZmXVRs{QP`HMFkAQw{PG6{P}ZiZ0y$7*5u@5cXzj!m)Gd% 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As the name suggests, a queue simulates the phenomenon of queuing, where newcomers keep joining the tail end of the queue and those at the head of the queue leave one by one. - -As shown in the figure below, we will call the head of the queue "head", the tail is called "tail", the operation of adding elements to the tail is called "queue", the operation of deleting elements from the head of the queue is called "queue". called "out of the queue". - -![First-in-first-out rule for queues](queue.assets/queue_operations.png) - -## Queue Common Operations - -The common operations of queues are shown in the table below. Note that method names may vary from one programming language to another. We use the same method names as stack here. - -

      Table   Efficiency of Queue Operations

      - -| method name | description | time complexity | -| ----------- | ------------------------------------------------------------- | --------------- | -| push() | Element in queue, i.e. adding element to the end of the queue | $O(1)$ | -| pop() | queue head element out | $O(1)$ | -| peek() | access the first element of the queue | $O(1)$ | - -We can just use the queue class that is readily available in the programming language. - -=== "Python" - - ```python title="queue.py" - # Initialize the queue - # In Python, we generally think of the double-ended queue class, deque, as a queue. - # Although queue.Queue() is a pure queue class, it's not very usable, so it's not recommended. - que: deque[int] = collections.deque() - - # Elements in the queue - que.append(1) - que.append(3) - que.append(2) - que.append(5) - que.append(4) - - # Access to the first element of the queue - front: int = que[0]; - - # Element out of line - pop: int = que.popleft() - - # Get the length of the queue - size: int = len(que) - - # Determine if the queue is empty - is_empty: bool = len(que) == 0 - ``` - -=== "C++" - - ```cpp title="queue.cpp" - /* Initialize Queue */ - queue queue; - - /* Element in queue */ - queue.push(1); - queue.push(3); - queue.push(2); - queue.push(5); - queue.push(4); - - /* Access to the first element of the queue */ - int front = queue.front(); - - /* Element out of line */ - queue.pop(); - - /* Get the length of the queue */ - int size = queue.size(); - - /* Determine if the queue is empty */ - bool empty = queue.empty(); - ``` - -=== "Java" - - ```java title="queue.java" - /* Initialize Queue */ - Queue queue = new LinkedList<>(); - - /* Element in queue */ - queue.offer(1); - queue.offer(3); - queue.offer(2); - queue.offer(5); - queue.offer(4); - - /* Access to the first element of the queue */ - int peek = queue.peek(); - - /* Element out of line */ - int pop = queue.poll(); - - /* Get the length of the queue */ - int size = queue.size(); - - /* Determine if the queue is empty */ - boolean isEmpty = queue.isEmpty(); - ``` - -=== "C#" - - ```csharp title="queue.cs" - /* Initialize Queue */ - Queue queue = new(); - - /* Element in queue */ - queue.Enqueue(1); - queue.Enqueue(3); - queue.Enqueue(2); - queue.Enqueue(5); - queue.Enqueue(4); - - /* Access to the first element of the queue */ - int peek = queue.Peek(); - - /* Element out of line */ - int pop = queue.Dequeue(); - - /* Get the length of the queue */ - int size = queue.Count; - - /* Determine if the queue is empty */ - bool isEmpty = queue.Count == 0; - ``` - -=== "Go" - - ```go title="queue_test.go" - /* Initialize Queue */ - // In Go, use lists as queues. - queue := list.New() - - /* Element in queue */ - queue.PushBack(1) - queue.PushBack(3) - queue.PushBack(2) - queue.PushBack(5) - queue.PushBack(4) - - /* Access to the first element of the queue */ - peek := queue.Front() - - /* Element out of line */ - pop := queue.Front() - queue.Remove(pop) - - /* Get the length of the queue */ - size := queue.Len() - - /* Determine if the queue is empty */ - isEmpty := queue.Len() == 0 - ``` - -=== "Swift" - - ```swift title="queue.swift" - /* Initialize Queue */ - // Swift doesn't have a built-in queue class, so you can use Array as a queue. - var queue: [Int] = [] - - /* Element in queue */ - queue.append(1) - queue.append(3) - queue.append(2) - queue.append(5) - queue.append(4) - - /* Access to the first element of the queue */ - let peek = queue.first! - - /* Element out of line */ - // The complexity of removeFirst is O(n) because it's an array. - let pool = queue.removeFirst() - - /* Get the length of the queue */ - let size = queue.count - - /* Determine if the queue is empty */ - let isEmpty = queue.isEmpty - ``` - -=== "JS" - - ```javascript title="queue.js" - /* Initialize Queue */ - // JavaScript doesn't have a built-in queue, you can use Array as a queue. - const queue = []; - - /* Element in queue */ - queue.push(1); - queue.push(3); - queue.push(2); - queue.push(5); - queue.push(4); - - /* Access to the first element of the queue */ - const peek = queue[0]; - - /* Element out of line */ - // The underlying level is an array, so the time complexity of the shift() method is O(n) - const pop = queue.shift(); - - /* Get the length of the queue */ - const size = queue.length; - - /* Determine if the queue is empty */ - const empty = queue.length === 0; - ``` - -=== "TS" - - ```typescript title="queue.ts" - /* Initialize Queue */ - // TypeScript doesn't have a built-in queue; you can use Array as a queue. - const queue: number[] = []; - - /* Element in queue */ - queue.push(1); - queue.push(3); - queue.push(2); - queue.push(5); - queue.push(4); - - /* Access to the first element of the queue */ - const peek = queue[0]; - - /* Element out of line */ - // The underlying level is an array, so the time complexity of the shift() method is O(n) - const pop = queue.shift(); - - /* Get the length of the queue */ - const size = queue.length; - - /* Determine if the queue is empty */ - const empty = queue.length === 0; - ``` - -=== "Dart" - - ```dart title="queue.dart" - /* Initialize Queue */ - // In Dart, the queue class Qeque is a double-ended queue and can also be used as a queue - Queue queue = Queue(); - - /* Element in queue */ - queue.add(1); - queue.add(3); - queue.add(2); - queue.add(5); - queue.add(4); - - /* Access to the first element of the queue */ - int peek = queue.first; - - /* Element out of line */ - int pop = queue.removeFirst(); - - /* Get the length of the queue */ - int size = queue.length; - - /* Determine if the queue is empty */ - bool isEmpty = queue.isEmpty; - ``` - -=== "Rust" - - ```rust title="queue.rs" - /* Initialize double-ended queue */ - // Use a double-ended queue as a regular queue in Rust - let mut deque: VecDeque = VecDeque::new(); - - /* Element in queue */ - deque.push_back(1); - deque.push_back(3); - deque.push_back(2); - deque.push_back(5); - deque.push_back(4); - - /* Access to the first element of the queue */ - if let Some(front) = deque.front() { - } - - /* Element out of line */ - if let Some(pop) = deque.pop_front() { - } - - /* Get the length of the queue */ - let size = deque.len(); - - /* Determine if the queue is empty */ - let is_empty = deque.is_empty(); - ``` - -=== "C" - - ```c title="queue.c" - // C does not provide a built-in queue - ``` - -=== "Zig" - - ```zig title="queue.zig" - - ``` - -## Queue Implementation - -To implement a queue, we need a data structure that can add elements at one end and remove elements at the other. Therefore, both linked lists and arrays can be used to implement queues. - -### A Linked List Based Implementation - -As shown in the figure below, we can consider the "head node" and "tail node" of the linked list as the "head" and "tail" respectively. "The tail node can only add nodes and the head node can only delete nodes. - -=== "LinkedListQueue" - ![Implementing queue entry and exit operations based on linked lists](queue.assets/linkedlist_queue.png) - -=== "push()" - ![linkedlist_queue_push](queue.assets/linkedlist_queue_push.png) - -=== "pop()" - ![linkedlist_queue_pop](queue.assets/linkedlist_queue_pop.png) - -Here is the code to implement a queue with a linked list. - -```src -[file]{linkedlist_queue}-[class]{linked_list_queue}-[func]{} -``` - -### An Array-Based Implementation - -Since the time complexity of deleting the first element of an array is $O(n)$ , this can lead to less efficient out-of-queue operations. However, we can avoid this problem by using the following clever approach. - -We can use a variable `front` to point to the index of the element at the head of the queue, and maintain a variable `size` to record the length of the queue. Define `rear = front + size` and this formula calculates `rear` to point to the next position after the end element of the queue. - -Based on this design, **the valid interval of the elements contained in the array is `[front, rear - 1]`**, and the various operations are implemented as shown in the figure below. - -- Queue entry operation: assign the input element at the `rear` index and increase `size` by 1. -- Operation: Simply increase `front` by 1 and decrease `size` by 1. - -It can be seen that both the queue-in and queue-out operations need to be performed only once, and both have a time complexity of $O(1)$ . - -=== "ArrayQueue" - ![Implement queue in/queue out operations based on arrays](queue.assets/array_queue.png) - -=== "push()" - ![array_queue_push](queue.assets/array_queue_push.png) - -=== "pop()" - ![array_queue_pop](queue.assets/array_queue_pop.png) - -You may notice a problem: `front` and `rear` are both moving to the right as they keep moving in and out of the array, **and when they reach the end of the array, they can't move any further**. To solve this problem, we can think of the array as a "circular array" with the first and last arrays connected. - -For circular arrays, we need to have `front` or `rear` return to the head of the array when it crosses the end of the array to continue traversing. This periodicity can be achieved by the "remainder operation", as shown in the following code. - -```src -[file]{array_queue}-[class]{array_queue}-[func]{} -``` - -The above implementation of the queue still has the limitation that its length is immutable. However, this problem is not difficult to solve, we can replace the array with a dynamic array, thus introducing an expansion mechanism. Interested students can try to implement their own. - -The conclusions of the comparison of the two implementations are consistent with the stack and will not be repeated here. - -## Typical Applications Of Queues - -- **Taobao Orders**. After a shopper places an order, the order will be added to the queue, and the system will then process the orders in the queue sequentially according to the order. During the double eleven period, a huge number of orders will be generated in a short period of time, and high concurrency becomes a problem that engineers need to focus on. -- **All kinds of to-do lists**. Any scenario that requires "first-come-first-served" functionality, such as a task queue for a printer, a queue for a restaurant, and so on. 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z(k*t+3(uTa<7KjXSheBv^-k8GMc3fq$>NWh7VT#Xq#e~eYx)P*T&;t2Tl$}HZs<>! zX$ctgCOFr|S{q~ycb+|~KV3vLy-qigNk5TBPyQ`bRwn%uGEXbq(RM;Ve&YzXw>e~8 IX%&$84@uSEqW}N^ diff --git a/docs-en/chapter_stack_and_queue/stack.md b/docs-en/chapter_stack_and_queue/stack.md deleted file mode 100644 index 77a23b4dfc..0000000000 --- a/docs-en/chapter_stack_and_queue/stack.md +++ /dev/null @@ -1,379 +0,0 @@ -# Stack - -A "stack" is a linear data structure that follows a first-in, first-out logic. - -We can analogize the stack to a stack of plates on a table, if we need to take out the bottom plate, we need to first remove the top plate in turn. We replace the plates with elements of various types (e.g., integers, characters, objects, etc.) to get the stack data structure. - -As shown in the figure below, we call the top of the stack the "top of the stack" and the bottom the "bottom of the stack". The operation of adding an element to the top of the stack is called "Into the stack", and the operation of removing an element from the top of the stack is called "Out of the stack". - -![First-in-first-out rule for stacks](stack.assets/stack_operations.png) - -## Common Operations On Stacks - -The common operations of the stack are shown in the table below, and the specific method names need to be determined according to the programming language used. Here, we take the common names `push()`, `pop()`, `peek()` as examples. - -

      Table   Efficiency of stack operations

      - -| Methods | Description | Time Complexity | -| ------- | -------------------------------------- | --------------- | -| push() | element to stack (add to top of stack) | $O(1)$ | -| pop() | top element of stack out | $O(1)$ | -| peek() | access top stack element | $O(1)$ | - -Usually, we can just use the built-in stack class of a programming language. However, some languages may not provide a stack class, so we can use the language's "arrays" or "linked lists" as stacks, and ignore stack-independent operations in the program logic. - -=== "Python" - - ```python title="stack.py" - # Initialize the stack - # Python doesn't have a built-in stack class, so you can use List as a stack. - stack: list[int] = [] - - # Element on the stack - stack.append(1) - stack.append(3) - stack.append(2) - stack.append(5) - stack.append(4) - - # Access the top element of the stack - peek: int = stack[-1] - - # Element out of the stack - pop: int = stack.pop() - - # Get the length of the stack - size: int = len(stack) - - # Determine if it's empty - is_empty: bool = len(stack) == 0 - ``` - -=== "C++" - - ```cpp title="stack.cpp" - /* Initialize the stack */ - stack stack; - - /* Element on stack */ - stack.push(1); - stack.push(3); - stack.push(2); - stack.push(5); - stack.push(4); - - /* Accesses the top element of the stack */ - int top = stack.top(); - - /* Element out of stack */ - stack.pop(); // no return value - - /* Get the length of the stack */ - int size = stack.size(); - - /* Determine if it is empty */ - bool empty = stack.empty(); - ``` - -=== "Java" - - ```java title="stack.java" - /* Initialize the stack */ - Stack stack = new Stack<>(); - - /* Element on stack */ - stack.push(1); - stack.push(3); - stack.push(2); - stack.push(5); - stack.push(4); - - /* Accesses the top element of the stack */ - int peek = stack.peek(); - - /* Element out of stack */ - int pop = stack.pop(); - - /* Get the length of the stack */ - int size = stack.size(); - - /* Determine if it is empty */ - boolean isEmpty = stack.isEmpty(); - ``` - -=== "C#" - - ```csharp title="stack.cs" - /* Initialize the stack */ - Stack stack = new(); - - /* Element on stack */ - stack.Push(1); - stack.Push(3); - stack.Push(2); - stack.Push(5); - stack.Push(4); - - /* Accesses the top element of the stack */ - int peek = stack.Peek(); - - /* Element out of stack */ - int pop = stack.Pop(); - - /* Get the length of the stack */ - int size = stack.Count; - - /* Determine if it is empty */ - bool isEmpty = stack.Count == 0; - ``` - -=== "Go" - - ```go title="stack_test.go" - /* Initialize the stack */ - // In Go, it is recommended to use Slice as a stack. - var stack []int - - /* Element on stack */ - stack = append(stack, 1) - stack = append(stack, 3) - stack = append(stack, 2) - stack = append(stack, 5) - stack = append(stack, 4) - - /* Accesses the top element of the stack */ - peek := stack[len(stack)-1] - - /* Element out of stack */ - pop := stack[len(stack)-1] - stack = stack[:len(stack)-1] - - /* Get the length of the stack */ - size := len(stack) - - /* Determine if it is empty */ - isEmpty := len(stack) == 0 - ``` - -=== "Swift" - - ```swift title="stack.swift" - /* Initialize the stack */ - // Swift doesn't have a built-in stack class, so you can use Array as a stack. - var stack: [Int] = [] - - /* Element on stack */ - stack.append(1) - stack.append(3) - stack.append(2) - stack.append(5) - stack.append(4) - - /* Accesses the top element of the stack */ - let peek = stack.last! - - /* Element out of stack */ - let pop = stack.removeLast() - - /* Get the length of the stack */ - let size = stack.count - - /* Determine if it is empty */ - let isEmpty = stack.isEmpty - ``` - -=== "JS" - - ```javascript title="stack.js" - /* Initialize the stack */ - // JavaScript doesn't have a built-in stack class, you can use Array as a stack. - const stack = []; - - /* Element on stack */ - stack.push(1); - stack.push(3); - stack.push(2); - stack.push(5); - stack.push(4); - - /* Accesses the top element of the stack */ - const peek = stack[stack.length-1]; - - /* Element out of stack */ - const pop = stack.pop(); - - /* Get the length of the stack */ - const size = stack.length; - - /* Determine if it is empty */ - const is_empty = stack.length === 0; - ``` - -=== "TS" - - ```typescript title="stack.ts" - /* Initialize the stack */ - // TypeScript doesn't have a built-in stack class, so you can use Array as a stack. - const stack: number[] = []; - - /* Element on stack */ - stack.push(1); - stack.push(3); - stack.push(2); - stack.push(5); - stack.push(4); - - /* Accesses the top element of the stack */ - const peek = stack[stack.length - 1]; - - /* Element out of stack */ - const pop = stack.pop(); - - /* Get the length of the stack */ - const size = stack.length; - - /* Determine if it is empty */ - const is_empty = stack.length === 0; - ``` - -=== "Dart" - - ```dart title="stack.dart" - /* Initialize the stack */ - // Dart doesn't have a built-in stack class, so you can use List as a stack. - List stack = []; - - /* Element on stack */ - stack.add(1); - stack.add(3); - stack.add(2); - stack.add(5); - stack.add(4); - - /* Accesses the top element of the stack */ - int peek = stack.last; - - /* Element out of stack */ - int pop = stack.removeLast(); - - /* Get the length of the stack */ - int size = stack.length; - - /* Determine if it is empty */ - bool isEmpty = stack.isEmpty; - ``` - -=== "Rust" - - ```rust title="stack.rs" - /* Initialize the stack */ - // Use the Vec as a stack. - let mut stack: Vec = Vec::new(); - - /* Element on stack */ - stack.push(1); - stack.push(3); - stack.push(2); - stack.push(5); - stack.push(4); - - /* Accesses the top element of the stack */ - let top = stack[stack.len() - 1]; - - /* Element out of stack */ - let pop = stack.pop().unwrap(); - - /* Get the length of the stack */ - let size = stack.len(); - - /* Determine if it is empty */ - let is_empty = stack.is_empty(); - ``` - -=== "C" - - ```c title="stack.c" - // C does not provide a built-in stack. - ``` - -=== "Zig" - - ```zig title="stack.zig" - - ``` - -## Stack Implementation - -In order to get a deeper understanding of how stacks work, let's try to implement a stack class ourselves. - -A stack follows the first-in-first-out principle, so we can only add or remove elements at the top of the stack. However, arrays and linked lists can add and remove elements anywhere, **so the stack can be thought of as a restricted array or linked list**. In other words, we can "mask" some of the extraneous operations of arrays or linked lists so that their external logic conforms to the properties of a stack. - -### A Linked List Based Implementation - -When using a linked list to implement a stack, we can consider the head node of the linked list as the top of the stack and the tail node as the bottom of the stack. - -As shown in the figure below, for the stack operation, we only need to insert the element into the head of the linked list node insertion method is called "head insertion method". For stack out operation, we just need to remove the head node from the linked list. - -=== "LinkedListStack" - ![Implementing stack-in-stack-out operations based on linked lists](stack.assets/linkedlist_stack.png) - -=== "push()" - ![linkedlist_stack_push](stack.assets/linkedlist_stack_push.png) - -=== "pop()" - ![linkedlist_stack_pop](stack.assets/linkedlist_stack_pop.png) - -The following is sample code for implementing a stack based on a linked list. - -```src -[file]{linkedlist_stack}-[class]{linked_list_stack}-[func]{} -``` - -### An Array-Based Implementation - -When using an array to implement a stack, we can use the tail of the array as the top of the stack. As shown in the figure below, the stack-in and stack-out operations correspond to adding and removing elements to the tail of the array, respectively, both with a time complexity of $O(1)$ . - -=== "ArrayStack" - ![Implementing stack in/out operations based on arrays](stack.assets/array_stack.png) - -=== "push()" - ![array_stack_push](stack.assets/array_stack_push.png) - -=== "pop()" - ![array_stack_pop](stack.assets/array_stack_pop.png) - -Since the elements of the stack may be constantly increasing, we can use dynamic arrays so that we do not have to deal with the array expansion problem. Here is the sample code. - -```src -[file]{array_stack}-[class]{array_stack}-[func]{} -``` - -## Comparison Of Two Implementations - -**Support Operations** - -Both implementations support the operations in the stack definition. The array implementation additionally supports random access, but this is beyond the scope of the stack definition, so it is generally not used. - -**Time Efficiency** - -In array-based implementations, both the stack-in and stack-out operations are performed in pre-allocated contiguous memory with good cache locality, and thus are more efficient. However, if the stack entry exceeds the capacity of the array, the expansion mechanism is triggered, causing the time complexity of that stack entry operation to become $O(n)$ . - -In the linked list implementation, the expansion of linked lists is very flexible and does not suffer from the efficiency degradation when expanding arrays as described above. However, the stack entry operation requires initialization of the node object and modification of the pointer, and is therefore relatively inefficient. However, if the stacked element is itself a node object, then the initialization step can be omitted, thus improving efficiency. - -In summary, when the elements of the in-stack and out-stack operations are basic data types, such as `int` or `double` , we can conclude the following. - -- Stacks implemented based on arrays are less efficient when expanding is triggered, but are more efficient on average because expanding is a low-frequency operation. -- Stacks implemented based on linked lists provide a more consistent performance in terms of efficiency. - -**Space efficiency** - -When initializing a list, the system assigns an "initial capacity" to the list, which may exceed the actual demand. Moreover, the expansion mechanism is usually based on a specific multiplier (e.g., 2x), and the expanded capacity may also exceed the actual demand. As a result, **stacks implemented based on arrays may cause a certain amount of wasted space**. - -However, since linked lists nodes require additional storage of pointers, **therefore linked nodes take up a relatively large amount of space**. - -In summary, we cannot simply determine which implementation is more memory-efficient; we need to analyze the situation on a case-by-case basis. - -## Typical Applications Of Stacks - -- **Backward and Forward in Browsers, Undo and Undoverse in Software**. Whenever we open a new web page, the browser executes the previous page into the stack so that we can go back to the previous page with the backward operation. The backward operation actually executes out of the stack. If we want to support both backward and forward, then we need two stacks to implement it together. -- **Program memory management**. Each time a function is called, the system adds a stack frame to the top of the stack to record the context information of the function. In a recursion function, the downward recursion phase performs constant in-stack operations, while the upward backward phase performs out-stack operations. diff --git a/docs-en/chapter_stack_and_queue/summary.md b/docs-en/chapter_stack_and_queue/summary.md deleted file mode 100644 index 263c66da99..0000000000 --- a/docs-en/chapter_stack_and_queue/summary.md +++ /dev/null @@ -1,31 +0,0 @@ -# Summary - -### Highlights - -- A stack is a data structure that follows the first-in-first-out principle and can be implemented as an array or a linked list. -- From a time efficiency perspective, array implementations of stacks have a high average efficiency, but the time complexity of a single stack entry operation deteriorates to $O(n)$ during the expansion process. In contrast, the linked list-based implementation of the stack has a more stable efficiency performance. -- In terms of space efficiency, an array implementation of a stack may result in a degree of space wastage. However, it should be noted that linked list nodes take up more memory space than array elements. -- A queue is a data structure that follows the first-in-first-out principle and can be implemented equally well with arrays or linked lists. The conclusions for queues are similar to the aforementioned conclusions for stacks in terms of the comparison of time efficiency and space efficiency. -- A double-ended queue is a type of queue with a higher degree of freedom that allows adding and removing operations of elements at both ends. - -### Q & A - -!!! question "Is the browser's forward and backward implemented as a two-way linked list?" - - The browser's forward and backward functions are essentially a manifestation of the "stack". When the user visits a new page, it is added to the top of the stack; when the user clicks the back button, the page is ejected from the top of the stack. Using a double-ended queue makes it easy to implement some additional operations, which are mentioned in the section on double-ended queues. - -!!! question "Is it necessary to free the memory of the outgoing node after going out of the stack?" - - If the popup node is still needed later, it does not need to be freed. If you don't need it later, languages like `Java` and `Python` have automatic garbage collection mechanisms, so you don't need to free the memory manually; in `C` and `C++` you need to free the memory manually. - -!!! Question "A double-ended queue looks like two stacks spliced together, what is its purpose?" - - A double-ended queue is like a combination of a stack and a queue, or two stacks put together. It shows the logic of the stack + queue, so it can realize all the applications of the stack and queue, and more flexible. - -!!! question "How exactly is undo (undo) and reverse undo (redo) implemented?" - - Two stacks are used, stack `A` for undo and stack `B` for anti-undo. - - 1. Whenever the user performs an operation, press this operation onto the stack `A` and empty the stack `B`. - 2. When the user executes "Undo", the most recent operation is popped from stack `A` and pressed into stack `B`. - 3. 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z3e<54UQA5c5@zpm`FwvC;jv7NrWAOKfLSI+#<+`gs;r{?dhX=jqX1y2Yo=4B tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15}; - ``` - -=== "Java" - - ```java title="" - /* Array representation of a binary tree */ - // Using Integer, a wrapper class for int, you can use null to mark the null position. - Integer[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 }; - ``` - -=== "C#" - - ```csharp title="" - /* Array representation of a binary tree */ - // With the int? nullable type, you can use null to mark the null position. - int?[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 }; - ``` - -=== "Go" - - ```go title="" - /* Array representation of a binary tree */ - // With any type of slice, you can use nil to mark empty spaces. - tree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15} - ``` - -=== "Swift" - - ```swift title="" - /* Array representation of a binary tree */ - // With the Int? nullable type, you can use nil to mark the null position. - let tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15] - ``` - -=== "JS" - - ```javascript title="" - /* Array representation of a binary tree */ - // Use null for empty space - let tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15]; - ``` - -=== "TS" - - ```typescript title="" - /* Array representation of a binary tree */ - // Use null for empty space - let tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15]; - ``` - -=== "Dart" - - ```dart title="" - /* Array representation of a binary tree */ - // With the int? nullable type, you can use null to mark the null position. - List tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15]; - ``` - -=== "Rust" - - ```rust title="" - /* Array representation of a binary tree */ - // Use None to mark the empty space - let tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)]; - ``` - -=== "C" - - ```c title="" - /* Array representation of a binary tree */ - // Use the int maximum value to mark empty spaces, thus requiring that the node value not be INT_MAX - int tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15}; - ``` - -=== "Zig" - - ```zig title="" - - ``` - -![Array representation of any type of binary tree](array_representation_of_tree.assets/array_representation_with_empty.png) - -It is worth stating that **complete binary trees are well suited to be represented using arrays**. Recall that the definition of a complete binary tree is that $\text{None}$ only occurs at the bottom and to the right, **so all $\text{None}$ must occur at the end of the level-order traversal sequence**. - -This means that when using an array to represent a complete binary tree, it is convenient to omit storing all $\text{None}$. An example is given in the figure below. - -![Array representation of complete binary tree](array_representation_of_tree.assets/array_representation_complete_binary_tree.png) - -The following code implements a binary tree based on an array representation and includes the following operations. - -- Given a node, get its value, left (right) child node, and parent node. -- Gets the sequence of preorder traversals, middle order traversals, postorder traversals, and level-order traversals. - -```src -[file]{array_binary_tree}-[class]{array_binary_tree}-[func]{} -``` - -## Strengths And Limitations - -The array representation of a binary tree has the following main advantages. - -- Arrays are stored in contiguous memory space, are cache friendly, and are faster to access and traverse. -- No need to store pointers, which is more space efficient. -- Allow random access to nodes. - -However, there are some limitations to the array representation. - -- Array storage requires contiguous memory 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right: None - })) - } - } - ``` - -=== "C" - - ```c title="" - /* AVL tree node structure */ - TreeNode struct TreeNode { - int val; - int height; - struct TreeNode *left; - struct TreeNode *right; - } TreeNode; - - /* Constructors */ - TreeNode *newTreeNode(int val) { - TreeNode *node; - - node = (TreeNode *)malloc(sizeof(TreeNode)); - node->val = val; - node->height = 0; - node->left = NULL; - node->right = NULL; - return node; - } - ``` - -=== "Zig" - - ```zig title="" - - ``` - -The "node height" is the distance from the node to its farthest leaf node, i.e. the number of "edges" it passes through. In particular, leaf nodes have a height of 0 and empty nodes have a height of -1. We will create two utility functions to get and update the height of the node. - -```src -[file]{avl_tree}-[class]{a_v_l_tree}-[func]{update_height} -``` - -### Node Balance Factor - -The balance factor of a node is defined as the height of the left subtree minus the height of the right subtree of the node, and the balance factor of an empty tree node is zero. We also encapsulate the function to get the balance factor of a node into a function for future use. - -```src -[file]{avl_tree}-[class]{a_v_l_tree}-[func]{balance_factor} -``` - -!!! note - - Let the balance factor be $f$ , then the balance factor of any node of an avl tree node satisfies $-1 \le f \le 1$ . - -## AVL Tree Rotation - -AVL trees are characterized by the "rotate" operation, which can rebalance the balanced binary tree nodes without affecting the middle-order traversal sequence of the binary tree. In other words, **rotation operation can maintain the nature of the "binary search tree", but also can make the tree back to "balanced binary tree"**. - -Nodes with absolute value of balance factor $> 1$ are called "out-of-balance nodes". Depending on the imbalance of a node, there are four types of rotation operations: right rotation, left rotation, right rotation followed by left rotation, and left rotation followed by right rotation. These rotations are described in detail below. - -### Right-Handed - -As shown in the figure below, the balance factor is below the node. From the bottom to the top, the first out-of-balance node in the binary tree is "node 3". We focus on the subtree rooted at this out-of-balance node, denote this node as `node` and its left-child node as `child`, and perform a "right-hand rotation". After the right-hand rotation, the subtree is balanced and still maintains the characteristics of a binary search tree. - -=== "<1>" - ![Right rotation step](avl_tree.assets/avltree_right_rotate_step1.png) - -=== "<2>" - ![avltree_right_rotate_step2](avl_tree.assets/avltree_right_rotate_step2.png) - -=== "<3>" - ![avltree_right_rotate_step3](avl_tree.assets/avltree_right_rotate_step3.png) - -=== "<4>" - ![avltree_right_rotate_step4](avl_tree.assets/avltree_right_rotate_step4.png) - -As shown in the figure below, when the node `child` has a right-child node (notated as `grandChild`), you need to add a step to the right-child node: make `grandChild` the left-child node of `node`. - -![Right rotation with grandChild](avl_tree.assets/avltree_right_rotate_with_grandchild.png) - -"Rotate to the right" is a figurative statement, in fact, you need to modify the node pointer to achieve, the code is shown below. - -```src -[file]{avl_tree}-[class]{a_v_l_tree}-[func]{right_rotate} -``` - -### Levitra - -Accordingly, if we consider the "mirror image" of the above imbalanced binary tree, we need to perform the "left-handed" operation shown in the figure below. - -![left-rotate operation](avl_tree.assets/avltree_left_rotate.png) - -Similarly, as shown in the figure below, when the node `child` has a left-child node (denoted as `grandChild`), you need to add a step in the left-hand rotation: make `grandChild` the right-child node of `node`. - -![Left rotation with grandChild](avl_tree.assets/avltree_left_rotate_with_grandchild.png) - -It can be observed that **the right-toration and left-rotation operations are logically mirror-symmetric, and the two imbalance cases they solve respectively are also symmetric**. Based on the symmetry, we simply replace all `left` with `right` and all `right` with `left` in the implementation code for right-handedness to obtain the implementation code for left-handedness. - -```src -[file]{avl_tree}-[class]{a_v_l_tree}-[func]{left_rotate} -``` - -### Left Then Right - -For the out-of-balance node 3 in the figure below, neither left-rotation nor right-rotation can be used to rebalance the tree node. In this case, you need to perform a "left spin" on `child` and then a "right spin" on `node`. - -![Rotate left then right](avl_tree.assets/avltree_left_right_rotate.png) - -### First Right, Then Left - -As shown in the figure below, for the mirror case of the above imbalanced binary tree, we need to perform a "right rotation" on `child` and then a "left rotation" on `node`. - -![Rotate right then left](avl_tree.assets/avltree_right_left_rotate.png) - -### Selection Of Rotation - -The four imbalance cases shown in the figure below correspond to each of the above cases, and require the use of right, left, right then left, and left then right rotation operations, respectively. - -![The four rotation cases of an avl tree](avl_tree.assets/avltree_rotation_cases.png) - -As shown in the table below, we determine which of the figure above the imbalanced node belongs to by determining the balance factor of the imbalanced node as well as the plus or minus sign of the balance factor of the child node on the higher side. - -

      Table   Selection conditions for the four rotation cases

      - -| Balance factor of imbalanced node | Balance factor of child node | Rotational method to be used | -| --------------------------------- | ---------------------------- | ---------------------------- | -| $> 1$ (i.e., left-biased tree) | $\geq 0$ | right-handed | -| $> 1$ (i.e., left-biased tree) | $<0$ | First left then right | -| $< -1$ (i.e., right biased tree) | $\leq 0$ | left-handed | -| $< -1$ (i.e., right-biased tree) | $>0$ | first right then left | - -For ease of use, we encapsulate the rotation operation into a function. **With this function, we can rotate various imbalance cases to bring the imbalanced nodes back into balance**. - -```src -[file]{avl_tree}-[class]{a_v_l_tree}-[func]{rotate} -``` - -## AVL Tree Common Operations - -### Insert Node - -The node insertion operation of an AVL tree is similar in principal to a binary search tree. The only difference is that after inserting a node in an AVL tree, a series of out-of-balance nodes may appear on the path from that node to the root node. Therefore, **we need to start from this node and perform a rotation operation from the bottom up to bring all the out-of-balance nodes back to balance**. - -```src -[file]{avl_tree}-[class]{a_v_l_tree}-[func]{insert_helper} -``` - -### Delete Node - -Similarly, based on the method of deleting nodes in a binary search tree, a rotation operation needs to be performed from the bottom to the top to bring back the balance to all the out-of-balance nodes. - -```src -[file]{avl_tree}-[class]{a_v_l_tree}-[func]{remove_helper} -``` - -### Find The Node - -The node finding operation of an avl tree is consistent with a binary search tree and will not be repeated here. - -## Typical Applications Of AVL Trees - -- Organize and store large data for high-frequency lookups and low-frequency additions and deletions. -- Used to build an indexing system in a database. -- Red-black trees are preferred over avl trees in many applications. 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The number of rounds is at most the height of the binary tree, using $O(\log n)$ time when the binary tree is balanced. - -```src -[file]{binary_search_tree}-[class]{binary_search_tree}-[func]{search} -``` - -### Insert Node - -Given an element `num` to be inserted, in order to maintain the nature of the binary search tree "left subtree < root node < right subtree", the insertion process is shown in the figure below. - -1. **Find insertion position**: similar to Find, start from the root node and loop down according to the relationship between the current node value and the size of `num` until the leaf node is crossed (traversing to $\text{None}$ ). -2. **Insert node at this position**: initialize node `num` and place the node at position $\text{None}$. - -![Insert node in binary search tree](binary_search_tree.assets/bst_insert.png) - -In the code implementation, the following two points need to be noted. - -- A binary search tree node is not allowed to have duplicate nodes, otherwise its definition would be violated. Therefore, if the node to be inserted already exists in the tree node, the insertion is not performed and it is returned directly. -- In order to insert a node, we need to save the node from the previous loop with the help of the node `pre`. This way, when traversing to $\text{None}$, we can get its parent node, and thus complete the node insertion operation. - -```src -[file]{binary_search_tree}-[class]{binary_search_tree}-[func]{insert} -``` - -As with finding nodes, inserting nodes uses $O(\log n)$ time. - -### Delete Node - -First find the target node in the binary tree and then remove it from the binary tree. - -Similar to inserting a node, we need to ensure that the "left subtree < root node < right subtree" property of the binary search tree is still satisfied after the deletion operation is completed. - -Therefore, we need to perform the corresponding node deletion operations according to the number of children of the target node, which are categorized into three cases: 0, 1 and 2. - -As shown in the figure below, when the degree of the node to be deleted is $0$, it means that the node is a leaf node and can be deleted directly. - -![Remove node in binary search tree (degree 0)](binary_search_tree.assets/bst_remove_case1.png) - -As shown in the figure below, when the degree of the node to be deleted is $1$, it is sufficient to replace the node to be deleted with its children. - -![Remove node in binary search tree (degree 1)](binary_search_tree.assets/bst_remove_case2.png) - -When the degree of the node to be deleted is $2$, we can't delete it directly, but need to replace it with a node. This node can be the smallest node of the right-child node or the largest node of the left-child node because we want to keep the nature of the binary search tree "left $<$ root $<$ right". - -Assuming that we select the smallest node of the right-child node (i.e., the next node in the middle-order traversal), the flow of the deletion operation is shown in the figure below. - -1. Find the next node in the "middle-order traversal sequence" of the node to be deleted, denoted as `tmp`. -2. Overwrite the value of `tmp` with the value of the node to be deleted and recursion tree node `tmp` . - -=== "<1>" - ![Remove node in binary search tree (degree 2)](binary_search_tree.assets/bst_remove_case3_step1.png) - -=== "<2>" - ![bst_remove_case3_step2](binary_search_tree.assets/bst_remove_case3_step2.png) - -=== "<3>" - ![bst_remove_case3_step3](binary_search_tree.assets/bst_remove_case3_step3.png) - -=== "<4>" - ![bst_remove_case3_step4](binary_search_tree.assets/bst_remove_case3_step4.png) - -The node deletion operation also uses $O(\log n)$ time, where $O(\log n)$ time is needed to find the node to be deleted, and $O(\log n)$ time is needed to get the succeeding node for mid-order traversal. - -```src -[file]{binary_search_tree}-[class]{binary_search_tree}-[func]{remove} -``` - -### Middle-Order Traversal Is Ordered - -As shown in the figure below, the middle-order traversal of a binary tree follows the traversal order of "left $\rightarrow$ root $\rightarrow$ right", and the binary search tree satisfies the size relationship of "left-child node $<$ root node $<$ right-child node". The binary search tree satisfies the "left child $<$ root $<$ right child" size relation. - -This means that when performing a mid-order traversal in a binary search tree, the next smallest node will always be traversed preferentially, leading to an important property:**The sequence of mid-order traversals of a binary search tree is ascending**. - -Using the ascending nature of the middle-order traversal, it takes only $O(n)$ time to fetch ordered data in a binary search tree without additional sorting operations, which is very efficient. - -![Sequence of intermediate order traversals of a binary search tree](binary_search_tree.assets/bst_inorder_traversal.png) - -## Efficiency Of Binary Search Trees - -Given a set of data, we consider storing it using an array or a binary search tree. Observing the table below, the time complexity of each operation of the binary search tree is of logarithmic degree, which provides a stable and efficient performance. Arrays are more efficient than binary search trees only in data applicability scenarios with high frequency additions and low frequency lookups for deletions. - -

      Table   Efficiency of arrays vs. search trees

      - -| | Unordered arrays | Binary search trees | -| -------------- | ---------------- | ------------------- | -| Find Elements | $O(n)$ | $O(\log n)$ | -| insert element | $O(1)$ | $O(\log n)$ | -| Delete Element | $O(n)$ | $O(\log n)$ | - -Ideally, the binary search tree is "balanced" so that any node can be found in a $\log n$ round-robin. - -However, if we keep inserting and deleting nodes in the binary search tree, it may cause the binary tree to degenerate into a linked list as shown in the figure below, at which point the time complexity of the various operations also degenerates to $O(n)$ . - -![Degradation of binary search tree](binary_search_tree.assets/bst_degradation.png) - -## Common Applications Of Binary Search Trees - -- Used as a multi-level index in the system for efficient find, insert, and delete operations. -- as the underlying data structure for certain searching algorithms. -- Used to store a stream of data to keep it in an organized state. diff --git 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Similar to a linked list, the basic unit of a binary tree is a node, and each node contains: a value, a reference to a left-child node, and a reference to a right-child node. - -=== "Python" - - ```python title="" - class TreeNode: - """Binary tree node classes"""" - def __init__(self, val: int): - self.val: int = val # node value - self.left: TreeNode | None = None # left-child node reference - self.right: TreeNode | None = None # right-child node reference - ``` - -=== "C++" - - ```cpp title="" - /* Binary tree node structure */ - struct TreeNode { - int val; // node value - TreeNode *left; // Pointer to left-child node. - TreeNode *right; // Pointer to right-child node. - TreeNode(int x) : val(x), left(nullptr), right(nullptr) {} - }; - ``` - -=== "Java" - - ```java title="" - /* Binary tree node class */ - class TreeNode { - int val; // node value - TreeNode left; // Reference to left-child node. - TreeNode right; // right-child node reference - TreeNode(int x) { val = x; } - } - ``` - -=== "C#" - - ```csharp title="" - /* Binary tree node class */ - class TreeNode { - int val; // node value - TreeNode? left; // Left-child node reference. - TreeNode? right; // Reference to right-child node. - TreeNode(int x) { val = x; } - } - ``` - -=== "Go" - - ```go title="" - /* Binary tree node structure */ - type TreeNode struct { - Val int - Left *TreeNode - Right *TreeNode - } - /* Constructor */ - func NewTreeNode(v int) *TreeNode { - return &TreeNode{ - Left: nil, // pointer to left-child node - Right: nil, // pointer to right-child node - Val: v, // node value - } - } - ``` - -=== "Swift" - - ```swift title="" - /* Binary tree node class */ - class TreeNode { - var val: Int // node value - var left: TreeNode? // Left-child node reference. - var right: TreeNode? // Right-child node reference. - - init(x: Int) { - val = x - } - } - ``` - -=== "JS" - - ```javascript title="" - /* Binary tree node class */ - class TreeNode { - val; // node value - left; // Pointer to the left-child node. - right; // pointer to right-child node - constructor(val, left, right) { - this.val = val === undefined ? 0 : val; - this.left = left === undefined ? null : left; - this.right = right === undefined ? null : right; - } - } - ``` - -=== "TS" - - ```typescript title="" - /* Binary tree node class */ - class TreeNode { - val: number; - left: TreeNode | null; - right: TreeNode | null; - - constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) { - this.val = val === undefined ? 0 : val; // node value - this.left = left === undefined ? null : left; // left-child node reference - this.right = right === undefined ? null : right; // right-child node reference - } - } - ``` - -=== "Dart" - - ```dart title="" - /* Binary tree node class */ - class TreeNode { - int val; // node value - TreeNode? left; // Left-child node reference. - TreeNode? right; // Reference to right-child node. - TreeNode(this.val, [this.left, this.right]); - } - ``` - -=== "Rust" - - ```rust title="" - use std::rc::Rc; - use std::cell::RefCell; - - /* Binary tree node structure */ - struct TreeNode { - val: i32, // node value - left: Option>>, // left-child node reference - right: Option>>, // right-child node reference - } - - impl TreeNode { - /* Constructor */ - fn new(val: i32) -> Rc> { - Rc::new(RefCell::new(Self { - val, - left: None, - right: None - })) - } - } - ``` - -=== "C" - - ```c title="" - /* Binary tree node structure */ - typedef struct TreeNode { - int val; // node value - int height; // height of the node - struct TreeNode *left; // Pointer to left-child node. - struct TreeNode *right; // Pointer to right-child node. - } TreeNode; - - /* Constructors */ - TreeNode *newTreeNode(int val) { - TreeNode *node; - - node = (TreeNode *)malloc(sizeof(TreeNode)); - node->val = val; - node->height = 0; - node->left = NULL; - node->right = NULL; - return node; - } - ``` - -=== "Zig" - - ```zig title="" - - ``` - -Each node has two references (pointers) to the left-child node and the right-child node, which are called the parent nodes of the two child nodes. When given a node in a binary tree, the tree formed by the left-child node and the following nodes of the node is called the "left subtree" of the node, and similarly, the "right subtree" of the node is called the "right subtree". - -**In a binary tree node, all nodes except leaf nodes contain children and non-empty subtrees**. As shown in the figure below, if "node 2" is considered as the parent node, its left-child node and right-child node are "node 4" and "node 5" respectively. The left-child node is "the tree formed by node 4 and its following nodes" and the right-child node is "the tree formed by node 5 and its following nodes". - -![parent node, child node, subtree](binary_tree.assets/binary_tree_definition.png) - -## Common Terminology For Binary Trees - -The common terminology for binary trees is shown in the figure below. - -- "root node root node": a node at the top level of a binary tree with no parent node. -- "leaf node": a node with no children, both pointers point to $\text{None}$. -- "edge": a line segment connecting two nodes, i.e. a node reference (pointer). -- The "level" of the node: increasing from top to bottom, the level of the root node is 1. -- Degree of a node degree: the number of children of a node. In a binary tree, the degree can take values in the range 0, 1, 2. -- Height of a binary tree: the number of edges from the root node to the farthest leaf node. -- Depth of a node depth: the number of edges that pass from the root node to the node. -- The "height" of a node: the number of edges passing from the farthest leaf node to the node. - -![Common terminology for binary trees](binary_tree.assets/binary_tree_terminology.png) - -!!! tip - - Note that we usually define "height" and "depth" as "number of edges traveled", but some topics or textbooks may define them as "the number of nodes traveled". In this case, add 1 to both height and depth. - -## Basic Binary Tree Operations - -### Initialize A Binary Tree - -Similar to a linked list, the nodes are initialized first and then the references (pointers) are constructed. - -=== "Python" - - ```python title="binary_tree.py" - # Initialize binary tree - # Initialize nodes - n1 = TreeNode(val=1) - n2 = TreeNode(val=2) - n3 = TreeNode(val=3) - n4 = TreeNode(val=4) - n5 = TreeNode(val=5) - # Construct references to point (i.e. pointers) - n1.left = n2 - n1.right = n3 - n2.left = n4 - n2.right = n5 - ``` - -=== "C++" - - ```cpp title="binary_tree.cpp" - /* Initialize the binary tree */ - // Initialize nodes - TreeNode* n1 = new TreeNode(1); - TreeNode* n2 = new TreeNode(2); - TreeNode* n3 = new TreeNode(3); - TreeNode* n4 = new TreeNode(4); - TreeNode* n5 = new TreeNode(5); - // Constructing reference pointers (i.e. pointers) - n1->left = n2; - n1->right = n3; - n2->left = n4; - n2->right = n5; - ``` - -=== "Java" - - ```java title="binary_tree.java" - // Initialize nodes - TreeNode n1 = new TreeNode(1); - TreeNode n2 = new TreeNode(2); - TreeNode n3 = new TreeNode(3); - TreeNode n4 = new TreeNode(4); - TreeNode n5 = new TreeNode(5); - // Constructing reference pointers (i.e. pointers) - n1.left = n2; - n1.right = n3; - n2.left = n4; - n2.right = n5; - ``` - -=== "C#" - - ```csharp title="binary_tree.cs" - /* Initialize the binary tree */ - // Initialize nodes - TreeNode n1 = new(1); - TreeNode n2 = new(2); - TreeNode n3 = new(3); - TreeNode n4 = new(4); - TreeNode n5 = new(5); - // Constructing reference pointers (i.e. pointers) - n1.left = n2; - n1.right = n3; - n2.left = n4; - n2.right = n5; - ``` - -=== "Go" - - ```go title="binary_tree.go" - /* Initialize the binary tree */ - // Initialize nodes - n1 := NewTreeNode(1) - n2 := NewTreeNode(2) - n3 := NewTreeNode(3) - n4 := NewTreeNode(4) - n5 := NewTreeNode(5) - // Constructing reference pointers (i.e. pointers) - n1.Left = n2 - n1.Right = n3 - n2.Left = n4 - n2.Right = n5 - ``` - -=== "Swift" - - ```swift title="binary_tree.swift" - // Initialize nodes - let n1 = TreeNode(x: 1) - let n2 = TreeNode(x: 2) - let n3 = TreeNode(x: 3) - let n4 = TreeNode(x: 4) - let n5 = TreeNode(x: 5) - // Constructing reference pointers (i.e. pointers) - n1.left = n2 - n1.right = n3 - n2.left = n4 - n2.right = n5 - ``` - -=== "JS" - - ```javascript title="binary_tree.js" - /* Initialize the binary tree */ - // Initialize nodes - let n1 = new TreeNode(1), - n2 = new TreeNode(2), - n3 = new TreeNode(3), - n4 = new TreeNode(4), - n5 = new TreeNode(5); - // Constructing reference pointers (i.e. pointers) - n1.left = n2; - n1.right = n3; - n2.left = n4; - n2.right = n5; - ``` - -=== "TS" - - ```typescript title="binary_tree.ts" - /* Initialize the binary tree */ - // Initialize nodes - let n1 = new TreeNode(1), - n2 = new TreeNode(2), - n3 = new TreeNode(3), - n4 = new TreeNode(4), - n5 = new TreeNode(5); - // Constructing reference pointers (i.e. pointers) - n1.left = n2; - n1.right = n3; - n2.left = n4; - n2.right = n5; - ``` - -=== "Dart" - - ```dart title="binary_tree.dart" - /* Initialize the binary tree */ - // Initialize nodes - TreeNode n1 = new TreeNode(1); - TreeNode n2 = new TreeNode(2); - TreeNode n3 = new TreeNode(3); - TreeNode n4 = new TreeNode(4); - TreeNode n5 = new TreeNode(5); - // Constructing reference pointers (i.e. pointers) - n1.left = n2; - n1.right = n3; - n2.left = n4; - n2.right = n5; - ``` - -=== "Rust" - - ```rust title="binary_tree.rs" - // Initialize nodes - let n1 = TreeNode::new(1); - let n2 = TreeNode::new(2); - let n3 = TreeNode::new(3); - let n4 = TreeNode::new(4); - let n5 = TreeNode::new(5); - // Constructing reference pointers (i.e. pointers) - n1.borrow_mut().left = Some(n2.clone()); - n1.borrow_mut().right = Some(n3); - n2.borrow_mut().left = Some(n4); - n2.borrow_mut().right = Some(n5); - ``` - -=== "C" - - ```c title="binary_tree.c" - /* Initialize the binary tree */ - // Initialize nodes - TreeNode *n1 = newTreeNode(1); - TreeNode *n2 = newTreeNode(2); - TreeNode *n3 = newTreeNode(3); - TreeNode *n4 = newTreeNode(4); - TreeNode *n5 = newTreeNode(5); - // Constructing reference pointers (i.e. pointers) - n1->left = n2; - n1->right = n3; - n2->left = n4; - n2->right = n5; - ``` - -=== "Zig" - - ```zig title="binary_tree.zig" - - ``` - -### Insertion And Deletion Of Nodes - -Similar to a linked list, insertion and deletion of nodes in a binary tree can be accomplished by modifying pointers. An example is given in the figure below. - -![Inserting and removing nodes in a binary tree](binary_tree.assets/binary_tree_add_remove.png) - -=== "Python" - - ```python title="binary_tree.py" - # Insertion and deletion of nodes - p = TreeNode(0) - # Insert node P in the middle of n1 -> n2 - n1.left = p - p.left = n2 - # Delete node P - n1.left = n2 - ``` - -=== "C++" - - ```cpp title="binary_tree.cpp" - /* Insertion and deletion of nodes */ - TreeNode* P = new TreeNode(0); - // Insert node P in the middle of n1 -> n2. - n1->left = P; - P->left = n2; - // Delete node P - n1->left = n2; - ``` - -=== "Java" - - ```java title="binary_tree.java" - TreeNode P = new TreeNode(0); - // Insert node P in the middle of n1 -> n2. - n1.left = P; - P.left = n2; - // Delete node P - n1.left = n2; - ``` - -=== "C#" - - ```csharp title="binary_tree.cs" - /* Insertion and deletion of nodes */ - TreeNode P = new(0); - // Insert node P in the middle of n1 -> n2. - n1.left = P; - P.left = n2; - // Delete node P - n1.left = n2; - ``` - -=== "Go" - - ```go title="binary_tree.go" - /* Insertion and deletion of nodes */ - // Insert node P in the middle of n1 -> n2. - p := NewTreeNode(0) - n1.Left = p - p.Left = n2 - // Delete node P - n1.Left = n2 - ``` - -=== "Swift" - - ```swift title="binary_tree.swift" - let P = TreeNode(x: 0) - // Insert node P in the middle of n1 -> n2. - n1.left = P - P.left = n2 - // Delete node P - n1.left = n2 - ``` - -=== "JS" - - ```javascript title="binary_tree.js" - /* Insertion and deletion of nodes */ - let P = new TreeNode(0); - // Insert node P in the middle of n1 -> n2. - n1.left = P; - P.left = n2; - // Delete node P - n1.left = n2; - ``` - -=== "TS" - - ```typescript title="binary_tree.ts" - /* Insertion and deletion of nodes */ - const P = new TreeNode(0); - // Insert node P in the middle of n1 -> n2. - n1.left = P; - P.left = n2; - // Delete node P - n1.left = n2; - ``` - -=== "Dart" - - ```dart title="binary_tree.dart" - /* Insertion and deletion of nodes */ - TreeNode P = new TreeNode(0); - // Insert node P in the middle of n1 -> n2. - n1.left = P; - P.left = n2; - // Delete node P - n1.left = n2; - ``` - -=== "Rust" - - ```rust title="binary_tree.rs" - let p = TreeNode::new(0); - // Insert node P in the middle of n1 -> n2. - n1.borrow_mut().left = Some(p.clone()); - p.borrow_mut().left = Some(n2.clone()); - // Delete node p - n1.borrow_mut().left = Some(n2); - ``` - -=== "C" - - ```c title="binary_tree.c" - /* Insertion and deletion of nodes */ - TreeNode *P = newTreeNode(0); - // Insert node P in the middle of n1 -> n2. - n1->left = P; - P->left = n2; - // Delete node P - n1->left = n2; - ``` - -=== "Zig" - - ```zig title="binary_tree.zig" - - ``` - -!!! note - - It is important to note that inserting a node may change the original logical structure of a binary tree, while deleting a node usually means deleting the node and all its subtrees. Therefore, in a binary tree, insertion and deletion operations are usually done in conjunction with a set of operations to achieve meaningful operations. - -## Common Binary Tree Types - -### Perfect Binary Tree - -The nodes in all levels of a "perfect binary tree" are completely filled. In a perfect binary tree, the degree of the leaf nodes is $0$, and the degree of all the other nodes is $2$; if the height of the tree is $h$, then the total number of nodes is $2^{h+1} - 1$, which shows the standard exponential relationship, reflecting the common phenomenon of cell division in nature. - -!!! tip - - Note that in the Chinese community, a perfect binary tree is often referred to as a "full binary tree". - -![perfect binary tree](binary_tree.assets/perfect_binary_tree.png) - -### Complete Binary Tree - -As shown in the figure below, only the nodes at the bottom level of a complete binary tree are not filled, and the nodes at the bottom level are filled as far to the left as possible. - -![complete binary tree](binary_tree.assets/complete_binary_tree.png) - -### Full Binary Tree - -As shown in the figure below, a "full binary tree full binary tree" has all nodes except leaf nodes with two children. - -![full binary tree](binary_tree.assets/full_binary_tree.png) - -### Balanced Binary Tree - -As shown in the figure below, the absolute value of the difference between the heights of the left-child node and the right subtree of any node in a balanced binary tree does not exceed 1. - -![balanced binary tree](binary_tree.assets/balanced_binary_tree.png) - -## Degeneration Of Binary Trees - -The figure below shows the ideal and degenerate states of a binary tree. A binary tree reaches a "perfect binary tree" when each level of nodes is filled, while a binary tree degenerates into a "linked list" when all nodes are skewed to one side. - -- A perfect binary tree is the ideal case to take full advantage of the "divide and conquer" of binary trees. -- Linked lists are the other extreme, where each operation becomes linear and the time complexity degrades to $O(n)$ . - -![Best and worst structures of binary trees](binary_tree.assets/binary_tree_best_worst_cases.png) - -As shown in the table below, the number of leaf nodes, the total number of nodes, and the height of the binary tree reaches extreme or minimal values under the best and worst structure. - -

      Table   Best and worst case scenarios for binary trees

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zN2wwUf-q_Jywe0>t6J8;xqrU)iHzae$vamF3N>VsoWfYi()-kk)hdu3(y%&a_IIG< zvyk9V-R69umk7_c!-n*b;wYDcvEVwaiZF?4*O2O;jYFk*=T?u@75F{sO7cC}M$}kQ zrHf48xO>3AZ&i`PMq0&VZ+i2d8HEaPpwslo87uVh2eCxEB+s$hzj+nMjiDdTbkIld z3Sb(ff~^##HFv+^(FjYGF^NrK=(i=n6v - -!!! abstract - - The tree is full of life, with its deep roots and branches. - - It shows us the vivid shape of the data divide and conquer. diff --git a/docs-en/chapter_tree/summary.md b/docs-en/chapter_tree/summary.md deleted file mode 100644 index 1d574d6c56..0000000000 --- a/docs-en/chapter_tree/summary.md +++ /dev/null @@ -1,54 +0,0 @@ -# Summary - -### Highlights - -- A binary tree is a non-linear data structure that embodies the logic of divide and conquer. Each binary tree node contains a value and two pointers to its left-child node and right-child node. -- For a node in a binary tree, its left (right) child node and the tree formed below it are called the left (right) child tree of that node. -- Terms related to binary trees include root node, leaf node, level, degree, edge, height and depth. -- Initialization, node insertion and node deletion operations for binary trees are performed in a similar way to linked list operations. -- The common types of binary trees are perfect binary tree, complete binary tree, full binary tree and balanced binary tree. A perfect binary tree is the most desirable state, while a linked list is the worst state after degradation. -- A binary tree can be represented as an array by arranging the node values and empty spaces in a level-order traversal order and implementing pointers based on the index mapping relationship between parent nodes and child nodes. -- Binary tree level-order traversal is a breadth-first search method, which reflects the "circle by circle outward" hierarchical traversal, usually through the queue to realize. -- Pre-order, middle-order, and back-order traversals are all depth-first searches, which embody the "go to the end, and then go back to continue" back traversal method, usually using recursion to achieve. -- A binary search tree is an efficient element-finding data structure with $O(\log n)$ time complexity for find, insert and delete operations. When the binary search tree degenerates into a linked list, the time complexity of each operation deteriorates to $O(n)$ . -- An AVL tree, also known as a balanced binary search tree, ensures that the tree remains balanced after constant insertion and deletion of nodes by means of a rotation operation. -- The rotation operations of AVL tree include right rotation, left rotation, right rotation then left rotation, and left rotation then right rotation. After a node is inserted or deleted, the AVL tree performs a rotation operation from the bottom to the top to rebalance the tree. - -### Q & A - -!!! question "For a binary tree with only one node, is the height of the tree and the depth of the root node both $0$?" - - Yes, because height and depth are usually defined as "the number of edges traveled". - -!!! Question "Insertion and deletion in a binary tree are generally accomplished by a set of operations, what is meant by "a set of operations" here? What does "one set of operations" mean? Can it be interpreted as the resource release of a child node of a resource?" - - Taking a binary search tree as an example, node deletion operations are handled in three cases, each of which requires multiple steps of node operations. - -!!! question "Why does DFS traverse a binary tree in first, middle and last order, and what is the use of each?" - - DFS traversal in the first, middle and last order is similar to the order of accessing an array, and it is the basic method of traversing a binary tree. Using these three traversal methods, we can get the result of traversal in a specific order. For example, in a binary search tree, since the node size satisfies `left-child node value < root node value < right-child node value`, we can obtain an ordered sequence of nodes by traversing the tree according to the priority of `left -> root -> right`. - -!!! question "The right-rotation operation deals with the relationship between the out-of-balance nodes `node`, `child`, and `grand_child`, so doesn't the connection between the parent node of `node` and the original connection of `node` need to be maintained? Wouldn't it be broken after the right-rotation operation?" - - We need to look at this from a recursion perspective. The right-rotate operation `right_rotate(root)` is passed in the root node of the subtree, and eventually `return child` returns the root node of the subtree after the rotation. The connection between the root node of the subtree and its parent node is done after the function returns, and is not part of the maintenance of the right-rotate operation. - -!!! question "In C++, functions are divided into `private` and `public`, is there any consideration for this? Why did you put the `height()` function and the `updateHeight()` function in `public` and `private` respectively?" - - It mainly depends on the scope of use of the method, if the method is only used inside the class, then it is designed as `private`. For example, it makes no sense for the user to call `updateHeight()` alone, it is just a step in the insertion and deletion operations. `height()` accesses the height of a node, similar to `vector.size()`, so it is set to `public` for ease of use. - -!!! Question "How to construct a binary search tree from a set of input data, please? Is the choice of root node important?" - - Yes, the method for building the tree is given in the `build_tree()` method in the binary search tree code. As for the selection of the root node, we usually sort the input data, then use the midpoint element as the root node, and then recursively build the left and right subtrees. Doing so maximizes the balance of the tree. - -!!! question "In Java, is the `equals()` method always used for string comparison?" - - In Java, for basic data types, `==` is used to compare whether the values of two variables are equal. For reference types, the two notations work differently. - - - `==` : Used to compare two variables to see if they point to the same object, i.e. if they are at the same location in memory. - - `equals()`: Used to compare whether the values of two objects are equal. - - So if we want to compare values, we would normally use `equals()` . However, the strings initialized by `String a = "hi"; String b = "hi";` are stored in a pool of string constants that point to the same object, so it is also possible to compare the contents of the two strings with `a == b`. - -!!! question "Is the number of nodes in the queue $2^h$ before breadth-first traversal to the bottom level?" - - Yes, for example, a full binary tree of height $h = 2$ with a total number of nodes $n = 7$ , then the number of nodes at the bottom level $4 = 2^h = (n + 1) / 2$ . diff --git a/mkdocs-en.yml b/mkdocs-en.yml index 5ab4de665d..2f72d3028b 100644 --- a/mkdocs-en.yml +++ b/mkdocs-en.yml @@ -44,121 +44,121 @@ nav: - 2.3 Time Complexity: chapter_computational_complexity/time_complexity.md - 2.4 Space Complexity: chapter_computational_complexity/space_complexity.md - 2.5 Summary: chapter_computational_complexity/summary.md - - Chapter 3. Data Structure: - # [icon: material/shape-outline] - - chapter_data_structure/index.md - - 3.1 Classification of Data Structure: chapter_data_structure/classification_of_data_structure.md - - 3.2 Basic Data Types: chapter_data_structure/basic_data_types.md - - 3.3 Number Encoding *: chapter_data_structure/number_encoding.md - - 3.4 Character Encoding *: chapter_data_structure/character_encoding.md - - 3.5 Summary: chapter_data_structure/summary.md - - Chapter 4. Array and Linked List: - # [icon: material/view-list-outline] - - chapter_array_and_linkedlist/index.md - - 4.1 Array: chapter_array_and_linkedlist/array.md - - 4.2 Linked List: chapter_array_and_linkedlist/linked_list.md - - 4.3 List: chapter_array_and_linkedlist/list.md - - 4.4 Summary: chapter_array_and_linkedlist/summary.md - - Chapter 5. Stack and Queue: - # [icon: material/stack-overflow] - - chapter_stack_and_queue/index.md - - 5.1 Stack: chapter_stack_and_queue/stack.md - - 5.2 Queue: chapter_stack_and_queue/queue.md - - 5.3 Double-ended Queue: chapter_stack_and_queue/deque.md - - 5.4 Summary: chapter_stack_and_queue/summary.md - - Chapter 6. Hash Table: - # [icon: material/table-search] - - chapter_hashing/index.md - - 6.1 Hash Table: chapter_hashing/hash_map.md - - 6.2 Hash Collision: chapter_hashing/hash_collision.md - - 6.3 Hash Algorithm: chapter_hashing/hash_algorithm.md - - 6.4 Summary: chapter_hashing/summary.md - - Chapter 7. Tree: - # [icon: material/graph-outline] - - chapter_tree/index.md - - 7.1 Binary Tree: chapter_tree/binary_tree.md - - 7.2 Binary Tree Traversal: chapter_tree/binary_tree_traversal.md - - 7.3 Array Representation of Tree: chapter_tree/array_representation_of_tree.md - - 7.4 Binary Search Tree: chapter_tree/binary_search_tree.md - - 7.5 AVL Tree *: chapter_tree/avl_tree.md - - 7.6 Summary: chapter_tree/summary.md - - Chapter 8. Heap: - # [icon: material/family-tree] - - chapter_heap/index.md - - 8.1 Heap: chapter_heap/heap.md - - 8.2 Building a Heap: chapter_heap/build_heap.md - - 8.3 Top-K Problem: chapter_heap/top_k.md - - 8.4 Summary: chapter_heap/summary.md - - Chapter 9. Graph: - # [icon: material/graphql] - - chapter_graph/index.md - - 9.1 Graph: chapter_graph/graph.md - - 9.2 Basic Graph Operations: chapter_graph/graph_operations.md - - 9.3 Graph Traversal: chapter_graph/graph_traversal.md - - 9.4 Summary: chapter_graph/summary.md - - Chapter 10. Searching: - # [icon: material/text-search] - - chapter_searching/index.md - - 10.1 Binary Search: chapter_searching/binary_search.md - - 10.2 Binary Search Insertion Point: chapter_searching/binary_search_insertion.md - - 10.3 Binary Search Boundaries: chapter_searching/binary_search_edge.md - - 10.4 Hashing Optimization Strategy: chapter_searching/replace_linear_by_hashing.md - - 10.5 Revisiting Search Algorithms: chapter_searching/searching_algorithm_revisited.md - - 10.6 Summary: chapter_searching/summary.md - - Chapter 11. Sorting: - # [icon: material/sort-ascending] - - chapter_sorting/index.md - - 11.1 Sorting Algorithms: chapter_sorting/sorting_algorithm.md - - 11.2 Selection Sort: chapter_sorting/selection_sort.md - - 11.3 Bubble Sort: chapter_sorting/bubble_sort.md - - 11.4 Insertion Sort: chapter_sorting/insertion_sort.md - - 11.5 Quick Sort: chapter_sorting/quick_sort.md - - 11.6 Merge Sort: chapter_sorting/merge_sort.md - - 11.7 Heap Sort: chapter_sorting/heap_sort.md - - 11.8 Bucket Sort: chapter_sorting/bucket_sort.md - - 11.9 Counting Sort: chapter_sorting/counting_sort.md - - 11.10 Radix Sort: chapter_sorting/radix_sort.md - - 11.11 Summary: chapter_sorting/summary.md - - Chapter 12. Divide and Conquer: - # [icon: material/set-split] - - chapter_divide_and_conquer/index.md - - 12.1 Divide and Conquer Algorithm: chapter_divide_and_conquer/divide_and_conquer.md - - 12.2 Divide and Conquer Search Strategy: chapter_divide_and_conquer/binary_search_recur.md - - 12.3 Building Tree Problems: chapter_divide_and_conquer/build_binary_tree_problem.md - - 12.4 Tower of Hanoi Problem: chapter_divide_and_conquer/hanota_problem.md - - 12.5 Summary: chapter_divide_and_conquer/summary.md - - Chapter 13. Backtracking: - # [icon: material/map-marker-path] - - chapter_backtracking/index.md - - 13.1 Backtracking Algorithm: chapter_backtracking/backtracking_algorithm.md - - 13.2 Permutations Problem: chapter_backtracking/permutations_problem.md - - 13.3 Subset Sum Problem: chapter_backtracking/subset_sum_problem.md - - 13.4 N-Queens Problem: chapter_backtracking/n_queens_problem.md - - 13.5 Summary: chapter_backtracking/summary.md - - Chapter 14. Dynamic Programming: - # [icon: material/table-pivot] - - chapter_dynamic_programming/index.md - - 14.1 Introduction to Dynamic Programming: chapter_dynamic_programming/intro_to_dynamic_programming.md - - 14.2 Features of DP Problems: chapter_dynamic_programming/dp_problem_features.md - - 14.3 DP Solution Approach: chapter_dynamic_programming/dp_solution_pipeline.md - - 14.4 0-1 Knapsack Problem: chapter_dynamic_programming/knapsack_problem.md - - 14.5 Unbounded Knapsack Problem: chapter_dynamic_programming/unbounded_knapsack_problem.md - - 14.6 Edit Distance Problem: chapter_dynamic_programming/edit_distance_problem.md - - 14.7 Summary: chapter_dynamic_programming/summary.md - - Chapter 15. Greedy: - # [icon: material/head-heart-outline] - - chapter_greedy/index.md - - 15.1 Greedy Algorithm: chapter_greedy/greedy_algorithm.md - - 15.2 Fractional Knapsack Problem: chapter_greedy/fractional_knapsack_problem.md - - 15.3 Maximum Capacity Problem: chapter_greedy/max_capacity_problem.md - - 15.4 Maximum Product Cutting Problem: chapter_greedy/max_product_cutting_problem.md - - 15.5 Summary: chapter_greedy/summary.md - - Chapter 16. Appendix: - # [icon: material/help-circle-outline] - - chapter_appendix/index.md - - 16.1 Installation of Programming Environment: chapter_appendix/installation.md - - 16.2 Contributing to the Creation: chapter_appendix/contribution.md - # [status: new] - - 16.3   Terminology: chapter_appendix/terminology.md + # - Chapter 3. Data Structure: + # # [icon: material/shape-outline] + # - chapter_data_structure/index.md + # - 3.1 Classification of Data Structure: chapter_data_structure/classification_of_data_structure.md + # - 3.2 Basic Data Types: chapter_data_structure/basic_data_types.md + # - 3.3 Number Encoding *: chapter_data_structure/number_encoding.md + # - 3.4 Character Encoding *: chapter_data_structure/character_encoding.md + # - 3.5 Summary: chapter_data_structure/summary.md + # - Chapter 4. Array and Linked List: + # # [icon: material/view-list-outline] + # - chapter_array_and_linkedlist/index.md + # - 4.1 Array: chapter_array_and_linkedlist/array.md + # - 4.2 Linked List: chapter_array_and_linkedlist/linked_list.md + # - 4.3 List: chapter_array_and_linkedlist/list.md + # - 4.4 Summary: chapter_array_and_linkedlist/summary.md + # - Chapter 5. Stack and Queue: + # # [icon: material/stack-overflow] + # - chapter_stack_and_queue/index.md + # - 5.1 Stack: chapter_stack_and_queue/stack.md + # - 5.2 Queue: chapter_stack_and_queue/queue.md + # - 5.3 Double-ended Queue: chapter_stack_and_queue/deque.md + # - 5.4 Summary: chapter_stack_and_queue/summary.md + # - Chapter 6. Hash Table: + # # [icon: material/table-search] + # - chapter_hashing/index.md + # - 6.1 Hash Table: chapter_hashing/hash_map.md + # - 6.2 Hash Collision: chapter_hashing/hash_collision.md + # - 6.3 Hash Algorithm: chapter_hashing/hash_algorithm.md + # - 6.4 Summary: chapter_hashing/summary.md + # - Chapter 7. Tree: + # # [icon: material/graph-outline] + # - chapter_tree/index.md + # - 7.1 Binary Tree: chapter_tree/binary_tree.md + # - 7.2 Binary Tree Traversal: chapter_tree/binary_tree_traversal.md + # - 7.3 Array Representation of Tree: chapter_tree/array_representation_of_tree.md + # - 7.4 Binary Search Tree: chapter_tree/binary_search_tree.md + # - 7.5 AVL Tree *: chapter_tree/avl_tree.md + # - 7.6 Summary: chapter_tree/summary.md + # - Chapter 8. Heap: + # # [icon: material/family-tree] + # - chapter_heap/index.md + # - 8.1 Heap: chapter_heap/heap.md + # - 8.2 Building a Heap: chapter_heap/build_heap.md + # - 8.3 Top-K Problem: chapter_heap/top_k.md + # - 8.4 Summary: chapter_heap/summary.md + # - Chapter 9. Graph: + # # [icon: material/graphql] + # - chapter_graph/index.md + # - 9.1 Graph: chapter_graph/graph.md + # - 9.2 Basic Graph Operations: chapter_graph/graph_operations.md + # - 9.3 Graph Traversal: chapter_graph/graph_traversal.md + # - 9.4 Summary: chapter_graph/summary.md + # - Chapter 10. Searching: + # # [icon: material/text-search] + # - chapter_searching/index.md + # - 10.1 Binary Search: chapter_searching/binary_search.md + # - 10.2 Binary Search Insertion Point: chapter_searching/binary_search_insertion.md + # - 10.3 Binary Search Boundaries: chapter_searching/binary_search_edge.md + # - 10.4 Hashing Optimization Strategy: chapter_searching/replace_linear_by_hashing.md + # - 10.5 Revisiting Search Algorithms: chapter_searching/searching_algorithm_revisited.md + # - 10.6 Summary: chapter_searching/summary.md + # - Chapter 11. Sorting: + # # [icon: material/sort-ascending] + # - chapter_sorting/index.md + # - 11.1 Sorting Algorithms: chapter_sorting/sorting_algorithm.md + # - 11.2 Selection Sort: chapter_sorting/selection_sort.md + # - 11.3 Bubble Sort: chapter_sorting/bubble_sort.md + # - 11.4 Insertion Sort: chapter_sorting/insertion_sort.md + # - 11.5 Quick Sort: chapter_sorting/quick_sort.md + # - 11.6 Merge Sort: chapter_sorting/merge_sort.md + # - 11.7 Heap Sort: chapter_sorting/heap_sort.md + # - 11.8 Bucket Sort: chapter_sorting/bucket_sort.md + # - 11.9 Counting Sort: chapter_sorting/counting_sort.md + # - 11.10 Radix Sort: chapter_sorting/radix_sort.md + # - 11.11 Summary: chapter_sorting/summary.md + # - Chapter 12. Divide and Conquer: + # # [icon: material/set-split] + # - chapter_divide_and_conquer/index.md + # - 12.1 Divide and Conquer Algorithm: chapter_divide_and_conquer/divide_and_conquer.md + # - 12.2 Divide and Conquer Search Strategy: chapter_divide_and_conquer/binary_search_recur.md + # - 12.3 Building Tree Problems: chapter_divide_and_conquer/build_binary_tree_problem.md + # - 12.4 Tower of Hanoi Problem: chapter_divide_and_conquer/hanota_problem.md + # - 12.5 Summary: chapter_divide_and_conquer/summary.md + # - Chapter 13. Backtracking: + # # [icon: material/map-marker-path] + # - chapter_backtracking/index.md + # - 13.1 Backtracking Algorithm: chapter_backtracking/backtracking_algorithm.md + # - 13.2 Permutations Problem: chapter_backtracking/permutations_problem.md + # - 13.3 Subset Sum Problem: chapter_backtracking/subset_sum_problem.md + # - 13.4 N-Queens Problem: chapter_backtracking/n_queens_problem.md + # - 13.5 Summary: chapter_backtracking/summary.md + # - Chapter 14. Dynamic Programming: + # # [icon: material/table-pivot] + # - chapter_dynamic_programming/index.md + # - 14.1 Introduction to Dynamic Programming: chapter_dynamic_programming/intro_to_dynamic_programming.md + # - 14.2 Features of DP Problems: chapter_dynamic_programming/dp_problem_features.md + # - 14.3 DP Solution Approach: chapter_dynamic_programming/dp_solution_pipeline.md + # - 14.4 0-1 Knapsack Problem: chapter_dynamic_programming/knapsack_problem.md + # - 14.5 Unbounded Knapsack Problem: chapter_dynamic_programming/unbounded_knapsack_problem.md + # - 14.6 Edit Distance Problem: chapter_dynamic_programming/edit_distance_problem.md + # - 14.7 Summary: chapter_dynamic_programming/summary.md + # - Chapter 15. Greedy: + # # [icon: material/head-heart-outline] + # - chapter_greedy/index.md + # - 15.1 Greedy Algorithm: chapter_greedy/greedy_algorithm.md + # - 15.2 Fractional Knapsack Problem: chapter_greedy/fractional_knapsack_problem.md + # - 15.3 Maximum Capacity Problem: chapter_greedy/max_capacity_problem.md + # - 15.4 Maximum Product Cutting Problem: chapter_greedy/max_product_cutting_problem.md + # - 15.5 Summary: chapter_greedy/summary.md + # - Chapter 16. Appendix: + # # [icon: material/help-circle-outline] + # - chapter_appendix/index.md + # - 16.1 Installation of Programming Environment: chapter_appendix/installation.md + # - 16.2 Contributing to the Creation: chapter_appendix/contribution.md + # # [status: new] + # - 16.3   Terminology: chapter_appendix/terminology.md - References: - chapter_reference/index.md From 7dcb3281d4e028e64cf29a495024f3ff68c27e6e Mon Sep 17 00:00:00 2001 From: krahets Date: Thu, 21 Dec 2023 23:19:44 +0800 Subject: [PATCH 24/30] Update buttons --- README-en.md | 18 ++++++++++-------- docs-en/index.assets/btn_chinese_edition.svg | 1 + .../index.assets/btn_chinese_edition_dark.svg | 1 + docs-en/index.assets/btn_download_pdf.svg | 1 + .../index.assets/btn_download_pdf_dark.svg | 2 +- docs-en/index.assets/btn_read_online.svg | 1 + docs-en/index.assets/btn_read_online_dark.svg | 2 +- docs/index.assets/btn_download_code_dark.png | Bin 2878 -> 0 bytes docs/index.assets/btn_download_code_light.png | Bin 3165 -> 0 bytes docs/index.assets/btn_download_pdf.svg | 1 + docs/index.assets/btn_download_pdf_dark.png | Bin 2283 -> 0 bytes docs/index.assets/btn_download_pdf_dark.svg | 1 + docs/index.assets/btn_download_pdf_light.png | Bin 2526 -> 0 bytes docs/index.assets/btn_english_edition.svg | 1 + .../index.assets/btn_english_edition_dark.svg | 1 + docs/index.assets/btn_read_online.svg | 1 + docs/index.assets/btn_read_online_dark.png | Bin 3284 -> 0 bytes docs/index.assets/btn_read_online_dark.svg | 1 + docs/index.assets/btn_read_online_light.png | Bin 4061 -> 0 bytes 19 files changed, 22 insertions(+), 10 deletions(-) create mode 100644 docs-en/index.assets/btn_chinese_edition.svg create mode 100644 docs-en/index.assets/btn_chinese_edition_dark.svg create mode 100644 docs-en/index.assets/btn_download_pdf.svg create mode 100644 docs-en/index.assets/btn_read_online.svg delete mode 100644 docs/index.assets/btn_download_code_dark.png delete mode 100644 docs/index.assets/btn_download_code_light.png create mode 100644 docs/index.assets/btn_download_pdf.svg delete mode 100644 docs/index.assets/btn_download_pdf_dark.png create mode 100644 docs/index.assets/btn_download_pdf_dark.svg delete mode 100644 docs/index.assets/btn_download_pdf_light.png create mode 100644 docs/index.assets/btn_english_edition.svg create mode 100644 docs/index.assets/btn_english_edition_dark.svg create mode 100644 docs/index.assets/btn_read_online.svg delete mode 100644 docs/index.assets/btn_read_online_dark.png create mode 100644 docs/index.assets/btn_read_online_dark.svg delete mode 100644 docs/index.assets/btn_read_online_light.png diff --git a/README-en.md b/README-en.md index d2e08210c1..2e942dca8a 100644 --- a/README-en.md +++ b/README-en.md @@ -1,8 +1,8 @@

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      @@ -68,7 +70,7 @@ If you find this book helpful, please give it a Star :star: to support us, thank This open-source book is continuously being updated, and we welcome your participation in this project to provide better learning content for readers. -- [Content Correction](https://www.hello-algo.com/chapter_appendix/contribution/): Please help us correct or point out mistakes in the comments section such as grammatical errors, missing content, ambiguities, invalid links, or code bugs. +- [Content Correction](https://www.hello-algo.com/en/chapter_appendix/contribution/): Please help us correct or point out mistakes in the comments section such as grammatical errors, missing content, ambiguities, invalid links, or code bugs. - [Code Transpilation](https://github.com/krahets/hello-algo/issues/15): We look forward to your contributions in various programming languages. 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--upgrade pip -RUN pip install -i https://pypi.tuna.tsinghua.edu.cn/simple mkdocs-material==9.4.1 mkdocs-glightbox +ENV PIP_INDEX_URL=https://pypi.tuna.tsinghua.edu.cn/simple +RUN pip install --upgrade pip +RUN pip install mkdocs-material==9.4.1 mkdocs-glightbox -WORKDIR /app +WORKDIR /hello-algo -COPY overrides /app/build/overrides -COPY docs /app/build/docs -COPY docs-en /app/build/docs-en -COPY mkdocs.yml /app/mkdocs.yml -COPY mkdocs-en.yml /app/mkdocs-en.yml +COPY overrides ./build/overrides +COPY docs ./build/docs +COPY docs-en ./build/docs-en +COPY mkdocs.yml ./mkdocs.yml +COPY mkdocs-en.yml ./mkdocs-en.yml RUN mkdocs build -f mkdocs.yml RUN mkdocs build -f mkdocs-en.yml From 108b909eaf57fdb9c1e91d1844cad61aa7de07a7 Mon Sep 17 00:00:00 2001 From: krahets Date: Thu, 21 Dec 2023 23:51:17 +0800 Subject: [PATCH 26/30] Update index.md --- README-en.md | 2 +- docs-en/index.md | 187 +++++++++++++++++++++++++++++------------------ 2 files changed, 118 insertions(+), 71 deletions(-) diff --git a/README-en.md b/README-en.md index 2e942dca8a..2f192cd223 100644 --- a/README-en.md +++ b/README-en.md @@ -62,7 +62,7 @@ If you find this book helpful, please give it a Star :star: to support us, thank > > **—— Junhui Deng, Professor of Computer Science, Tsinghua University** -> "If I had *Hello Algo* when I was learning data structures and algorithms, it would have been 10 times easier!" +> "If I had 'Hello Algo' when I was learning data structures and algorithms, it would have been 10 times easier!" > > **—— Mu Li, Senior Principal Scientist, Amazon** diff --git a/docs-en/index.md b/docs-en/index.md index b39090eec9..349df8daf2 100644 --- a/docs-en/index.md +++ b/docs-en/index.md @@ -3,22 +3,26 @@ comments: true glightbox: false hide: - footer + - toc + - edit --- -

      - -

      - - -

      +
      + + +

      Hello Algo

      -

      Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code

      +

      Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code

      - GitHub Repo stars + GitHub repo stars + +   + + GitHub all releases   @@ -26,101 +30,144 @@ hide:

      ---- - -

      Praise

      - -!!! quote - - "An accessible and easy-to-understand introduction to data structures and algorithms, guiding readers to learn actively and hands-on. Highly recommended for beginners in algorithms." - - **— Junhui Deng, Professor of Computer Science at Tsinghua University** - -!!! quote - - "If I'd had 'Hello Algo' back when I was learning Data Structures and Algorithms, it would have been 10 times easier to learn!" +

      + + + Dive In + + + + Clone Repo + + + + Get PDF + +

      - **— Mu Li, Senior Principal Scientist at Amazon** +

      + + + + + + + + + + + + + +

      --- -

      Animated illustrations throughout the book

      - -

      Clear and easy-to-understand content with a smooth learning curve
      Computer, tablet, and cell phone all-terminal reading

      - -![algorithm_animation](index.assets/animation.gif) - -!!! quote "" - -

      "A picture is worth a thousand words."

      +

      Endorsements

      + +
      +
      +

      Quote

      +

      "An easy-to-understand introductory book on data structures and algorithms, which guides readers to learn by minds-on and hands-on. Strongly recommended for algorithm beginners!"

      +

      —— Junhui Deng, Professor of Computer Science, Tsinghua University

      +
      +
      +

      Quote

      +

      "If I had 'Hello Algo' when I was learning data structures and algorithms, it would have been 10 times easier!"

      +

      —— Mu Li, Senior Principal Scientist, Amazon

      +
      +
      --- -

      Code Runs in One Click

      - -

      Provides concise implementations and test samples of algorithms and data structures that are ready to run
      Support for Java, C++, Python, Go, JS, TS, C#, Swift, Zig and more

      . - -![running_code](index.assets/running_code.gif) - -!!! quote "" - -

      "Talk is cheap. Show me the code."

      - ---- +
      +
      +
      +

      Animated illustrations

      + +
      +

      Easy to understand
      Smooth learning curve

      +
      + +
      -

      Possibility of discussion and questions

      +
      +

      "A picture is worth a thousand words."

      +
      -

      Encourage your friends to help each other grow
      Questions and comments are generally answered within two days

      . +
      + +
      +
      +

      Off-the-Shelf Code

      + +
      +

      Multi programming languages
      Run with one click

      +
      +
      -![comment](index.assets/comment.gif) +
      +

      "Talk is cheap. Show me the code."

      +
      -!!! quote "" +
      +
      +
      +

      Discussion

      + +
      +

      Discussion and questions welcome
      Readers progress together

      +
      + +
      -

      "Chase the wind and catch the moon and don't stop, the flat barren place is the spring mountains"

      -

      Cheer together!

      +
      +

      "Chase the wind and moon, never stopping"

      +

      "Beyond the plains, there are spring mountains"

      +
      --- -

      Procedure

      +

      Preface

      + +Two years ago, I shared the "Sword Offer" series of problem solutions on LeetCode, which received much love and support from many students. During my interactions with readers, the most common question I encountered was "How to get started with algorithms." Gradually, I developed a deep interest in this question. -Two years ago, I shared the "Sword to Offer" series on LeetCode, which was loved and supported by many students. During the communication with readers, one of the most common questions I received was "how to start learning algorithms". I have gradually developed a strong interest in this question. +Blindly solving problems seems to be the most popular method, being simple, direct, and effective. However, problem-solving is like playing a "Minesweeper" game, where students with strong self-learning abilities can successfully clear the mines one by one, but those with insufficient foundations may end up bruised from explosions, retreating step by step in frustration. Thoroughly reading textbooks is also common, but for students aiming for job applications, the energy consumed by graduation, resume submissions, and preparing for written tests and interviews makes tackling thick books a daunting challenge. -Brushing up on problems with a blank slate seems to be the most popular method, which is simple, direct and effective. Brushing questions is like playing the game of "minesweeper", students with strong self-learning ability can successfully remove the mines one by one, while students with insufficient foundation are likely to be blown up all over their heads and retreat in frustration. Reading through textbooks and books is also a common practice, but for job-seeking students, the graduation season, submitting resumes, and preparing for written tests and interviews have taken up most of their energy, and heavy books often become a daunting challenge. +If you are facing similar troubles, then you are lucky to have found this book. This book is my answer to this question, not necessarily the best solution, but at least an active attempt. Although this book won't directly land you an Offer, it will guide you through the "knowledge map" of data structures and algorithms, help you understand the shape, size, and distribution of different "mines," and equip you with various "demining methods." With these skills, I believe you can more comfortably solve problems and read literature, gradually building a complete knowledge system. -If you are facing similar struggles, then luckily this book has found you. This book is my answer to this problem, if not the optimal solution, at least a positive attempt. Although this book is not enough for you to get an offer directly, it will guide you to explore the "knowledge map" of data structures and algorithms, bring you to understand the shapes, sizes, and distributions of different "landmines", and let you master various "demining methods". demining methods". With these skills, I believe you can be more comfortable to deal with the questions and read the literature, and gradually build up a complete knowledge system. +I deeply agree with Professor Feynman's saying: "Knowledge isn't free. You have to pay attention." In this sense, this book is not entirely "free." To not disappoint the precious "attention" you pay to this book, I will do my utmost, investing the greatest "attention" to complete the creation of this book. -

      The Author

      +

      Author

      -Yudong Jin([Krahets](https://leetcode.cn/u/jyd/)), Senior Algorithmic Engineer of Big Factory, Master of Shanghai Jiaotong University. LeetCode network reads the highest number of bloggers, published by the ["Diagramming of Algorithmic Data Structures"](https://leetcode.cn/leetbook/detail/illustration-of-algorithm/) has been subscribed to 270,000 books. +Yudong Jin([Krahets](https://leetcode.cn/u/jyd/)), Senior Algorithm Engineer in a top tech company, Master's degree from Shanghai Jiao Tong University. The highest-read blogger across the entire LeetCode, his published ["Illustration of Algorithm Data Structures"](https://leetcode.cn/leetbook/detail/illustration-of-algorithm/) has been subscribed to by over 300k. --- -

      Acknowledgments

      +

      Contribution

      -This book has grown through the combined efforts of many contributors in the open source community. Thank you to each of the contributors who invested their time and effort to make this book better through their selfless contributions, they are (in order of GitHub auto-generation): +This book is continuously improved with the joint efforts of many contributors from the open-source community. Thanks to each writer who invested their time and energy, listed in the order generated by GitHub:

      - +

      -The code in this book were reviewed by Gonglja, gvenusleo, justin-tse, krahets, night-cruise, nuomi1, Reanon, sjinzh (in alphabetical order). We thank them for their time and effort in ensuring that the code is standardized and consistent across languages. +The code review work for this book was completed by Gonglja, gvenusleo, hpstory, justin‐tse, krahets, night-cruise, nuomi1, Reanon, and sjinzh (listed in alphabetical order). Thanks to them for their time and effort, ensuring the standardization and uniformity of the code in various languages.
      - +
      - - - - - - - - - + + + + + + + + +
      Gonglja
      Gonglja

      C, C++
      gvenusleo
      gvenusleo

      Dart
      hpstory
      hpstory

      C#
      justin-tse
      justin-tse

      JS, TS
      krahets
      krahets

      Java, Python
      night-cruise
      night-cruise

      Rust
      nuomi1
      nuomi1

      Swift
      Reanon
      Reanon

      Go, C
      sjinzh
      sjinzh

      Rust, Zig
      Gonglja
      Gonglja

      C, C++
      gvenusleo
      gvenusleo

      Dart
      hpstory
      hpstory

      C#
      justin-tse
      justin-tse

      JS, TS
      krahets
      krahets

      Java, Python
      night-cruise
      night-cruise

      Rust
      nuomi1
      nuomi1

      Swift
      Reanon
      Reanon

      Go, C
      sjinzh
      sjinzh

      Rust, Zig
      - ---- From cda57c5985e442e1366e4a10e7f1ee097dbcb80e Mon Sep 17 00:00:00 2001 From: krahets Date: Thu, 21 Dec 2023 23:59:32 +0800 Subject: [PATCH 27/30] Update index.md --- docs-en/index.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/docs-en/index.md b/docs-en/index.md index 349df8daf2..305ff4c89c 100644 --- a/docs-en/index.md +++ b/docs-en/index.md @@ -31,7 +31,7 @@ hide:

      - + Dive In @@ -68,7 +68,7 @@ hide:

      Quote

      -

      "An easy-to-understand introductory book on data structures and algorithms, which guides readers to learn by minds-on and hands-on. Strongly recommended for algorithm beginners!"

      +

      "An easy-to-understand book on data structures and algorithms, which guides readers to learn by minds-on and hands-on. Strongly recommended for algorithm beginners!"

      —— Junhui Deng, Professor of Computer Science, Tsinghua University

      @@ -113,7 +113,7 @@ hide:
      -

      Discussion

      +

      Learning Together

      Discussion and questions welcome
      Readers progress together

      From f241711ebbb60dfe989091430ca44b8535a85244 Mon Sep 17 00:00:00 2001 From: krahets Date: Fri, 22 Dec 2023 00:05:40 +0800 Subject: [PATCH 28/30] Update README --- README-en.md | 10 ++++------ README.md | 2 ++ 2 files changed, 6 insertions(+), 6 deletions(-) diff --git a/README-en.md b/README-en.md index 2f192cd223..684946889b 100644 --- a/README-en.md +++ b/README-en.md @@ -1,5 +1,3 @@ -

      English | 中文

      -

      @@ -14,11 +12,11 @@

      - - - + - + + +

      diff --git a/README.md b/README.md index 1d5792f76a..55d8b35a31 100644 --- a/README.md +++ b/README.md @@ -14,6 +14,8 @@ + +

      From ad9ab298dee13ff549b10ec7d65af4c5c96f786a Mon Sep 17 00:00:00 2001 From: krahets Date: Fri, 22 Dec 2023 00:08:40 +0800 Subject: [PATCH 29/30] Fix index.md --- docs-en/chapter_computational_complexity/index.md | 2 +- docs-en/chapter_data_structure/index.md | 2 +- docs-en/chapter_introduction/index.md | 2 +- docs-en/chapter_preface/index.md | 2 +- 4 files changed, 4 insertions(+), 4 deletions(-) diff --git a/docs-en/chapter_computational_complexity/index.md b/docs-en/chapter_computational_complexity/index.md index ce38a2b4d6..501564c6af 100644 --- a/docs-en/chapter_computational_complexity/index.md +++ b/docs-en/chapter_computational_complexity/index.md @@ -2,7 +2,7 @@

      -![complexity_analysis](../assets/covers/chapter_complexity_analysis.jpg){ width="600" } +![complexity_analysis](../assets/covers/chapter_complexity_analysis.jpg)
      diff --git a/docs-en/chapter_data_structure/index.md b/docs-en/chapter_data_structure/index.md index 7a655a2c86..9d332b8231 100644 --- a/docs-en/chapter_data_structure/index.md +++ b/docs-en/chapter_data_structure/index.md @@ -2,7 +2,7 @@
      -![data structure](../assets/covers/chapter_data_structure.jpg){ width="600" } +![data structure](../assets/covers/chapter_data_structure.jpg)
      diff --git a/docs-en/chapter_introduction/index.md b/docs-en/chapter_introduction/index.md index 2f61cea8c4..e85a23b368 100644 --- a/docs-en/chapter_introduction/index.md +++ b/docs-en/chapter_introduction/index.md @@ -2,7 +2,7 @@
      -![A first look at the algorithm](../assets/covers/chapter_introduction.jpg){ width="600" } +![A first look at the algorithm](../assets/covers/chapter_introduction.jpg)
      diff --git a/docs-en/chapter_preface/index.md b/docs-en/chapter_preface/index.md index 9a6b66aa6e..bfe6ffda7d 100644 --- a/docs-en/chapter_preface/index.md +++ b/docs-en/chapter_preface/index.md @@ -2,7 +2,7 @@
      -![Preface](../assets/covers/chapter_preface.jpg){ width="600" } +![Preface](../assets/covers/chapter_preface.jpg)
      From bcc836fc6d9a33797119c23427570bd8ff9367eb Mon Sep 17 00:00:00 2001 From: krahets Date: Fri, 22 Dec 2023 00:14:15 +0800 Subject: [PATCH 30/30] Fix mkdocs-en.yml --- README-en.md | 2 +- mkdocs-en.yml | 4 ++-- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/README-en.md b/README-en.md index 684946889b..b977a98fbb 100644 --- a/README-en.md +++ b/README-en.md @@ -56,7 +56,7 @@ If you find this book helpful, please give it a Star :star: to support us, thank ## Endorsements -> "An easy-to-understand introductory book on data structures and algorithms, which guides readers to learn by minds-on and hands-on. Strongly recommended for algorithm beginners!" +> "An easy-to-understand book on data structures and algorithms, which guides readers to learn by minds-on and hands-on. Strongly recommended for algorithm beginners!" > > **—— Junhui Deng, Professor of Computer Science, Tsinghua University** diff --git a/mkdocs-en.yml b/mkdocs-en.yml index 2f72d3028b..1dc3d6f549 100644 --- a/mkdocs-en.yml +++ b/mkdocs-en.yml @@ -160,5 +160,5 @@ nav: # - 16.2 Contributing to the Creation: chapter_appendix/contribution.md # # [status: new] # - 16.3   Terminology: chapter_appendix/terminology.md - - References: - - chapter_reference/index.md + # - References: + # - chapter_reference/index.md

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a/docs-en/chapter_tree/binary_tree_traversal.md +++ /dev/null @@ -1,89 +0,0 @@ -# Binary Tree Traversal - -From a physical structure point of view, a tree is a data structure based on a linked list, and therefore its traversal is done by accessing the nodes one by one through pointers. However, a tree is a nonlinear data structure, which makes traversing a tree more complex than traversing a linked list, and needs to be implemented with the help of a searching algorithm. - -Common traversals of binary trees include level-order traversal, preorder traversal, middle-order traversal, and postorder traversal. - -## Level-Order Traversal - -As shown in the figure below, "level-order traversal level-order traversal" traverses the binary tree level by level from the top to the bottom and visits the nodes at each level in the order from left to right. - -Level-order traversal essentially belongs to the "breadth-first traversal breadth-first traversal", which embodies a "circle by circle outward" layer-by-layer traversal. - -![Level-order traversal of binary trees](binary_tree_traversal.assets/binary_tree_bfs.png) - -### Code Implementation - -Breadth-first traversal is usually implemented with the help of a "queue". While queues follow the "first in, first out" rule, breadth-first traversals follow the "progress by level" rule, but the idea behind both is the same. - -```src -[file]{binary_tree_bfs}-[class]{}-[func]{level_order} -``` - -### Complexity Analysis - -- **Time complexity $O(n)$** : All nodes are visited once, using $O(n)$ time, where $n$ is the number of nodes. -- **Space complexity $O(n)$** : In the worst case, i.e., a full binary tree, there are at most $(n + 1) / 2$ nodes in the queue at the same time before traversing to the bottom level, occupying $O(n)$ space. - -## Pre-Order, Middle-Order, And Back-Order Traversal - -Accordingly, preorder, midorder and postorder traversal are all "depth-first traversal depth-first traversal", which embodies a traversal method of "going to the end first and then backtracking to continue". - -The figure below shows how depth-first traversal of a binary tree works. **Depth-first traversal is like "walking" around the entire perimeter of the binary tree**, encountering three positions at each node, corresponding to a preorder traversal, a middle-order traversal, and a postorder traversal. - -![Pre-, mid-, and postorder traversal of a binary search tree](binary_tree_traversal.assets/binary_tree_dfs.png) - -### Code Implementation - -Depth-first search is usually implemented based on recursion: - -```src -[file]{binary_tree_dfs}-[class]{}-[func]{post_order} -``` - -!!! note - - Depth-first search can also be implemented based on iteration, so interested students can do their own research. - -The figure below illustrates the recursion of a forward traversal of a binary tree, which can be divided into two inverse parts, "recursion" and "recursion". - -1. "Pass" indicates that a new method is opened and the program visits the next node in the process. -2. "Return" means that the function returns and the current node has been visited. - -=== "<1>" - ![Recursion process for preorder traversal](binary_tree_traversal.assets/preorder_step1.png) - -=== "<2>" - ![preorder_step2](binary_tree_traversal.assets/preorder_step2.png) - -=== "<3>" - ![preorder_step3](binary_tree_traversal.assets/preorder_step3.png) - -=== "<4>" - ![preorder_step4](binary_tree_traversal.assets/preorder_step4.png) - -=== "<5>" - ![preorder_step5](binary_tree_traversal.assets/preorder_step5.png) - -=== "<6>" - ![preorder_step6](binary_tree_traversal.assets/preorder_step6.png) - -=== "<7>" - ![preorder_step7](binary_tree_traversal.assets/preorder_step7.png) - -=== "<8>" - ![preorder_step8](binary_tree_traversal.assets/preorder_step8.png) - -=== "<9>" - ![preorder_step9](binary_tree_traversal.assets/preorder_step9.png) - -=== "<10>" - ![preorder_step10](binary_tree_traversal.assets/preorder_step10.png) - -=== "<11>" - ![preorder_step11](binary_tree_traversal.assets/preorder_step11.png) - -### Complexity Analysis - -- **Time complexity $O(n)$** : All nodes are visited once, using $O(n)$ time. -- **Space complexity $O(n)$** : In the worst case, i.e., when the tree degenerates to a linked list, the depth of recursion reaches $n$ , and the system occupies $O(n)$ stack frame space. diff --git a/docs-en/chapter_tree/index.md b/docs-en/chapter_tree/index.md deleted file mode 100644 index 8374ad99ee..0000000000 --- a/docs-en/chapter_tree/index.md +++ /dev/null @@ -1,13 +0,0 @@ -# Tree - -

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      7dyZ6dx8Y8pl);P>+tPl zNr&&bKTe@D{}UQEIH8!8@Ck~xkeT%guibThdHx($zi#P zJJtVp4x?hEd(oTqYU@;lY3O+g1^4D1|7w)5KI6w(}xslMJjaIlI-} z>nDpm&_*m%wy*?>7G7c|gm7YSAb^cGbOsBAu>c1l0EQNr;Q@?50VZxx0iYmTQ)_(3 z$G3N?M9H}E22B`G3574$lLw`J`8x)$U-zG&#eyqp?o<>K`Fl$hKz!Qyn2PQ%pSCR4M&HuHGW8F9uS9Ah$he=}heW?v z9E`3SGMDW@#Qona##t*NWOO5G989&tn?v=JeD?m-O)I zDwlL-@7IloYvbK57@V2n*}*3q{tfbCM0Vb5F=v;>HjrW1Ptc-a`FS?^UIEOoK6Aq` z$)f9O`dtxIBG1~Tn4!gyY=a3(WL^AJj8)tVALwO<`Cc!4^=GEczo2}ehQEsA<_eUA z_ll{zknqBF_PVRZrgTJX6=s7u&~KzffHo5HmyjfLrLy9Pe(@Dn;f=tuH%d&4_RXZ- zzhilDdaCB6v0pJ%URA4ppg2Ev?&b1Y{|7p$WW$w6>0c*vDt-3xCZ|NOqeqS$Qp!UP0QCoL+){iH9s zxmp%EWeedY`p7q$>JrKOB`0rMbxWIzxvnHp!<}8m_t9L;qt)$oJQsm;`%9p{`Hs=U z!D*p(NMxn(hbI@l<>#MZ+a~RMxaDJ2T8~s;bigHH$3pYtq6P~22PyG$SRsuEY!=lc-_gtT=Mt=<53(OW-m&!;k zl1(t>N>D>z^N%LE0`U@^u+hoD3syJ}UvMk~^4^O#VC&DJZ`xMKVg8QjyMJ?P?IlF zF8z^EU7er(bVZ}=mB>Q$s%VqT+lA4@f}?7sQ@AraHuL>InbWGaUWU1-M&r^ z$89h98H+cHqcSwrJ{O$DbMntd4U{c@)o4{e^eF}(QDi_&KsZrcFITnpCpk>Ns?f1U z4oi}E4(}M54ry0bQe#E^OJ4Sf&-gf<4TU@YVgbj^zt1nA}WfQbI(j<7eJh zAz_Q<5P{O!yEd=y>AOTje-;nf?L2=>k?7SCDd^1jO4pdAlt(2u&#=UIkZ01$*+fRhqn4|!OU z^M0&F4n1IM---sPJ;R8};Z2Gwr=RbHMmm#bo08&M$N8dANNzZ!T`nP^TPZ-Kf*iXp z3QBGetk?eFc!zegY$U;+jc1oG`p~=23GWB(6*JAkjw@v$%3XY@46;@)@SboyDI)a)W60C1M^GAd5go{OB%j z;}Wqci4&BRL&aeFmh0SNiLP=S5voD7(6M?;oc(Xm8>Bdu<84tnO%wta+qqtjWkic$O?Upx5)9#X8{KDh7FW4j%@jtuRo$qO~mzhu#f^4W)}lI^5OZ2hc6{#x*njU z$YoPVWLTu#CMsfXkK|vH*?&z6r%Ikv_+C!JMuyvW58wXQg^RNlPoqUL@#P#{H|dJ{JAxEVX#k+nPZhsGBXHu@ z6;SYJ@F{;uzdL#A6@zSvA@NmWZ<)5l6g9{B=7qWFYScyO^+iOpr4tT^1kpO#BVxP8 zNBSlw=-&gv&v)7BL#jp`woM!UTs>|_R zfSqTzH!fF2Tg)g79+|fvo<96N<{%)x_$^2@OHEB?_tPY;(@FjmXNwyn3%On~+mpx1 zxy3H?G)3QD_xGaXZ$3@q=gn15HNT*bGTjt8@~KPDMyjY)QD&>^9h0f4NfGxwe|-Lq TVozWi5dXB*4b" - ![merge sort step](merge_sort.assets/merge_sort_step1.png) - -=== "<2>" - ![merge_sort_step2](merge_sort.assets/merge_sort_step2.png) - -=== "<3>" - ![merge_sort_step3](merge_sort.assets/merge_sort_step3.png) - -=== "<4>" - ![merge_sort_step4](merge_sort.assets/merge_sort_step4.png) - -=== "<5>" - ![merge_sort_step5](merge_sort.assets/merge_sort_step5.png) - -=== "<6>" - ![merge_sort_step6](merge_sort.assets/merge_sort_step6.png) - -=== "<7>" - ![merge_sort_step7](merge_sort.assets/merge_sort_step7.png) - -=== "<8>" - ![merge_sort_step8](merge_sort.assets/merge_sort_step8.png) - -=== "<9>" - ![merge_sort_step9](merge_sort.assets/merge_sort_step9.png) - -=== "<10>" - ![merge_sort_step10](merge_sort.assets/merge_sort_step10.png) - -Observe that the recursion sort is consistent with the recursion order of the backward traversal of the binary tree. - -- **Posterior order traversal**: recursion to the left-child node, then to the right-child node, and finally deal with the root node. -- **Recursion sort**: recursion on the left subarray, then recursion on the right subarray, and finally process the merge. - -```src -[file]{merge_sort}-[class]{}-[func]{merge_sort} -``` - -It is worth noting that the interval to be merged for `nums` is `[left, right]`, while the corresponding interval for `tmp` is `[0, right - left]`. - -## Algorithm Properties - -- **Time complexity $O(n \log n)$, Non-adaptive sorting**: division produces a recursion tree of height $\log n$, and the total number of operations for merging at each level is $n$ , so the overall time complexity is $O(n \log n)$ . -- **Space complexity $O(n)$, non-in-place sort**: recursion depth is $\log n$, using $O(\log n)$ size stack frame space. The merge operation needs to be implemented with the help of an auxiliary array, using extra space of size $O(n)$. -- **Stable ordering**: the order of equal elements remains the same during the merge process. - -## Linked List Sorting - -For linked lists, merge sort has a significant advantage over other sorting algorithms, **It can optimize the space complexity of the linked list sorting task to $O(1)$** . - -- **Division Phase**: The division of the linked list can be achieved by using "iteration" instead of "recursion", thus eliminating the use of stack frames for recursion. -- **Merge Phase**: In a linked list, node addition and deletion operations can be accomplished by simply changing references (pointers), so the merge phase (combining two short ordered linked lists into one long ordered linked list) does not require the creation of additional linked lists. - -Specific implementation details are more complex, interested students can consult the relevant information to learn. 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Let's take the help of an algorithm problem for better solution. - -!!! question - - Given an array of integers `nums` and a target element `target`, search the array for two elements whose "and" is `target` and return their array indexes. It is sufficient to return any one of the solutions. - -## Linear Lookups: Trading Time For Space - -Consider directly traversing all possible combinations. As shown in the figure below, we open a two-level loop that determines in each round whether the sum of two integers is `target` , and if so, returns their indexes. - -![Linear lookup to solve for sum of two numbers](replace_linear_by_hashing.assets/two_sum_brute_force.png) - -```src -[file]{two_sum}-[class]{}-[func]{two_sum_brute_force} -``` - -The time complexity of this method is $O(n^2)$ and the space complexity is $O(1)$ , which is very time-consuming in large data volume. - -## Hash Lookups: Trading Space For Time - -Consider with the help of a hash table where the key-value pairs are array elements and element indexes respectively. Traverse the array in a loop, performing the steps shown in the figure below each round. - -1. Determine whether the number `target - nums[i]` is in the hash table, if so, return the indexes of the two elements directly. -2. Add the key-value pair `nums[i]` and the index `i` to the hash table. - -=== "<1>" - ![Auxiliary hash table to solve for the sum of two numbers](replace_linear_by_hashing.assets/two_sum_hashtable_step1.png) - -=== "<2>" - ![two_sum_hashtable_step2](replace_linear_by_hashing.assets/two_sum_hashtable_step2.png) - -=== "<3>" - ![two_sum_hashtable_step3](replace_linear_by_hashing.assets/two_sum_hashtable_step3.png) - -The implementation code is shown below and requires only a single level of looping. - -```src -[file]{two_sum}-[class]{}-[func]{two_sum_hash_table} -``` - -This method reduces the time complexity from $O(n^2)$ to $O(n)$ by hash lookup, which significantly improves the operation efficiency. - -The space complexity is $O(n)$ due to the need to maintain an additional hash table. **Nevertheless, the overall space-time efficiency of this method is more balanced and therefore it is the optimal solution to this problem**. diff --git 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a/docs-en/chapter_heap/top_k.md +++ /dev/null @@ -1,71 +0,0 @@ -# Top-K Problem - -!!! question - - Given an unordered array of length $n$ `nums` , return the first $k$ largest element in the array. - -For this problem, we present two solutions that are more straightforward in their thinking, followed by the more efficient heap solution. - -## Method 1: Iterate Through The Selection - -We can perform the $k$ rounds of traversal shown in the figure below, extracting the $1$, $2$, $\dots$, and $k$ largest element in each round, respectively, with a time complexity of $O(nk)$ . - -This method is only applicable to the case of $k \ll n$, because when $k$ is close to $n$, its time complexity tends to $O(n^2)$, which is very time-consuming. - -![Traversal to find the largest k elements](top_k.assets/top_k_traversal.png) - -!!! tip - - When $k = n$, we get a complete ordered sequence, which is equivalent to a "selection sort" algorithm. - -## Method 2: Sorting - -As shown in the figure below, we can sort the array `nums` and then return the rightmost $k$ elements with a time complexity of $O(n \log n)$ . - -Obviously, this method "overshoots" the task, since we only need to find the largest $k$ element and not sort the others. - -![Sorting to find the largest k elements](top_k.assets/top_k_sorting.png) - -## Method 3: Heap - -We can solve the Top-K problem more efficiently based on heaps, the process is shown in the figure below. - -1. Initialize a min heap with the smallest top element. -2. The first $k$ elements of the array are first put into the heap in order. -3. Starting from the $k + 1$th element, if the current element is larger than the top element of the heap, then the top element of the heap is taken out of the heap and the current element is put into the heap. -4. After the traversal is complete, it is the largest $k$ element that is kept in the heap. - -=== "<1>" - ![Find the largest k elements based on heap](top_k.assets/top_k_heap_step1.png) - -=== "<2>" - ![top_k_heap_step2](top_k.assets/top_k_heap_step2.png) - -=== "<3>" - ![top_k_heap_step3](top_k.assets/top_k_heap_step3.png) - -=== "<4>" - ![top_k_heap_step4](top_k.assets/top_k_heap_step4.png) - -=== "<5>" - ![top_k_heap_step5](top_k.assets/top_k_heap_step5.png) - -=== "<6>" - ![top_k_heap_step6](top_k.assets/top_k_heap_step6.png) - -=== "<7>" - ![top_k_heap_step7](top_k.assets/top_k_heap_step7.png) - -=== "<8>" - ![top_k_heap_step8](top_k.assets/top_k_heap_step8.png) - -=== "<9>" - ![top_k_heap_step9](top_k.assets/top_k_heap_step9.png) - -A total of $n$ rounds of heap entry and heap exit are performed, and the maximum length of the heap is $k$ , so the time complexity is $O(n \log k)$ . The method is very efficient, and the time complexity tends to $O(n)$ when $k$ is small, and does not exceed $O(n \log n)$ when $k$ is large. - -In addition, this method is suitable for dynamic data streaming usage scenarios. We can continuously maintain the elements in the heap as we keep adding data, thus enabling dynamic updates of up to $k$ elements. - -```src -[file]{top_k}-[class]{}-[func]{top_k_heap} -``` diff --git a/docs-en/chapter_reference/index.md b/docs-en/chapter_reference/index.md deleted file mode 100644 index 008092ebb4..0000000000 --- a/docs-en/chapter_reference/index.md +++ /dev/null @@ -1,19 +0,0 @@ -# References - -[1] Thomas H. Cormen, et al. Introduction to Algorithms (3rd Edition). - -[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1st Edition). - -[3] Yan Weimin. Data structure (C language version). - -[4] Deng, J. F.. Data structure (C++ language edition, 3rd edition). - -[5] Mark Allen Weiss. Data structures and algorithm analysis: a description of the Java language (3rd ed.). - -[6] J. Cheng. Big talk about data structures. - -[7] Wang Q. Data structures and the beauty of algorithms. - -[8] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6th Edition). - -[9] Aston Zhang, et al. 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It utilizes the ordered nature of data to reduce the search range by half in each round until the target element is found or the search interval is empty. - -!!! question - - Given an array `nums` of length $n$ with elements in descending order, the array contains no duplicate elements. Find and return the index of the element `target` in the array. If the array does not contain the element, return $-1$. - -![binary search example data](binary_search.assets/binary_search_example.png) - -As shown in the figure below, we first initialize pointers $i = 0$ and $j = n - 1$ , pointing to the first and last elements of the array, representing the search interval $[0, n - 1]$ . Note that the middle bracket denotes the closed interval, which contains the boundary values themselves. - -Next, loop through the following two steps. - -1. Calculate the midpoint index $m = \lfloor {(i + j) / 2} \rfloor$ , where $\lfloor \: \rfloor$ denotes the downward rounding operation. -2. To determine the size relationship between `nums[m]` and `target`, there are three cases. - 1. When `nums[m] < target`, it means that `target` is in the interval $[m + 1, j]$, so execute $i = m + 1$ . - 2. When `nums[m] > target` , it means that `target` is in the interval $[i, m - 1]$ , so execute $j = m - 1$ . - 3. When `nums[m] = target`, it means that `target` is found, so it returns the index $m$. - -If the array does not contain the target element, the search interval is eventually reduced to empty. This returns $-1$ . - -=== "<1>" - ![binary search process](binary_search.assets/binary_search_step1.png) - -=== "<2>" - ![binary_search_step2](binary_search.assets/binary_search_step2.png) - -=== "<3>" - ![binary_search_step3](binary_search.assets/binary_search_step3.png) - -=== "<4>" - ![binary_search_step4](binary_search.assets/binary_search_step4.png) - -=== "<5>" - ![binary_search_step5](binary_search.assets/binary_search_step5.png) - -=== "<6>" - ![binary_search_step6](binary_search.assets/binary_search_step6.png) - -=== "<7>" - ![binary_search_step7](binary_search.assets/binary_search_step7.png) - -It is worth noting that since $i$ and $j$ are both of type `int`, **so $i + j$ may be out of the range of values of type `int`**. In order to avoid large numbers crossing the boundary, we usually use the formula $m = \lfloor {i + (j - i) / 2} \rfloor$ to calculate the midpoint. - -```src -[file]{binary_search}-[class]{}-[func]{binary_search} -``` - -**Time complexity $O(\log n)$** : In a bisection loop, the interval is reduced by half in each round and the number of loops is $\log_2 n$ . - -**Space complexity $O(1)$** : Pointers $i$ and $j$ use constant size space. - -## Interval Representation - -In addition to the double-closed intervals mentioned above, a common interval representation is the "left-closed-right-open" interval, defined as $[0, n)$, where the left edge contains itself and the right edge does not contain itself. In this representation, the interval $[i, j]$ is empty when $i = j$. - -We can implement a binary search algorithm with the same functionality based on this representation. - -```src -[file]{binary_search}-[class]{}-[func]{binary_search_lcro} -``` - -As shown in the figure below, the initialization, loop condition and interval reduction operations of the binary search algorithm are different in the two interval representations. - -Since both the left and right edges of the "double-closed interval" representation are defined as closed intervals, the narrowing of the interval by the pointers $i$ and $j$ is also symmetric. This is less prone to errors, **so the "double-closed interval" style is generally recommended**. - -![two interval definitions](binary_search.assets/binary_search_ranges.png) - -## Strengths And Limitations - -Binary search has better performance in both time and space. - -- Binary search is time efficient. The logarithmic order has a significant time complexity advantage in large data sizes. For example, when the data size $n = 2^{20}$, linear search requires $2^{20} = 1048576$ rounds of cycles, while binary search requires only $\log_2 2^{20} = 20$ rounds of cycles. -- Binary search requires no extra space. Compared to searching algorithms that require extra space (such as hash search), binary search is more space-efficient. - -However, binary search is not suitable for all cases, mainly for the following reasons. - -- Binary search only works with ordered data. If the input data is unordered, it is not worthwhile to specialize in sorting in order to use binary search. This is because the time complexity of the sorting algorithm is usually $O(n \log n)$ , which is higher than both linear and binary search. For scenarios where elements are frequently inserted into specific locations to keep the array organized, the time complexity is $O(n)$ , which is also very expensive. -- Binary search is only applicable to arrays. Binary search requires jumping (non-contiguous) access to elements, and performing jumping accesses in a linked list is inefficient, so it is not suitable for application to linked lists or data structures based on linked list implementations. -- Linear lookup performs better with small data size. 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zSk?p9xd`B6O%TOIg?DSi``{66_ar)z6(V&<{sVN1zBjFawCW&<4TU|G!$c6~|Iqzm z4nB0)S|=oSU@(Was*$XVE$mu=D zAF)*!J9`Ytd|nazRvI)aC&`a$dJdD)ss+Z2>S#_ryQX zCw~Y4H&=z|>hq4kq_^6gXqmCwM<~L1`(WXx!?4pOe}tW>EPpggMpr zu*>$3_@Lp8CEk(1^MtDfVjCyN0pkMJzptHMwd0xa1ryhP^ut=#z$fF-qbB42WM8k8ypF=qf5)#C66PM;0-OWcG7We3z@-R)0M|4E_ z`3z$t+Hcr|jAcOHw@QXKxc_?+CQ(iCA;F{w9>a)bTe|Tzf=t2Un}m&IM@kHFUDAqs zJUstaixPRJCnV+tyeWvWeW8ji#V{bE*}?)2JSh>It9(>rgQi1&GwPy5JT$NN029?A zo_)S01CUB2BZYk)ds~r!Fr0)=Rc~$jj*fmC#H3!uV7yyS0v7KBF%Jqyk^>2|=iQ~x zHW-R}1uRFsK@=-jCfX(c$CvY)+~LTY`zZ1AyK9*pG02!-i*MGpHQpekG*k7WUBqHu zFm#6_zrQ_9(d5cP+NEguT-o?4M8Yf#%hk}5BCn8&_jJS_+_`69R#p3qww0rLt4b^1 zjRkE=gcGT!<78(#@;Q>AkO#>+ZB3|UJOB?bmK4;1UZ_qE#Bk2da7L|B?o0F^<)%!H z9^y)6KFh1uJjOF3<5gv4o9)w8Dh|B#HY2gJ!S^<89TBZaRZ}|h*PEwVpK#$F`=l*8 z6TT@n6&-!+p" - ![Greedy process for the maximum capacity problem](max_capacity_problem.assets/max_capacity_greedy_step1.png) - -=== "<2>" - ![max_capacity_greedy_step2](max_capacity_problem.assets/max_capacity_greedy_step2.png) - -=== "<3>" - ![max_capacity_greedy_step3](max_capacity_problem.assets/max_capacity_greedy_step3.png) - -=== "<4>" - ![max_capacity_greedy_step4](max_capacity_problem.assets/max_capacity_greedy_step4.png) - -=== "<5>" - ![max_capacity_greedy_step5](max_capacity_problem.assets/max_capacity_greedy_step5.png) - -=== "<6>" - ![max_capacity_greedy_step6](max_capacity_problem.assets/max_capacity_greedy_step6.png) - -=== "<7>" - ![max_capacity_greedy_step7](max_capacity_problem.assets/max_capacity_greedy_step7.png) - -=== "<8>" - ![max_capacity_greedy_step8](max_capacity_problem.assets/max_capacity_greedy_step8.png) - -=== "<9>" - ![max_capacity_greedy_step9](max_capacity_problem.assets/max_capacity_greedy_step9.png) - -### Code Implementation - -The code loops for at most $n$ rounds, **so the time complexity is $O(n)$**. - -The variables $i$, $j$, $res$ use constant size extra space, **so the space complexity is $O(1)$** . - -```src -[file]{max_capacity}-[class]{}-[func]{max_capacity} -``` - -### Proof Of Correctness - -The reason why greedy is faster than exhaustive is that each round of greedy choices "skips" some states. - -For example, in state $cap[i, j]$, $i$ is the short board and $j$ is the long board. Greedily moving the short board $i$ inward by one frame will cause the states shown in the figure below to be "skipped". **This means that the capacity of these states cannot be verified later**. - -$$ -cap[i, i+1], cap[i, i+2], \dots, cap[i, j-2], cap[i, j-1] -$$ - -![Moving short boards causes skipped states](max_capacity_problem.assets/max_capacity_skipped_states.png) - -Observe that these skipped states are actually all the states that move the longboard $j$ inwards. And in the second step, we have shown that moving the longboard inward must result in a smaller capacity. That is, **none of the skipped states can be optimal solutions skipping them will not result in missing the optimal solution**. - -The above analysis shows that **moving the short board is "safe"** and that the greedy strategy is effective. diff --git a/docs-en/chapter_greedy/max_product_cutting_problem.assets/max_product_cutting_definition.png b/docs-en/chapter_greedy/max_product_cutting_problem.assets/max_product_cutting_definition.png deleted file mode 100644 index 11a291bcf22dcf359c51a5fe0ed47834fe8686e0..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 13139 zcmeHtcT`l*muJ21CL?% zBuS1<21(oa{r=9(?EKEmo;_!FXV2L`y6fG#_uYHzQ@1|ny?Pa(qIB;Z@daW4z&QnZ zSyce=aF?(yLO8D3Pvw4!6V_A|)$X32o?@}sfPjFbqod>FlexLMrKP3K&CRK)shypj zy}iBl_4VJse_Q)ZtUWV*Q{}bq^$3~1tXf?tE-sG2VAMaTiN5n+eL&2u^keoSLqb9V ztyf!IT-?IKLPJBt;o;%L#KghD!Q$d#V`Jm(+qb8ur}y{wS5{UA1_oAFSJ&3oW@cta zM@MI8XIolYwzs#pwzj@~`_|XjS6p1&)YKFj5+W}zFD)(I-QC^R)|Qu-_w(n^p`oGJ 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They are implemented differently in the "adjacency matrix" and "adjacency list" representations. - -## Implementation Based On Adjacency Matrices - -Given an undirected graph with $n$ number of vertices, the various operations are implemented as shown in the figure below. - -- **Add or remove edges**: just modify the specified edges directly in the adjacency matrix, using $O(1)$ time. And since it is an undirected graph, you need to update the edges in both directions at the same time. -- **Add vertex**: add a row and a column to the end of the adjacency matrix and fill them all $0$ that's it, using $O(n)$ time. -- **Delete vertex**: deletes a row and a column in the adjacency matrix. The worst case is reached when deleting the first row and column, which requires $(n-1)^2$ elements to be "shifted up to the left", thus using $O(n^2)$ time. -- **Initialize**: pass $n$ vertices, initialize the list of vertices `vertices` of length $n$, using $O(n)$ time; initialize the adjacency matrix `adjMat` of size $n \times n$, using $O(n^2)$ time. - -=== "Initialize the adjacency matrix." - ![Initialization of adjacency matrix, adding and removing edges, adding and removing vertices](graph_operations.assets/adjacency_matrix_initialization.png) - -=== "Add edge" - ![adjacency_matrix_add_edge](graph_operations.assets/adjacency_matrix_add_edge.png) - -=== "Delete edge" - ![adjacency_matrix_remove_edge](graph_operations.assets/adjacency_matrix_remove_edge.png) - -=== "Add vertex" - ![adjacency_matrix_add_vertex](graph_operations.assets/adjacency_matrix_add_vertex.png) - -=== "Delete vertex" - ![adjacency_matrix_remove_vertex](graph_operations.assets/adjacency_matrix_remove_vertex.png) - -The following is the implementation code based on the adjacency matrix representation of the graph. - -```src -[file]{graph_adjacency_matrix}-[class]{graph_adj_mat}-[func]{} -``` - -## Implementation Based On Adjacency Lists - -Let the total number of vertices of an undirected graph be $n$ and the total number of edges be $m$ , then various operations can be realized according to the method shown in the figure below. - -- **Add Edge**: just add the edge at the end of the linked list corresponding to the vertex, using $O(1)$ time. Since it is an undirected graph, you need to add edges in both directions. -- **Delete Edge**: finds and deletes the specified edge in the linked list corresponding to the vertex, using $O(m)$ time. In an undirected graph, edges in both directions need to be deleted simultaneously. -- **Add vertex**: add a linked list to the adjacency list and use the added vertex as the linked list node, using $O(1)$ time. -- **Delete vertex**: need to traverse the entire adjacency list and delete all edges containing the specified vertex, using $O(n + m)$ time. -- **Initialization**: create $n$ vertices and $2m$ edges in the adjacency list, using $O(n + m)$ time. - -=== "Initialize the adjacency list." - ![Initialization of adjacency list, adding and removing edges, adding and removing vertices](graph_operations.assets/adjacency_list_initialization.png) - -=== "Add edge" - ![adjacency_list_add_edge](graph_operations.assets/adjacency_list_add_edge.png) - -=== "Delete edge" - ![adjacency_list_remove_edge](graph_operations.assets/adjacency_list_remove_edge.png) - -=== "Add vertex" - ![adjacency_list_add_vertex](graph_operations.assets/adjacency_list_add_vertex.png) - -=== "Delete vertex" - ![adjacency_list_remove_vertex](graph_operations.assets/adjacency_list_remove_vertex.png) - -The following is the code implementation of the adjacency list. Comparing to the figure above, the actual code has the following differences. - -- To make it easier to add and remove vertices, and to simplify the code, we use lists (dynamic arrays) instead of linked lists. -- A hash table is used to store the adjacency table, `key` is the vertex instance, and `value` is the list of adjacency vertices (linked list) for that vertex. - -In addition, we use the `Vertex` class to represent the vertices in the adjacency table because if we use the list index to distinguish different vertices as in the adjacency matrix, then suppose we want to delete a vertex with index $i$, we need to traverse the whole adjacency list and subtract all indexes greater than $i$ from $1$, which is very inefficient. If each vertex is a unique instance of `Vertex`, then after deleting a vertex, there is no need to change the other vertices. - -```src -[file]{graph_adjacency_list}-[class]{graph_adj_list}-[func]{} -``` - -## Efficiency Comparison - -Given that there are $n$ vertices and $m$ edges in the graph, the table below compares the time and space efficiency of the adjacency matrix and adjacency list. - -

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z52}_AEJ|!E{ng%Nh7Y?JsgP6P6>F$0&WMlm$9NM7?mUw%BxF>5iA_U$cNQ)e8jkOqGxZmfw*6gNKw!46&KiDRTNCfj~2ypBtPhpQ+yy7KL9aq3y=zB z;-1#(nst)(Rpl}8OfnBCA zJAd+of1k2q7^(WAfVq=s92M1D{dOYDeQ%>rQV2i^``UPYr^H}yY3I1eHeZA<9{uY8 NYD(IQwes+n{{c*a@st1n diff --git a/docs-en/chapter_dynamic_programming/dp_solution_pipeline.md b/docs-en/chapter_dynamic_programming/dp_solution_pipeline.md deleted file mode 100644 index 9f46af5473..0000000000 --- a/docs-en/chapter_dynamic_programming/dp_solution_pipeline.md +++ /dev/null @@ -1,181 +0,0 @@ -# Dynamic Programming Solutions - -The last two sections introduced the main features of dynamic programming problems, and next we explore two more practical problems together. - -1. How can you tell if a problem is a dynamic programming problem? -2. Where to start and what are the complete steps for solving dynamic programming problems? - -## Problem Determination - -In general, if a problem contains overlapping subproblems, optimal substructures, and satisfies no posteriority, then it is usually suitable for solution by dynamic programming. However, it is difficult to extract these properties directly from the problem description. **Therefore we usually relax the conditions and first observe whether the problem is suitable to be solved using backtracking (exhaustive)**. - -**Problems suitable for solution by backtracking usually satisfy the "Decision Tree Model"**, which can be described using a tree structure where each node represents a decision and each path represents a sequence of decisions. - -In other words, if the problem contains explicit decision concepts and the solution is generated through a series of decisions, then it satisfies the decision tree model and can usually be solved using backtracking. - -On this basis, there are some judgmental "plus points" for dynamic programming problems. - -- The problem contains optimization descriptions such as maximum (small) or maximum (small). -- The states of a problem can be represented using a list, a multidimensional matrix, or a tree, and there is a recursive relationship between a state and its surrounding states. - -Accordingly, there are some "minuses". - -- The goal of the problem is to find all possible solutions, not to find the optimal solution. -- The problem description is characterized by distinct permutations and combinations that need to be returned for specific multiple scenarios. - -If a problem satisfies the decision tree model and has more obvious "plus points", we can assume that it is a dynamic programming problem and verify it in the solution process. - -## Problem Solving Steps - -The solution process for dynamic programming will vary depending on the nature and difficulty of the problem, but usually follows the following steps: describe the decision, define the state, build the $dp$ table, derive the state-transition equation, determine the boundary conditions, etc. - -In order to show the solution step by step, we use a classic problem "Minimum Path Sum" as an example. - -!!! question - - Given a $n \times m$ two-dimensional grid `grid` , each cell in the grid contains a non-negative integer indicating the cost of that cell. The robot starts at the top-left cell and can only move one step down or to the right at a time until it reaches the bottom-right cell. Please return the minimum path sum from the upper-left corner to the lower-right corner. - -The figure below shows an example where the minimum path sum for a given grid is $13$ . - -![Minimum path and example data](dp_solution_pipeline.assets/min_path_sum_example.png) - -**Step 1: Think about the decisions made in each round and define the state that leads to the $dp$ table** - -The decision in each round of this problem is to take one step down or to the right from the current lattice. Let the row and column index of the current lattice be $[i, j]$ , then the index becomes $[i+1, j]$ or $[i, j+1]$ after taking a step down or right. Therefore, the state should contain both row index and column index variables, denoted as $[i, j]$ . - -The subproblem corresponding to the state $[i, j]$ is the minimum path sum from the start point $[0, 0]$ to $[i, j]$ and the solution is denoted as $dp[i, j]$ . - -At this point, we have the two-dimensional $dp$ matrix shown in the figure below, with the same dimensions as the input grid $grid$. - -![state definition with dp table](dp_solution_pipeline.assets/min_path_sum_solution_step1.png) - -!!! note - - The dynamic programming and backtracking process can be described as a sequence of decisions, and the state consists of all the decision variables. It should contain all the variables describing the progress of the solution, and it contains enough information that can be used to derive the next state. - - Each state corresponds to a subproblem, and we will define a $dp$ table to store the solutions of all the subproblems, with each independent variable of the state being a dimension of the $dp$ table. Essentially, the $dp$ table is a mapping between the states and the solutions of the subproblems. - -**Step 2: Identify the optimal substructure and hence derive the state-transition equation** - -For the state $[i, j]$ , it can only be transferred from the upper edge lattice $[i-1, j]$ and the left lattice $[i, j-1]$ . Therefore the optimal substructure is: the minimum path sum to reach $[i, j]$ is determined by the smaller of the minimum path sum of $[i, j-1]$ and the minimum path sum of $[i-1, j]$. - -Based on the above analysis, the state-transition equation shown in the figure below can be introduced: - -$$ -dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j] -$$ - -![Optimal substructure and state-trasition equation](dp_solution_pipeline.assets/min_path_sum_solution_step2.png) - -!!! note - - Based on the defined $dp$ table, think about the relationship between the original problem and the subproblems, and find out the way to construct the optimal solution of the original problem through the optimal solution of the subproblems, i.e., the optimal substructure. - - Once we have found the optimal substructure, we can use it to construct the state-transition equation. - -**Step 3: Determine boundary conditions and state transfer order** - -In this problem, the state of the first row can only be derived from the state to its left, and the state of the first column can only be derived from the state above it, so the first row $i = 0$ and the first column $j = 0$ are boundary conditions. - -As shown in the figure below, since each grid is transferred from its left and top grids, we use employ a loop to traverse the matrix, an outer loop to traverse the rows and an inner loop to traverse the columns. - -![Boundary conditions and state transfer order](dp_solution_pipeline.assets/min_path_sum_solution_step3.png) - -!!! note - - Boundary conditions are used in dynamic programming to initialize $dp$ tables and in search for pruning. - - The core of the state transfer order is to ensure that by the time the solution to the current problem is computed, all the solutions to the smaller subproblems on which it depends have already been computed correctly. - -Based on the above analysis, we can already write dynamic programming code directly. However, subproblem decomposition is a kind of top-to-bottom idea, so it is more in line with the habit of thinking to implement it in the order of "Brute-force Search $\rightarrow$ Memorized Search $\rightarrow$ Dynamic Programming". - -### Method One: Brute Force Search - -Starting the search from the state $[i, j]$ and continuously decomposing into smaller states $[i-1, j]$ and $[i, j-1]$ , the recursion function includes the following elements. - -- **Recursion parameter**: state $[i, j]$ . -- **Return value**: the minimum path from $[0, 0]$ to $[i, j]$ and $dp[i, j]$ . -- **Termination condition**: return cost $grid[0, 0]$ when $i = 0$ and $j = 0$ . -- **pruning**: index crossing when $i < 0$ or $j < 0$, in which case the cost $+\infty$ is returned, representing infeasibility. - -```src -[file]{min_path_sum}-[class]{}-[func]{min_path_sum_dfs} -``` - -The figure below gives a recursion tree node with $dp[2, 1]$ as the root node, which contains a number of overlapping subproblems, the number of which increases dramatically as the size of the grid `grid` becomes larger. - -Essentially, the cause of the overlapping sub-problem is that **there exist multiple paths to reach a cell from the upper left corner**. - -![Brute-force search recursion tree](dp_solution_pipeline.assets/min_path_sum_dfs.png) - -Each state has two choices, down and right, and it takes a total of $m + n - 2$ steps to walk from the top left corner to the bottom right corner, so the worst time complexity is $O(2^{m + n})$ . Note that this calculation does not take into account the case of approaching grid boundaries, which leaves only one choice when the network boundary is reached. Therefore the actual number of paths will be less. - -### Method 2: Memorized Search - -We introduce a memoized list `mem` of the same size as the grid `grid` for recording the solutions of individual subproblems and pruning the overlapping subproblems. - -```src -[file]{min_path_sum}-[class]{}-[func]{min_path_sum_dfs_mem} -``` - -As shown in the figure below, with the introduction of memoization, the solution to all subproblems is computed only once, so the time complexity depends on the total number of states, i.e., the grid size $O(nm)$ . - -![Memorized search recursion tree](dp_solution_pipeline.assets/min_path_sum_dfs_mem.png) - -### Method 3: Dynamic Programming - -Iteration-based implementation of dynamic programming solutions. - -```src -[file]{min_path_sum}-[class]{}-[func]{min_path_sum_dp} -``` - -The figure below illustrates the state transfer process for the minimum path sum, which traverses the entire lattice, **hence the time complexity is $O(nm)$** . - -The size of the array `dp` is $n \times m$ , **hence the space complexity is $O(nm)$** . - -=== "<1>" - ![Dynamic programming process for minimum path sum](dp_solution_pipeline.assets/min_path_sum_dp_step1.png) - -=== "<2>" - ![min_path_sum_dp_step2](dp_solution_pipeline.assets/min_path_sum_dp_step2.png) - -=== "<3>" - ![min_path_sum_dp_step3](dp_solution_pipeline.assets/min_path_sum_dp_step3.png) - -=== "<4>" - ![min_path_sum_dp_step4](dp_solution_pipeline.assets/min_path_sum_dp_step4.png) - -=== "<5>" - ![min_path_sum_dp_step5](dp_solution_pipeline.assets/min_path_sum_dp_step5.png) - -=== "<6>" - ![min_path_sum_dp_step6](dp_solution_pipeline.assets/min_path_sum_dp_step6.png) - -=== "<7>" - ![min_path_sum_dp_step7](dp_solution_pipeline.assets/min_path_sum_dp_step7.png) - -=== "<8>" - ![min_path_sum_dp_step8](dp_solution_pipeline.assets/min_path_sum_dp_step8.png) - -=== "<9>" - ![min_path_sum_dp_step9](dp_solution_pipeline.assets/min_path_sum_dp_step9.png) - -=== "<10>" - ![min_path_sum_dp_step10](dp_solution_pipeline.assets/min_path_sum_dp_step10.png) - -=== "<11>" - ![min_path_sum_dp_step11](dp_solution_pipeline.assets/min_path_sum_dp_step11.png) - -=== "<12>" - ![min_path_sum_dp_step12](dp_solution_pipeline.assets/min_path_sum_dp_step12.png) - -### Space Optimization - -Since each grid is only related to the grid to its left and above, we can implement the $dp$ table with only a single row array. - -Note that since the array `dp` can only represent the state of one row, we can't initialize the first column's state ahead of time, but instead update it as we traverse each row. - -```src -[file]{min_path_sum}-[class]{}-[func]{min_path_sum_dp_comp} -``` diff --git a/docs-en/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_decision_tree.png b/docs-en/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_decision_tree.png deleted file mode 100644 index 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a/docs-en/chapter_backtracking/permutations_problem.md b/docs-en/chapter_backtracking/permutations_problem.md deleted file mode 100644 index c283d1fc38..0000000000 --- a/docs-en/chapter_backtracking/permutations_problem.md +++ /dev/null @@ -1,95 +0,0 @@ -# Permutations Problem - -The permutations problem is a typical application of the backtracking algorithm. It is defined as finding all possible permutations of the elements of a set (e.g., an array or a string) given that set. - -The table below lists several example data, including the input array and all the corresponding alignments. - -

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0000000000..8d712f3657 --- /dev/null +++ b/docs-en/chapter_tree/binary_tree_traversal.md @@ -0,0 +1,89 @@ +# Binary Tree Traversal + +From a physical structure point of view, a tree is a data structure based on a linked list, and therefore its traversal is done by accessing the nodes one by one through pointers. However, a tree is a nonlinear data structure, which makes traversing a tree more complex than traversing a linked list, and needs to be implemented with the help of a searching algorithm. + +Common traversals of binary trees include level-order traversal, preorder traversal, middle-order traversal, and postorder traversal. + +## Level-Order Traversal + +As shown in the figure below, "level-order traversal level-order traversal" traverses the binary tree level by level from the top to the bottom and visits the nodes at each level in the order from left to right. + +Level-order traversal essentially belongs to the "breadth-first traversal breadth-first traversal", which embodies a "circle by circle outward" layer-by-layer traversal. + +![Level-order traversal of binary trees](binary_tree_traversal.assets/binary_tree_bfs.png) + +### Code Implementation + +Breadth-first traversal is usually implemented with the help of a "queue". While queues follow the "first in, first out" rule, breadth-first traversals follow the "progress by level" rule, but the idea behind both is the same. + +```src +[file]{binary_tree_bfs}-[class]{}-[func]{level_order} +``` + +### Complexity Analysis + +- **Time complexity $O(n)$** : All nodes are visited once, using $O(n)$ time, where $n$ is the number of nodes. +- **Space complexity $O(n)$** : In the worst case, i.e., a full binary tree, there are at most $(n + 1) / 2$ nodes in the queue at the same time before traversing to the bottom level, occupying $O(n)$ space. + +## Pre-Order, Middle-Order, And Back-Order Traversal + +Accordingly, preorder, midorder and postorder traversal are all "depth-first traversal depth-first traversal", which embodies a traversal method of "going to the end first and then backtracking to continue". + +The figure below shows how depth-first traversal of a binary tree works. **Depth-first traversal is like "walking" around the entire perimeter of the binary tree**, encountering three positions at each node, corresponding to a preorder traversal, a middle-order traversal, and a postorder traversal. + +![Pre-, mid-, and postorder traversal of a binary search tree](binary_tree_traversal.assets/binary_tree_dfs.png) + +### Code Implementation + +Depth-first search is usually implemented based on recursion: + +```src +[file]{binary_tree_dfs}-[class]{}-[func]{post_order} +``` + +!!! note + + Depth-first search can also be implemented based on iteration, so interested students can do their own research. + +The figure below illustrates the recursion of a forward traversal of a binary tree, which can be divided into two inverse parts, "recursion" and "recursion". + +1. "Pass" indicates that a new method is opened and the program visits the next node in the process. +2. "Return" means that the function returns and the current node has been visited. + +=== "<1>" + ![Recursion process for preorder traversal](binary_tree_traversal.assets/preorder_step1.png) + +=== "<2>" + ![preorder_step2](binary_tree_traversal.assets/preorder_step2.png) + +=== "<3>" + ![preorder_step3](binary_tree_traversal.assets/preorder_step3.png) + +=== "<4>" + ![preorder_step4](binary_tree_traversal.assets/preorder_step4.png) + +=== "<5>" + ![preorder_step5](binary_tree_traversal.assets/preorder_step5.png) + +=== "<6>" + ![preorder_step6](binary_tree_traversal.assets/preorder_step6.png) + +=== "<7>" + ![preorder_step7](binary_tree_traversal.assets/preorder_step7.png) + +=== "<8>" + ![preorder_step8](binary_tree_traversal.assets/preorder_step8.png) + +=== "<9>" + ![preorder_step9](binary_tree_traversal.assets/preorder_step9.png) + +=== "<10>" + ![preorder_step10](binary_tree_traversal.assets/preorder_step10.png) + +=== "<11>" + ![preorder_step11](binary_tree_traversal.assets/preorder_step11.png) + +### Complexity Analysis + +- **Time complexity $O(n)$** : All nodes are visited once, using $O(n)$ time. +- **Space complexity $O(n)$** : In the worst case, i.e., when the tree degenerates to a linked list, the depth of recursion reaches $n$ , and the system occupies $O(n)$ stack frame space. diff --git a/docs-en/chapter_tree/index.md b/docs-en/chapter_tree/index.md new file mode 100644 index 0000000000..8374ad99ee --- /dev/null +++ b/docs-en/chapter_tree/index.md @@ -0,0 +1,13 @@ +# Tree + +

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    7dyZ6dx8Y8pl);P>+tPl zNr&&bKTe@D{}UQEIH8!8@Ck~xkeT%guibThdHx($zi#P zJJtVp4x?hEd(oTqYU@;lY3O+g1^4D1|7w)5KI6w(}xslMJjaIlI-} z>nDpm&_*m%wy*?>7G7c|gm7YSAb^cGbOsBAu>c1l0EQNr;Q@?50VZxx0iYmTQ)_(3 z$G3N?M9H}E22B`G3574$lLw`J`8x)$U-zG&#eyqp?o<>K`Fl$hKz!Qyn2PQ%pSCR4M&HuHGW8F9uS9Ah$he=}heW?v z9E`3SGMDW@#Qona##t*NWOO5G989&tn?v=JeD?m-O)I zDwlL-@7IloYvbK57@V2n*}*3q{tfbCM0Vb5F=v;>HjrW1Ptc-a`FS?^UIEOoK6Aq` z$)f9O`dtxIBG1~Tn4!gyY=a3(WL^AJj8)tVALwO<`Cc!4^=GEczo2}ehQEsA<_eUA z_ll{zknqBF_PVRZrgTJX6=s7u&~KzffHo5HmyjfLrLy9Pe(@Dn;f=tuH%d&4_RXZ- zzhilDdaCB6v0pJ%URA4ppg2Ev?&b1Y{|7p$WW$w6>0c*vDt-3xCZ|NOqeqS$Qp!UP0QCoL+){iH9s zxmp%EWeedY`p7q$>JrKOB`0rMbxWIzxvnHp!<}8m_t9L;qt)$oJQsm;`%9p{`Hs=U z!D*p(NMxn(hbI@l<>#MZ+a~RMxaDJ2T8~s;bigHH$3pYtq6P~22PyG$SRsuEY!=lc-_gtT=Mt=<53(OW-m&!;k zl1(t>N>D>z^N%LE0`U@^u+hoD3syJ}UvMk~^4^O#VC&DJZ`xMKVg8QjyMJ?P?IlF zF8z^EU7er(bVZ}=mB>Q$s%VqT+lA4@f}?7sQ@AraHuL>InbWGaUWU1-M&r^ z$89h98H+cHqcSwrJ{O$DbMntd4U{c@)o4{e^eF}(QDi_&KsZrcFITnpCpk>Ns?f1U z4oi}E4(}M54ry0bQe#E^OJ4Sf&-gf<4TU@YVgbj^zt1nA}WfQbI(j<7eJh zAz_Q<5P{O!yEd=y>AOTje-;nf?L2=>k?7SCDd^1jO4pdAlt(2u&#=UIkZ01$*+fRhqn4|!OU z^M0&F4n1IM---sPJ;R8};Z2Gwr=RbHMmm#bo08&M$N8dANNzZ!T`nP^TPZ-Kf*iXp z3QBGetk?eFc!zegY$U;+jc1oG`p~=23GWB(6*JAkjw@v$%3XY@46;@)@SboyDI)a)W60C1M^GAd5go{OB%j z;}Wqci4&BRL&aeFmh0SNiLP=S5voD7(6M?;oc(Xm8>Bdu<84tnO%wta+qqtjWkic$O?Upx5)9#X8{KDh7FW4j%@jtuRo$qO~mzhu#f^4W)}lI^5OZ2hc6{#x*njU z$YoPVWLTu#CMsfXkK|vH*?&z6r%Ikv_+C!JMuyvW58wXQg^RNlPoqUL@#P#{H|dJ{JAxEVX#k+nPZhsGBXHu@ z6;SYJ@F{;uzdL#A6@zSvA@NmWZ<)5l6g9{B=7qWFYScyO^+iOpr4tT^1kpO#BVxP8 zNBSlw=-&gv&v)7BL#jp`woM!UTs>|_R zfSqTzH!fF2Tg)g79+|fvo<96N<{%)x_$^2@OHEB?_tPY;(@FjmXNwyn3%On~+mpx1 zxy3H?G)3QD_xGaXZ$3@q=gn15HNT*bGTjt8@~KPDMyjY)QD&>^9h0f4NfGxwe|-Lq TVozWi5dXB*4b" + ![merge sort step](merge_sort.assets/merge_sort_step1.png) + +=== "<2>" + ![merge_sort_step2](merge_sort.assets/merge_sort_step2.png) + +=== "<3>" + ![merge_sort_step3](merge_sort.assets/merge_sort_step3.png) + +=== "<4>" + ![merge_sort_step4](merge_sort.assets/merge_sort_step4.png) + +=== "<5>" + ![merge_sort_step5](merge_sort.assets/merge_sort_step5.png) + +=== "<6>" + ![merge_sort_step6](merge_sort.assets/merge_sort_step6.png) + +=== "<7>" + ![merge_sort_step7](merge_sort.assets/merge_sort_step7.png) + +=== "<8>" + ![merge_sort_step8](merge_sort.assets/merge_sort_step8.png) + +=== "<9>" + ![merge_sort_step9](merge_sort.assets/merge_sort_step9.png) + +=== "<10>" + ![merge_sort_step10](merge_sort.assets/merge_sort_step10.png) + +Observe that the recursion sort is consistent with the recursion order of the backward traversal of the binary tree. + +- **Posterior order traversal**: recursion to the left-child node, then to the right-child node, and finally deal with the root node. +- **Recursion sort**: recursion on the left subarray, then recursion on the right subarray, and finally process the merge. + +```src +[file]{merge_sort}-[class]{}-[func]{merge_sort} +``` + +It is worth noting that the interval to be merged for `nums` is `[left, right]`, while the corresponding interval for `tmp` is `[0, right - left]`. + +## Algorithm Properties + +- **Time complexity $O(n \log n)$, Non-adaptive sorting**: division produces a recursion tree of height $\log n$, and the total number of operations for merging at each level is $n$ , so the overall time complexity is $O(n \log n)$ . +- **Space complexity $O(n)$, non-in-place sort**: recursion depth is $\log n$, using $O(\log n)$ size stack frame space. The merge operation needs to be implemented with the help of an auxiliary array, using extra space of size $O(n)$. +- **Stable ordering**: the order of equal elements remains the same during the merge process. + +## Linked List Sorting + +For linked lists, merge sort has a significant advantage over other sorting algorithms, **It can optimize the space complexity of the linked list sorting task to $O(1)$** . + +- **Division Phase**: The division of the linked list can be achieved by using "iteration" instead of "recursion", thus eliminating the use of stack frames for recursion. +- **Merge Phase**: In a linked list, node addition and deletion operations can be accomplished by simply changing references (pointers), so the merge phase (combining two short ordered linked lists into one long ordered linked list) does not require the creation of additional linked lists. + +Specific implementation details are more complex, interested students can consult the relevant information to learn. 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Let's take the help of an algorithm problem for better solution. + +!!! question + + Given an array of integers `nums` and a target element `target`, search the array for two elements whose "and" is `target` and return their array indexes. It is sufficient to return any one of the solutions. + +## Linear Lookups: Trading Time For Space + +Consider directly traversing all possible combinations. As shown in the figure below, we open a two-level loop that determines in each round whether the sum of two integers is `target` , and if so, returns their indexes. + +![Linear lookup to solve for sum of two numbers](replace_linear_by_hashing.assets/two_sum_brute_force.png) + +```src +[file]{two_sum}-[class]{}-[func]{two_sum_brute_force} +``` + +The time complexity of this method is $O(n^2)$ and the space complexity is $O(1)$ , which is very time-consuming in large data volume. + +## Hash Lookups: Trading Space For Time + +Consider with the help of a hash table where the key-value pairs are array elements and element indexes respectively. Traverse the array in a loop, performing the steps shown in the figure below each round. + +1. Determine whether the number `target - nums[i]` is in the hash table, if so, return the indexes of the two elements directly. +2. Add the key-value pair `nums[i]` and the index `i` to the hash table. + +=== "<1>" + ![Auxiliary hash table to solve for the sum of two numbers](replace_linear_by_hashing.assets/two_sum_hashtable_step1.png) + +=== "<2>" + ![two_sum_hashtable_step2](replace_linear_by_hashing.assets/two_sum_hashtable_step2.png) + +=== "<3>" + ![two_sum_hashtable_step3](replace_linear_by_hashing.assets/two_sum_hashtable_step3.png) + +The implementation code is shown below and requires only a single level of looping. + +```src +[file]{two_sum}-[class]{}-[func]{two_sum_hash_table} +``` + +This method reduces the time complexity from $O(n^2)$ to $O(n)$ by hash lookup, which significantly improves the operation efficiency. + +The space complexity is $O(n)$ due to the need to maintain an additional hash table. **Nevertheless, the overall space-time efficiency of this method is more balanced and therefore it is the optimal solution to this problem**. diff --git 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that the pointers $i$ and $j$ are closer to `target`**. +- When `nums[m] == target`, it means that the element smaller than `target` is in the interval $[i, m - 1]$, so $j = m - 1$ is used to narrow the interval, **so that the pointer $j$ is closer to the element smaller than `target`**. + +When the loop finishes, $i$ points to the leftmost `target` and $j$ points to the first element that is smaller than `target`, **so index $i$ is the insertion point**. + +=== "<1>" + ![Steps for binary search for insertion points of duplicate elements](binary_search_insertion.assets/binary_search_insertion_step1.png) + +=== "<2>" + ![binary_search_insertion_step2](binary_search_insertion.assets/binary_search_insertion_step2.png) + +=== "<3>" + ![binary_search_insertion_step3](binary_search_insertion.assets/binary_search_insertion_step3.png) + +=== "<4>" + ![binary_search_insertion_step4](binary_search_insertion.assets/binary_search_insertion_step4.png) + +=== "<5>" + ![binary_search_insertion_step5](binary_search_insertion.assets/binary_search_insertion_step5.png) + +=== "<6>" + ![binary_search_insertion_step6](binary_search_insertion.assets/binary_search_insertion_step6.png) + +=== "<7>" + ![binary_search_insertion_step7](binary_search_insertion.assets/binary_search_insertion_step7.png) + +=== "<8>" + ![binary_search_insertion_step8](binary_search_insertion.assets/binary_search_insertion_step8.png) + +Observe the following code and determine that the branches `nums[m] > target` and `nums[m] == target` operate the same way, so both can be merged. + +Even so, we can still keep the judgment condition unfolded because its logic is clearer and more readable. + +```src +[file]{binary_search_insertion}-[class]{}-[func]{binary_search_insertion} +``` + +!!! tip + + The code in this section is written in the "double-closed interval" style. If you are interested, you can implement the "left-closed-right-open" method by yourself. + +In a nutshell, a binary search is nothing more than setting a target for pointers $i$ and $j$, which may be a specific element (e.g., `target`) or a range of elements (e.g., elements less than `target`). + +In a continuous loop of bisection, the pointers $i$ and $j$ are both approaching a predefined goal. Eventually, they either succeed in finding the answer, or they cross the edge and stop. diff --git a/docs-en/chapter_searching/index.md b/docs-en/chapter_searching/index.md new file mode 100644 index 0000000000..b9a0aae2c8 --- /dev/null +++ b/docs-en/chapter_searching/index.md @@ -0,0 +1,13 @@ +# Search + +

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--- /dev/null +++ b/docs-en/chapter_heap/top_k.md @@ -0,0 +1,71 @@ +# Top-K Problem + +!!! question + + Given an unordered array of length $n$ `nums` , return the first $k$ largest element in the array. + +For this problem, we present two solutions that are more straightforward in their thinking, followed by the more efficient heap solution. + +## Method 1: Iterate Through The Selection + +We can perform the $k$ rounds of traversal shown in the figure below, extracting the $1$, $2$, $\dots$, and $k$ largest element in each round, respectively, with a time complexity of $O(nk)$ . + +This method is only applicable to the case of $k \ll n$, because when $k$ is close to $n$, its time complexity tends to $O(n^2)$, which is very time-consuming. + +![Traversal to find the largest k elements](top_k.assets/top_k_traversal.png) + +!!! tip + + When $k = n$, we get a complete ordered sequence, which is equivalent to a "selection sort" algorithm. + +## Method 2: Sorting + +As shown in the figure below, we can sort the array `nums` and then return the rightmost $k$ elements with a time complexity of $O(n \log n)$ . + +Obviously, this method "overshoots" the task, since we only need to find the largest $k$ element and not sort the others. + +![Sorting to find the largest k elements](top_k.assets/top_k_sorting.png) + +## Method 3: Heap + +We can solve the Top-K problem more efficiently based on heaps, the process is shown in the figure below. + +1. Initialize a min heap with the smallest top element. +2. The first $k$ elements of the array are first put into the heap in order. +3. Starting from the $k + 1$th element, if the current element is larger than the top element of the heap, then the top element of the heap is taken out of the heap and the current element is put into the heap. +4. After the traversal is complete, it is the largest $k$ element that is kept in the heap. + +=== "<1>" + ![Find the largest k elements based on heap](top_k.assets/top_k_heap_step1.png) + +=== "<2>" + ![top_k_heap_step2](top_k.assets/top_k_heap_step2.png) + +=== "<3>" + ![top_k_heap_step3](top_k.assets/top_k_heap_step3.png) + +=== "<4>" + ![top_k_heap_step4](top_k.assets/top_k_heap_step4.png) + +=== "<5>" + ![top_k_heap_step5](top_k.assets/top_k_heap_step5.png) + +=== "<6>" + ![top_k_heap_step6](top_k.assets/top_k_heap_step6.png) + +=== "<7>" + ![top_k_heap_step7](top_k.assets/top_k_heap_step7.png) + +=== "<8>" + ![top_k_heap_step8](top_k.assets/top_k_heap_step8.png) + +=== "<9>" + ![top_k_heap_step9](top_k.assets/top_k_heap_step9.png) + +A total of $n$ rounds of heap entry and heap exit are performed, and the maximum length of the heap is $k$ , so the time complexity is $O(n \log k)$ . The method is very efficient, and the time complexity tends to $O(n)$ when $k$ is small, and does not exceed $O(n \log n)$ when $k$ is large. + +In addition, this method is suitable for dynamic data streaming usage scenarios. We can continuously maintain the elements in the heap as we keep adding data, thus enabling dynamic updates of up to $k$ elements. + +```src +[file]{top_k}-[class]{}-[func]{top_k_heap} +``` diff --git a/docs-en/chapter_introduction/algorithms_are_everywhere.assets/binary_search_dictionary_step1.png b/docs-en/chapter_introduction/algorithms_are_everywhere.assets/binary_search_dictionary_step1.png new file mode 100644 index 0000000000000000000000000000000000000000..d1f59f0e126741beb41ca5f026bc18c850bc0cdb GIT binary patch literal 15329 zcmd6ObySt#*XMJK?oR0x5b4w_Ez%%@)TKK`O5y>eL=Z$8kp=+~>EOtlrG#=4SLPdU0FL?t-X0qp`!-xWJUko~71hwtP+VLbA0PkW!v{r0#n#r=ii(Pq zl$6EA#kjb*pFe-j&d!c+j89Kb%gD&&=jUr^XgqrK=6@7}+EZ(v|hSy`EumKGQoC>|R=I5^n7)b;M&J3l|a(b3VkmXDR=6_%Ekrx&L! zEiLZu?sj%|o5!1uj*dAwIm2ti;f-P5-roHy{WUc;hi8Ya^R2t5y9sUav9Yls4R6!C z)5}JGC@U-H_2;H^BuD>@a&>iGJy@C9o@tzE_&Qjis;Ziqnc3IZ=j`m<-rinaU0pL- zQ#DcL-`nP0R-8XKIl8wUTo+W-)8X0u>(~14mDA(I@xi2@jp(rv6LeMo+)UI+uR~Ez 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100644 index 0000000000..6c7d691a44 --- /dev/null +++ b/docs-en/chapter_introduction/algorithms_are_everywhere.md @@ -0,0 +1,56 @@ +# Algorithms Are Everywhere + +When we hear the word "algorithm", we naturally think of math. In reality, however, many algorithms do not involve complex math, but rely more on basic logic, which can be found in our daily lives. + +Before formally exploring algorithms, there is an interesting fact worth sharing:**You have already learned many algorithms without realizing it and are used to applying them in your daily life**. Below, I will give a few concrete examples to confirm this. + +**Example 1: Consulting the dictionary**. In a dictionary, each Chinese character corresponds to a Pinyin character, and the dictionary is organized by Pinyin alphabetical order. Suppose we need to look up a character whose pinyin initial letter is $r$, we usually realize it as shown in the figure below. + +1. Go through about half of the pages of the dictionary to see what the first letter of the page is, assuming the first letter is $m$ . +2. Since $r$ comes after $m$ in the pinyin alphabet, the first half of the dictionary is excluded and the search is narrowed down to the second half. +3. Repeat step `1.` and step `2.` until you find the page number with the initial letter $r$. + +=== "<1>" + ![Dictionary search step](algorithms_are_everywhere.assets/binary_search_dictionary_step1.png) + +=== "<2>" + ![binary_search_dictionary_step2](algorithms_are_everywhere.assets/binary_search_dictionary_step2.png) + +=== "<3>" + ![binary_search_dictionary_step3](algorithms_are_everywhere.assets/binary_search_dictionary_step3.png) + +=== "<4>" + ![binary_search_dictionary_step4](algorithms_are_everywhere.assets/binary_search_dictionary_step4.png) + +=== "<5>" + ![binary_search_dictionary_step5](algorithms_are_everywhere.assets/binary_search_dictionary_step5.png) + +Looking up the dictionary, a necessary skill for elementary school students, is actually the famous binary search algorithm. From the data structure point of view, we can regard the dictionary as a sorting algorithm, and from the algorithm point of view, we can regard the above series of operations as "binary search". + +**Example 2: Organize Poker**. When we play poker, we need to organize the playing cards so that they are arranged from smallest to largest in each game, the implementation process is shown in the figure below. + +1. Divide the playing cards into "ordered" and "unordered" parts, and assume that the leftmost card in the initial state is already ordered. +2. Draw a playing card from the unordered section and insert it into the correct position in the ordered section; when finished, the leftmost 2 playing cards are in order. +3. Keep cycling through the steps `2.` , inserting a playing card from the unordered part to the ordered part each round until all cards are in order. + +![Poker sorting steps](algorithms_are_everywhere.assets/playing_cards_sorting.png) + +The above method of organizing playing cards is essentially an "insertion sort" algorithm, which is very efficient when dealing with small data sets. Insertion sort can be found in the sorting library functions of many programming languages. + +**Example 3: Currency change**. Suppose we purchased $69$ yuan of goods at the supermarket and gave the cashier $100$ yuan, then the cashier needs to find us $31$ yuan. He will naturally complete the thinking shown in the figure below. + +1. Options are currencies with smaller denominations than $31$ yuan, including $1$ yuan, $5$ yuan, $10$ yuan, $20$ yuan. +2. Take the largest $20$ dollars from the available options, leaving $31 - 20 = 11$ dollars. +3. Take the largest $10$ from the remaining available options, leaving $11 - 10 = 1$. +4. Take the largest $1$ from the remaining available options, leaving $1 - 1 = 0$. +5. Complete the change and the solution is $20 + 10 + 1 = 31$. + +![Currency change process](algorithms_are_everywhere.assets/greedy_change.png) + +In each of the above steps, we take the best option available at the time (using as large a denomination of currency as possible) and end up with a feasible solution for finding change. From a data structure and algorithm perspective, this approach is essentially a "greedy algorithm". + +Algorithms are essential to the solution of almost any problem, from cooking a dish to interstellar voyages. The advent of computers has allowed us to program data structures to be stored in memory while writing code that calls the CPU and GPU to execute algorithms. In this way, we are able to transfer the problems of our lives to computers and solve all kinds of complex problems in a more efficient way. + +!!! tip + + If you still feel half-aware of concepts such as data structures, algorithms, arrays, and binary searches, read on as this book will guide you through the halls of knowledge of data structures and algorithms. diff --git a/docs-en/chapter_introduction/index.md b/docs-en/chapter_introduction/index.md new file mode 100644 index 0000000000..9d4fa14b83 --- /dev/null +++ b/docs-en/chapter_introduction/index.md @@ -0,0 +1,13 @@ +# Getting To Know The Algorithm For The First Time + +

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+=== "<3>" + ![max_capacity_greedy_step3](max_capacity_problem.assets/max_capacity_greedy_step3.png) + +=== "<4>" + ![max_capacity_greedy_step4](max_capacity_problem.assets/max_capacity_greedy_step4.png) + +=== "<5>" + ![max_capacity_greedy_step5](max_capacity_problem.assets/max_capacity_greedy_step5.png) + +=== "<6>" + ![max_capacity_greedy_step6](max_capacity_problem.assets/max_capacity_greedy_step6.png) + +=== "<7>" + ![max_capacity_greedy_step7](max_capacity_problem.assets/max_capacity_greedy_step7.png) + +=== "<8>" + ![max_capacity_greedy_step8](max_capacity_problem.assets/max_capacity_greedy_step8.png) + +=== "<9>" + ![max_capacity_greedy_step9](max_capacity_problem.assets/max_capacity_greedy_step9.png) + +### Code Implementation + +The code loops for at most $n$ rounds, **so the time complexity is $O(n)$**. + +The variables $i$, $j$, $res$ use constant size extra space, **so the space complexity is $O(1)$** . + +```src +[file]{max_capacity}-[class]{}-[func]{max_capacity} +``` + +### Proof Of Correctness + +The reason why greedy is faster than exhaustive is that each round of greedy choices "skips" some states. + +For example, in state $cap[i, j]$, $i$ is the short board and $j$ is the long board. Greedily moving the short board $i$ inward by one frame will cause the states shown in the figure below to be "skipped". **This means that the capacity of these states cannot be verified later**. + +$$ +cap[i, i+1], cap[i, i+2], \dots, cap[i, j-2], cap[i, j-1] +$$ + +![Moving short boards causes skipped states](max_capacity_problem.assets/max_capacity_skipped_states.png) + +Observe that these skipped states are actually all the states that move the longboard $j$ inwards. And in the second step, we have shown that moving the longboard inward must result in a smaller capacity. 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They are implemented differently in the "adjacency matrix" and "adjacency list" representations. + +## Implementation Based On Adjacency Matrices + +Given an undirected graph with $n$ number of vertices, the various operations are implemented as shown in the figure below. + +- **Add or remove edges**: just modify the specified edges directly in the adjacency matrix, using $O(1)$ time. And since it is an undirected graph, you need to update the edges in both directions at the same time. +- **Add vertex**: add a row and a column to the end of the adjacency matrix and fill them all $0$ that's it, using $O(n)$ time. +- **Delete vertex**: deletes a row and a column in the adjacency matrix. The worst case is reached when deleting the first row and column, which requires $(n-1)^2$ elements to be "shifted up to the left", thus using $O(n^2)$ time. +- **Initialize**: pass $n$ vertices, initialize the list of vertices `vertices` of length $n$, using $O(n)$ time; initialize the adjacency matrix `adjMat` of size $n \times n$, using $O(n^2)$ time. + +=== "Initialize the adjacency matrix." + ![Initialization of adjacency matrix, adding and removing edges, adding and removing vertices](graph_operations.assets/adjacency_matrix_initialization.png) + +=== "Add edge" + ![adjacency_matrix_add_edge](graph_operations.assets/adjacency_matrix_add_edge.png) + +=== "Delete edge" + ![adjacency_matrix_remove_edge](graph_operations.assets/adjacency_matrix_remove_edge.png) + +=== "Add vertex" + ![adjacency_matrix_add_vertex](graph_operations.assets/adjacency_matrix_add_vertex.png) + +=== "Delete vertex" + ![adjacency_matrix_remove_vertex](graph_operations.assets/adjacency_matrix_remove_vertex.png) + +The following is the implementation code based on the adjacency matrix representation of the graph. + +```src +[file]{graph_adjacency_matrix}-[class]{graph_adj_mat}-[func]{} +``` + +## Implementation Based On Adjacency Lists + +Let the total number of vertices of an undirected graph be $n$ and the total number of edges be $m$ , then various operations can be realized according to the method shown in the figure below. + +- **Add Edge**: just add the edge at the end of the linked list corresponding to the vertex, using $O(1)$ time. Since it is an undirected graph, you need to add edges in both directions. +- **Delete Edge**: finds and deletes the specified edge in the linked list corresponding to the vertex, using $O(m)$ time. In an undirected graph, edges in both directions need to be deleted simultaneously. +- **Add vertex**: add a linked list to the adjacency list and use the added vertex as the linked list node, using $O(1)$ time. +- **Delete vertex**: need to traverse the entire adjacency list and delete all edges containing the specified vertex, using $O(n + m)$ time. +- **Initialization**: create $n$ vertices and $2m$ edges in the adjacency list, using $O(n + m)$ time. + +=== "Initialize the adjacency list." + ![Initialization of adjacency list, adding and removing edges, adding and removing vertices](graph_operations.assets/adjacency_list_initialization.png) + +=== "Add edge" + ![adjacency_list_add_edge](graph_operations.assets/adjacency_list_add_edge.png) + +=== "Delete edge" + ![adjacency_list_remove_edge](graph_operations.assets/adjacency_list_remove_edge.png) + +=== "Add vertex" + ![adjacency_list_add_vertex](graph_operations.assets/adjacency_list_add_vertex.png) + +=== "Delete vertex" + ![adjacency_list_remove_vertex](graph_operations.assets/adjacency_list_remove_vertex.png) + +The following is the code implementation of the adjacency list. Comparing to the figure above, the actual code has the following differences. + +- To make it easier to add and remove vertices, and to simplify the code, we use lists (dynamic arrays) instead of linked lists. +- A hash table is used to store the adjacency table, `key` is the vertex instance, and `value` is the list of adjacency vertices (linked list) for that vertex. + +In addition, we use the `Vertex` class to represent the vertices in the adjacency table because if we use the list index to distinguish different vertices as in the adjacency matrix, then suppose we want to delete a vertex with index $i$, we need to traverse the whole adjacency list and subtract all indexes greater than $i$ from $1$, which is very inefficient. If each vertex is a unique instance of `Vertex`, then after deleting a vertex, there is no need to change the other vertices. + +```src +[file]{graph_adjacency_list}-[class]{graph_adj_list}-[func]{} +``` + +## Efficiency Comparison + +Given that there are $n$ vertices and $m$ edges in the graph, the table below compares the time and space efficiency of the adjacency matrix and adjacency list. + +

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z52}_AEJ|!E{ng%Nh7Y?JsgP6P6>F$0&WMlm$9NM7?mUw%BxF>5iA_U$cNQ)e8jkOqGxZmfw*6gNKw!46&KiDRTNCfj~2ypBtPhpQ+yy7KL9aq3y=zB z;-1#(nst)(Rpl}8OfnBCA zJAd+of1k2q7^(WAfVq=s92M1D{dOYDeQ%>rQV2i^``UPYr^H}yY3I1eHeZA<9{uY8 NYD(IQwes+n{{c*a@st1n literal 0 HcmV?d00001 diff --git a/docs-en/chapter_dynamic_programming/dp_solution_pipeline.md b/docs-en/chapter_dynamic_programming/dp_solution_pipeline.md new file mode 100644 index 0000000000..9f46af5473 --- /dev/null +++ b/docs-en/chapter_dynamic_programming/dp_solution_pipeline.md @@ -0,0 +1,181 @@ +# Dynamic Programming Solutions + +The last two sections introduced the main features of dynamic programming problems, and next we explore two more practical problems together. + +1. How can you tell if a problem is a dynamic programming problem? +2. Where to start and what are the complete steps for solving dynamic programming problems? + +## Problem Determination + +In general, if a problem contains overlapping subproblems, optimal substructures, and satisfies no posteriority, then it is usually suitable for solution by dynamic programming. However, it is difficult to extract these properties directly from the problem description. **Therefore we usually relax the conditions and first observe whether the problem is suitable to be solved using backtracking (exhaustive)**. + +**Problems suitable for solution by backtracking usually satisfy the "Decision Tree Model"**, which can be described using a tree structure where each node represents a decision and each path represents a sequence of decisions. + +In other words, if the problem contains explicit decision concepts and the solution is generated through a series of decisions, then it satisfies the decision tree model and can usually be solved using backtracking. + +On this basis, there are some judgmental "plus points" for dynamic programming problems. + +- The problem contains optimization descriptions such as maximum (small) or maximum (small). +- The states of a problem can be represented using a list, a multidimensional matrix, or a tree, and there is a recursive relationship between a state and its surrounding states. + +Accordingly, there are some "minuses". + +- The goal of the problem is to find all possible solutions, not to find the optimal solution. +- The problem description is characterized by distinct permutations and combinations that need to be returned for specific multiple scenarios. + +If a problem satisfies the decision tree model and has more obvious "plus points", we can assume that it is a dynamic programming problem and verify it in the solution process. + +## Problem Solving Steps + +The solution process for dynamic programming will vary depending on the nature and difficulty of the problem, but usually follows the following steps: describe the decision, define the state, build the $dp$ table, derive the state-transition equation, determine the boundary conditions, etc. + +In order to show the solution step by step, we use a classic problem "Minimum Path Sum" as an example. + +!!! question + + Given a $n \times m$ two-dimensional grid `grid` , each cell in the grid contains a non-negative integer indicating the cost of that cell. The robot starts at the top-left cell and can only move one step down or to the right at a time until it reaches the bottom-right cell. Please return the minimum path sum from the upper-left corner to the lower-right corner. + +The figure below shows an example where the minimum path sum for a given grid is $13$ . + +![Minimum path and example data](dp_solution_pipeline.assets/min_path_sum_example.png) + +**Step 1: Think about the decisions made in each round and define the state that leads to the $dp$ table** + +The decision in each round of this problem is to take one step down or to the right from the current lattice. Let the row and column index of the current lattice be $[i, j]$ , then the index becomes $[i+1, j]$ or $[i, j+1]$ after taking a step down or right. Therefore, the state should contain both row index and column index variables, denoted as $[i, j]$ . + +The subproblem corresponding to the state $[i, j]$ is the minimum path sum from the start point $[0, 0]$ to $[i, j]$ and the solution is denoted as $dp[i, j]$ . + +At this point, we have the two-dimensional $dp$ matrix shown in the figure below, with the same dimensions as the input grid $grid$. + +![state definition with dp table](dp_solution_pipeline.assets/min_path_sum_solution_step1.png) + +!!! note + + The dynamic programming and backtracking process can be described as a sequence of decisions, and the state consists of all the decision variables. It should contain all the variables describing the progress of the solution, and it contains enough information that can be used to derive the next state. + + Each state corresponds to a subproblem, and we will define a $dp$ table to store the solutions of all the subproblems, with each independent variable of the state being a dimension of the $dp$ table. Essentially, the $dp$ table is a mapping between the states and the solutions of the subproblems. + +**Step 2: Identify the optimal substructure and hence derive the state-transition equation** + +For the state $[i, j]$ , it can only be transferred from the upper edge lattice $[i-1, j]$ and the left lattice $[i, j-1]$ . Therefore the optimal substructure is: the minimum path sum to reach $[i, j]$ is determined by the smaller of the minimum path sum of $[i, j-1]$ and the minimum path sum of $[i-1, j]$. + +Based on the above analysis, the state-transition equation shown in the figure below can be introduced: + +$$ +dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j] +$$ + +![Optimal substructure and state-trasition equation](dp_solution_pipeline.assets/min_path_sum_solution_step2.png) + +!!! note + + Based on the defined $dp$ table, think about the relationship between the original problem and the subproblems, and find out the way to construct the optimal solution of the original problem through the optimal solution of the subproblems, i.e., the optimal substructure. + + Once we have found the optimal substructure, we can use it to construct the state-transition equation. + +**Step 3: Determine boundary conditions and state transfer order** + +In this problem, the state of the first row can only be derived from the state to its left, and the state of the first column can only be derived from the state above it, so the first row $i = 0$ and the first column $j = 0$ are boundary conditions. + +As shown in the figure below, since each grid is transferred from its left and top grids, we use employ a loop to traverse the matrix, an outer loop to traverse the rows and an inner loop to traverse the columns. + +![Boundary conditions and state transfer order](dp_solution_pipeline.assets/min_path_sum_solution_step3.png) + +!!! note + + Boundary conditions are used in dynamic programming to initialize $dp$ tables and in search for pruning. + + The core of the state transfer order is to ensure that by the time the solution to the current problem is computed, all the solutions to the smaller subproblems on which it depends have already been computed correctly. + +Based on the above analysis, we can already write dynamic programming code directly. However, subproblem decomposition is a kind of top-to-bottom idea, so it is more in line with the habit of thinking to implement it in the order of "Brute-force Search $\rightarrow$ Memorized Search $\rightarrow$ Dynamic Programming". + +### Method One: Brute Force Search + +Starting the search from the state $[i, j]$ and continuously decomposing into smaller states $[i-1, j]$ and $[i, j-1]$ , the recursion function includes the following elements. + +- **Recursion parameter**: state $[i, j]$ . +- **Return value**: the minimum path from $[0, 0]$ to $[i, j]$ and $dp[i, j]$ . +- **Termination condition**: return cost $grid[0, 0]$ when $i = 0$ and $j = 0$ . +- **pruning**: index crossing when $i < 0$ or $j < 0$, in which case the cost $+\infty$ is returned, representing infeasibility. + +```src +[file]{min_path_sum}-[class]{}-[func]{min_path_sum_dfs} +``` + +The figure below gives a recursion tree node with $dp[2, 1]$ as the root node, which contains a number of overlapping subproblems, the number of which increases dramatically as the size of the grid `grid` becomes larger. + +Essentially, the cause of the overlapping sub-problem is that **there exist multiple paths to reach a cell from the upper left corner**. + +![Brute-force search recursion tree](dp_solution_pipeline.assets/min_path_sum_dfs.png) + +Each state has two choices, down and right, and it takes a total of $m + n - 2$ steps to walk from the top left corner to the bottom right corner, so the worst time complexity is $O(2^{m + n})$ . Note that this calculation does not take into account the case of approaching grid boundaries, which leaves only one choice when the network boundary is reached. Therefore the actual number of paths will be less. + +### Method 2: Memorized Search + +We introduce a memoized list `mem` of the same size as the grid `grid` for recording the solutions of individual subproblems and pruning the overlapping subproblems. + +```src +[file]{min_path_sum}-[class]{}-[func]{min_path_sum_dfs_mem} +``` + +As shown in the figure below, with the introduction of memoization, the solution to all subproblems is computed only once, so the time complexity depends on the total number of states, i.e., the grid size $O(nm)$ . + +![Memorized search recursion tree](dp_solution_pipeline.assets/min_path_sum_dfs_mem.png) + +### Method 3: Dynamic Programming + +Iteration-based implementation of dynamic programming solutions. + +```src +[file]{min_path_sum}-[class]{}-[func]{min_path_sum_dp} +``` + +The figure below illustrates the state transfer process for the minimum path sum, which traverses the entire lattice, **hence the time complexity is $O(nm)$** . + +The size of the array `dp` is $n \times m$ , **hence the space complexity is $O(nm)$** . + +=== "<1>" + ![Dynamic programming process for minimum path sum](dp_solution_pipeline.assets/min_path_sum_dp_step1.png) + +=== "<2>" + ![min_path_sum_dp_step2](dp_solution_pipeline.assets/min_path_sum_dp_step2.png) + +=== "<3>" + ![min_path_sum_dp_step3](dp_solution_pipeline.assets/min_path_sum_dp_step3.png) + +=== "<4>" + ![min_path_sum_dp_step4](dp_solution_pipeline.assets/min_path_sum_dp_step4.png) + +=== "<5>" + ![min_path_sum_dp_step5](dp_solution_pipeline.assets/min_path_sum_dp_step5.png) + +=== "<6>" + ![min_path_sum_dp_step6](dp_solution_pipeline.assets/min_path_sum_dp_step6.png) + +=== "<7>" + ![min_path_sum_dp_step7](dp_solution_pipeline.assets/min_path_sum_dp_step7.png) + +=== "<8>" + ![min_path_sum_dp_step8](dp_solution_pipeline.assets/min_path_sum_dp_step8.png) + +=== "<9>" + ![min_path_sum_dp_step9](dp_solution_pipeline.assets/min_path_sum_dp_step9.png) + +=== "<10>" + ![min_path_sum_dp_step10](dp_solution_pipeline.assets/min_path_sum_dp_step10.png) + +=== "<11>" + ![min_path_sum_dp_step11](dp_solution_pipeline.assets/min_path_sum_dp_step11.png) + +=== "<12>" + ![min_path_sum_dp_step12](dp_solution_pipeline.assets/min_path_sum_dp_step12.png) + +### Space Optimization + +Since each grid is only related to the grid to its left and above, we can implement the $dp$ table with only a single row array. + +Note that since the array `dp` can only represent the state of one row, we can't initialize the first column's state ahead of time, but instead update it as we traverse each row. + +```src +[file]{min_path_sum}-[class]{}-[func]{min_path_sum_dp_comp} +``` diff --git a/docs-en/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_decision_tree.png b/docs-en/chapter_dynamic_programming/edit_distance_problem.assets/edit_distance_decision_tree.png new file mode 100644 index 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This concept is very similar to time complexity, just replace "runtime" with "memory space". + +## Algorithmic Correlation Space + +The main types of memory space used by algorithms during operation include the following. + +- **Input Space**: used to store the input data for the algorithm. +- **Staging Space**: used to store data such as variables, objects, function contexts, etc. of the algorithm during runtime. +- **Output Space**: used to store the output data of the algorithm. + +In general, the statistical range of space complexity is "staging space" plus "output space". + +The staging space can be further divided into three sections. + +- **Staging Data**: used to save various constants, variables, objects, etc. during the running of the algorithm. +- **Stack frame space**: used to hold the context data of the called function. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is freed when the function returns. +- **Instruction space**: used to hold compiled program instructions, usually ignored in practical statistics. + +When analyzing the space complexity of a piece of program, **we usually count three parts**: transient data, stack frame space and output data. + +![Associated spaces used by the algorithm](space_complexity.assets/space_types.png) + +=== "Python" + + ```python title="" + class Node: + """Classes"""" + def __init__(self, x: int): + self.val: int = x # node value + self.next: Node | None = None # reference to the next node + + def function() -> int: + """"Function""""" + # Perform certain operations... + return 0 + + def algorithm(n) -> int: # input data + A = 0 # Temporary storage of data (constant, usually in uppercase) + b = 0 # Temporary data (variable) + node = Node(0) # temporary data (object) + c = function() # Stack frame space (call function) + return A + b + c # output data + ``` + +=== "C++" + + ```cpp title="" + /* Structures */ + struct Node { + int val; + Node *next; + Node(int x) : val(x), next(nullptr) {} + }; + + /* Functions */ + int func() { + // Perform certain operations... + return 0; + } + + int algorithm(int n) { // input data + const int a = 0; // temporary data (constant) + int b = 0; // temporary data (variable) + Node* node = new Node(0); // temporary data (object) + int c = func(); // stack frame space (call function) + return a + b + c; // output data + } + ``` + +=== "Java" + + ```java title="" + /* Classes */ + class Node { + int val; + Node next; + Node(int x) { val = x; } + } + + /* Functions */ + int function() { + // Perform certain operations... + return 0; + } + + int algorithm(int n) { // input data + final int a = 0; // temporary data (constant) + int b = 0; // temporary data (variable) + Node node = new Node(0); // temporary data (object) + int c = function(); // stack frame space (call function) + return a + b + c; // output data + } + ``` + +=== "C#" + + ```csharp title="" + /* Classes */ + class Node { + int val; + Node next; + Node(int x) { val = x; } + } + + /* Functions */ + int Function() { + // Perform certain operations... + return 0; + } + + int Algorithm(int n) { // input data + const int a = 0; // temporary data (constant) + int b = 0; // temporary data (variable) + Node node = new(0); // temporary data (object) + int c = Function(); // stack frame space (call function) + return a + b + c; // output data + } + ``` + +=== "Go" + + ```go title="" + /* Structures */ + type node struct { + val int + next *node + } + + /* Create node structure */ + func newNode(val int) *node { + return &node{val: val} + } + + /* Functions */ + func function() int { + // Perform certain operations... + return 0 + } + + func algorithm(n int) int { // input data + const a = 0 // temporary data (constant) + b := 0 // temporary storage of data (variable) + newNode(0) // temporary data (object) + c := function() // stack frame space (call function) + return a + b + c // output data + } + ``` + +=== "Swift" + + ```swift title="" + /* Classes */ + class Node { + var val: Int + var next: Node? + + init(x: Int) { + val = x + } + } + + /* Functions */ + func function() -> Int { + // Perform certain operations... + return 0 + } + + func algorithm(n: Int) -> Int { // input data + let a = 0 // temporary data (constant) + var b = 0 // temporary data (variable) + let node = Node(x: 0) // temporary data (object) + let c = function() // stack frame space (call function) + return a + b + c // output data + } + ``` + +=== "JS" + + ```javascript title="" + /* Classes */ + class Node { + val; + next; + constructor(val) { + this.val = val === undefined ? 0 : val; // node value + this.next = null; // reference to the next node + } + } + + /* Functions */ + function constFunc() { + // Perform certain operations + return 0; + } + + function algorithm(n) { // input data + const a = 0; // temporary data (constant) + let b = 0; // temporary data (variable) + const node = new Node(0); // temporary data (object) + const c = constFunc(); // Stack frame space (calling function) + return a + b + c; // output data + } + ``` + +=== "TS" + + ```typescript title="" + /* Classes */ + class Node { + val: number; + next: Node | null; + constructor(val?: number) { + this.val = val === undefined ? 0 : val; // node value + this.next = null; // reference to the next node + } + } + + /* Functions */ + function constFunc(): number { + // Perform certain operations + return 0; + } + + function algorithm(n: number): number { // input data + const a = 0; // temporary data (constant) + let b = 0; // temporary data (variable) + const node = new Node(0); // temporary data (object) + const c = constFunc(); // Stack frame space (calling function) + return a + b + c; // output data + } + ``` + +=== "Dart" + + ```dart title="" + /* Classes */ + class Node { + int val; + Node next; + Node(this.val, [this.next]); + } + + /* Functions */ + int function() { + // Perform certain operations... + return 0; + } + + int algorithm(int n) { // input data + const int a = 0; // temporary data (constant) + int b = 0; // temporary data (variable) + Node node = Node(0); // temporary data (object) + int c = function(); // stack frame space (call function) + return a + b + c; // output data + } + ``` + +=== "Rust" + + ```rust title="" + use std::rc::Rc; + use std::cell::RefCell; + + /* Structures */ + struct Node { + val: i32, + next: Option>>, + } + + /* Creating a Node structure */ + impl Node { + fn new(val: i32) -> Self { + Self { val: val, next: None } + } + } + + /* Functions */ + fn function() -> i32 { + // Perform certain operations... + return 0; + } + + fn algorithm(n: i32) -> i32 { // input data + const a: i32 = 0; // temporary data (constant) + let mut b = 0; // temporary data (variable) + let node = Node::new(0); // temporary data (object) + let c = function(); // stack frame space (call function) + return a + b + c; // output data + } + ``` + +=== "C" + + ```c title="" + /* Functions */ + int func() { + // Perform certain operations... + return 0; + } + + int algorithm(int n) { // input data + const int a = 0; // temporary data (constant) + int b = 0; // temporary data (variable) + int c = func(); // stack frame space (call function) + return a + b + c; // output data + } + ``` + +=== "Zig" + + ```zig title="" + + ``` + +## Method Of Projection + +Space complexity is derived in much the same way as time complexity, simply by changing the statistic from "number of operations" to "amount of space used". + +And unlike time complexity, **we usually focus only on the worst space complexity**. This is because memory space is a hard requirement and we have to make sure that there is enough memory space reserved under all input data. + +Looking at the following code, the "worst" in worst space complexity has two levels of meaning. + +1. **whichever is the worst input data**: when $n < 10$, the space complexity is $O(1)$; however, when $n > 10$, the initialized array `nums` occupies $O(n)$ space; thus the worst space complexity is $O(n)$ . +2. **in terms of peak memory during the run of the algorithm**: for example, the program occupies $O(1)$ space until the last line is executed; when the array `nums` is initialized, the program occupies $O(n)$ space; thus the worst space complexity is $O(n)$ . + +=== "Python" + + ```python title="" + def algorithm(n: int): + a = 0 # O(1) + b = [0] * 10000 # O(1) + if n > 10: + nums = [0] * n # O(n) + ``` + +=== "C++" + + ```cpp title="" + void algorithm(int n) { + int a = 0; // O(1) + vector b(10000); // O(1) + if (n > 10) + vector nums(n); // O(n) + } + ``` + +=== "Java" + + ```java title="" + void algorithm(int n) { + int a = 0; // O(1) + int[] b = new int[10000]; // O(1) + if (n > 10) + int[] nums = new int[n]; // O(n) + } + ``` + +=== "C#" + + ```csharp title="" + void Algorithm(int n) { + int a = 0; // O(1) + int[] b = new int[10000]; // O(1) + if (n > 10) { + int[] nums = new int[n]; // O(n) + } + } + ``` + +=== "Go" + + ```go title="" + func algorithm(n int) { + a := 0 // O(1) + b := make([]int, 10000) // O(1) + var nums []int + if n > 10 { + nums := make([]int, n) // O(n) + } + fmt.Println(a, b, nums) + } + ``` + +=== "Swift" + + ```swift title="" + func algorithm(n: Int) { + let a = 0 // O(1) + let b = Array(repeating: 0, count: 10000) // O(1) + if n > 10 { + let nums = Array(repeating: 0, count: n) // O(n) + } + } + ``` + +=== "JS" + + ```javascript title="" + function algorithm(n) { + const a = 0; // O(1) + const b = new Array(10000); // O(1) + if (n > 10) { + const nums = new Array(n); // O(n) + } + } + ``` + +=== "TS" + + ```typescript title="" + function algorithm(n: number): void { + const a = 0; // O(1) + const b = new Array(10000); // O(1) + if (n > 10) { + const nums = new Array(n); // O(n) + } + } + ``` + +=== "Dart" + + ```dart title="" + void algorithm(int n) { + int a = 0; // O(1) + List b = List.filled(10000, 0); // O(1) + if (n > 10) { + List nums = List.filled(n, 0); // O(n) + } + } + ``` + +=== "Rust" + + ```rust title="" + fn algorithm(n: i32) { + let a = 0; // O(1) + let b = [0; 10000]; // O(1) + if n > 10 { + let nums = vec![0; n as usize]; // O(n) + } + } + ``` + +=== "C" + + ```c title="" + void algorithm(int n) { + int a = 0; // O(1) + int b[10000]; // O(1) + if (n > 10) + int nums[n] = {0}; // O(n) + } + ``` + +=== "Zig" + + ```zig title="" + + ``` + +**In recursion functions, care needs to be taken to count the stack frame space**. For example in the following code: + +- The function `loop()` calls $n$ times `function()` in a loop, and each round of `function()` returns and frees stack frame space, so the space complexity is still $O(1)$. +- The recursion function `recur()` will have $n$ unreturned `recur()` at the same time during runtime, thus occupying $O(n)$ of stack frame space. + +=== "Python" + + ```python title="" + def function() -> int: + # Perform certain operations + return 0 + + def loop(n: int): + """Loop O(1)""""" + for _ in range(n): + function() + + def recur(n: int) -> int: + """Recursion O(n)""""" + if n == 1: return + return recur(n - 1) + ``` + +=== "C++" + + ```cpp title="" + int func() { + // Perform certain operations + return 0; + } + /* Cycle O(1) */ + void loop(int n) { + for (int i = 0; i < n; i++) { + func(); + } + } + /* Recursion O(n) */ + void recur(int n) { + if (n == 1) return; + return recur(n - 1); + } + ``` + +=== "Java" + + ```java title="" + int function() { + // Perform certain operations + return 0; + } + /* Cycle O(1) */ + void loop(int n) { + for (int i = 0; i < n; i++) { + function(); + } + } + /* Recursion O(n) */ + void recur(int n) { + if (n == 1) return; + return recur(n - 1); + } + ``` + +=== "C#" + + ```csharp title="" + int Function() { + // Perform certain operations + return 0; + } + /* Cycle O(1) */ + void Loop(int n) { + for (int i = 0; i < n; i++) { + Function(); + } + } + /* Recursion O(n) */ + int Recur(int n) { + if (n == 1) return 1; + return Recur(n - 1); + } + ``` + +=== "Go" + + ```go title="" + func function() int { + // Perform certain operations + return 0 + } + + /* Cycle O(1) */ + func loop(n int) { + for i := 0; i < n; i++ { + function() + } + } + + /* Recursion O(n) */ + func recur(n int) { + if n == 1 { + return + } + recur(n - 1) + } + ``` + +=== "Swift" + + ```swift title="" + @discardableResult + func function() -> Int { + // Perform certain operations + return 0 + } + + /* Cycle O(1) */ + func loop(n: Int) { + for _ in 0 ..< n { + function() + } + } + + /* Recursion O(n) */ + func recur(n: Int) { + if n == 1 { + return + } + recur(n: n - 1) + } + ``` + +=== "JS" + + ```javascript title="" + function constFunc() { + // Perform certain operations + return 0; + } + /* Cycle O(1) */ + function loop(n) { + for (let i = 0; i < n; i++) { + constFunc(); + } + } + /* Recursion O(n) */ + function recur(n) { + if (n === 1) return; + return recur(n - 1); + } + ``` + +=== "TS" + + ```typescript title="" + function constFunc(): number { + // Perform certain operations + return 0; + } + /* Cycle O(1) */ + function loop(n: number): void { + for (let i = 0; i < n; i++) { + constFunc(); + } + } + /* Recursion O(n) */ + function recur(n: number): void { + if (n === 1) return; + return recur(n - 1); + } + ``` + +=== "Dart" + + ```dart title="" + int function() { + // Perform certain operations + return 0; + } + /* Cycle O(1) */ + void loop(int n) { + for (int i = 0; i < n; i++) { + function(); + } + } + /* Recursion O(n) */ + void recur(int n) { + if (n == 1) return; + return recur(n - 1); + } + ``` + +=== "Rust" + + ```rust title="" + fn function() -> i32 { + // Perform certain operations + return 0; + } + /* Cycle O(1) */ + fn loop(n: i32) { + for i in 0..n { + function(); + } + } + /* Recursion O(n) */ + void recur(n: i32) { + if n == 1 { + return; + } + recur(n - 1); + } + ``` + +=== "C" + + ```c title="" + int func() { + // Perform certain operations + return 0; + } + /* Cycle O(1) */ + void loop(int n) { + for (int i = 0; i < n; i++) { + func(); + } + } + /* Recursion O(n) */ + void recur(int n) { + if (n == 1) return; + return recur(n - 1); + } + ``` + +=== "Zig" + + ```zig title="" + + ``` + +## Common Types + +Let the input data size be $n$ , the figure below shows the common types of space complexity (ordered from low to high). + +$$ +\begin{aligned} +O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline +\text{constant order} < \text{logarithmic order} < \text{linear order} < \text{square order} < \text{exponential order} +\end{aligned} +$$ + +![Common space complexity types](space_complexity.assets/space_complexity_common_types.png) + +### Constant Order $O(1)$ + +Constant orders are common for constants, variables, and objects whose quantity is unrelated to the size of the input data $n$. + +Note that memory occupied by initializing a variable or calling a function in a loop is freed when it enters the next loop, so there is no accumulation of occupied space and the space complexity remains $O(1)$ : + +```src +[file]{space_complexity}-[class]{}-[func]{constant} +``` + +### Linear Order $O(N)$ + +Linear orders are commonly found in arrays, linked lists, stacks, queues, etc. where the number of elements is proportional to $n$: + +```src +[file]{space_complexity}-[class]{}-[func]{linear} +``` + +As shown in the figure below, the depth of recursion for this function is $n$, which means that there are $n$ unreturned `linear_recur()` functions at the same time, using $O(n)$ size stack frame space: + +```src +[file]{space_complexity}-[class]{}-[func]{linear_recur} +``` + +![Linear order space complexity generated by recursion function](space_complexity.assets/space_complexity_recursive_linear.png) + +### Squared Order $O(N^2)$ + +The squared order is common in matrices and graphs, where the number of elements is squared with $n$: + +```src +[file]{space_complexity}-[class]{}-[func]{quadratic} +``` + +As shown in the figure below, the recursion depth of this function is $n$ , and an array is initialized in each recursion function with lengths $n$, $n-1$, $\dots$, $2$, $1$ , and an average length of $n / 2$ , thus occupying $O(n^2)$ space overall: + +```src +[file]{space_complexity}-[class]{}-[func]{quadratic_recur} +``` + +![Square-order space complexity generated by the recursion function](space_complexity.assets/space_complexity_recursive_quadratic.png) + +### Exponential Order $O(2^N)$ + +Exponential order is common in binary trees. Looking at the figure below, a "full binary tree" of degree $n$ has $2^n - 1$ nodes, occupying $O(2^n)$ space: + +```src +[file]{space_complexity}-[class]{}-[func]{build_tree} +``` + +![Exponential order space complexity generated by a full binary tree](space_complexity.assets/space_complexity_exponential.png) + +### Logarithmic Order $O(\Log N)$ + +Logarithmic order is commonly used in divide and conquer algorithms. For example, in a merge sort, an array of length $n$ is input, and each round of recursion divides the array in half from its midpoint to form a recursion tree of height $\log n$, using $O(\log n)$ stack frame space. + +Another example is to convert a number into a string, input a positive integer $n$ which has the number of bits $\log_{10} n + 1$, i.e., the corresponding length of the string is $\log_{10} n + 1$, so the space complexity is $O(\log_{10} n + 1) = O(\log n)$. + +## Weighing Time And Space + +Ideally, we would like to optimize both the time complexity and the space complexity of an algorithm. In practice, however, optimizing both time complexity and space complexity at the same time is usually very difficult. + +**Reducing time complexity usually comes at the cost of increasing space complexity, and vice versa**. We refer to the idea of sacrificing memory space to increase the speed of an algorithm as "space for time"; the opposite is called "time for space". + +The choice of thinking depends on what we value more. In most cases, time is more valuable than space, so "space for time" is usually the more common strategy. Of course, it is also important to control the space complexity in case of large amount of data. diff --git a/docs-en/chapter_computational_complexity/summary.md b/docs-en/chapter_computational_complexity/summary.md new file mode 100644 index 0000000000..1ffad3777c --- /dev/null +++ b/docs-en/chapter_computational_complexity/summary.md @@ -0,0 +1,49 @@ +# Summary + +### Highlights + +**Evaluation of the efficiency of the algorithm** + +- Time efficiency and space efficiency are the two main evaluation metrics for measuring the merit of an algorithm. +- We can evaluate the efficiency of the algorithm through real-world testing, but it is difficult to eliminate the effect of the testing environment and it consumes a lot of computational resources. +- Complexity analysis overcomes the drawbacks of real-world testing, the results of the analysis are applicable to all operating platforms and reveal the efficiency of the algorithm at different data sizes. + +**Time complexity** + +- Time complexity is used to measure the trend of the algorithm's running time with the growth of data volume, which can effectively evaluate the efficiency of the algorithm, but it may fail in some cases, such as when the input data volume is small or the time complexity is the same, it is not possible to accurately compare the advantages and disadvantages of the efficiency of the algorithm. +- The worst time complexity is denoted by a large $O$ symbol, which corresponds to the asymptotic upper bound of the function, reflecting the level of growth in the number of operations $T(n)$ as $n$ tends to positive infinity. +- Extrapolating the time complexity is done in two steps, first counting the number of operations and then determining the asymptotic upper bound. +- Common time complexities are $O(1)$, $O(\log n)$, $O(n)$, $O(n \log n)$, $O(n^2)$, $O(2^n)$, and $O(n!)$, etc., in order of their degree. +- The time complexity of certain algorithms is non-fixed and is related to the distribution of the input data. The time complexity is categorized into worst, best, and average time complexity, with the best time complexity hardly ever being used because the input data generally needs to satisfy stringent conditions to achieve the best case. +- The average time complexity reflects the efficiency of an algorithm's operation under random data inputs, and is closest to the performance of algorithms in real-world applications. Calculating the average time complexity requires a statistical input data distribution as well as a synthesized mathematical expectation. + +**space complexity** + +- Space complexity acts similarly to time complexity and is used to measure the tendency of an algorithm's space occupation to grow with the amount of data. +- The memory space associated with the operation of an algorithm can be categorized into input space, staging space, and output space. Normally, the input space is not counted in the space complexity calculation. The transient space can be divided into instruction space, data space, and stack frame space, of which the stack frame space usually affects the space complexity only in the recursion function. +- We usually focus only on the worst-case space complexity, i.e., the space complexity of the statistical algorithm with the worst-case input data and the worst-case running time complexity. +- Common space complexities are $O(1)$, $O(\log n)$, $O(n)$, $O(n^2)$ and $O(2^n)$ in descending order. + +### Q & A + +!!! question "Is the space complexity of tail recursion $O(1)$?" + + Theoretically, the space complexity of a tail recursion function can be optimized to $O(1)$ . However, most programming languages (e.g. Java, Python, C++, Go, C#, etc.) do not support automatic optimization of tail recursion, so the space complexity is usually considered to be $O(n)$ . + +!!! question "What is the difference between the terms function and method?" + + A function can be executed independently and all arguments are passed explicitly. A method is associated with an object, is implicitly passed to the object that calls it, and can manipulate data contained in instances of the class. + + Here are a few common programming languages to illustrate. + + - C is a procedural programming language and has no object-oriented concepts, so it has only functions. However, we can simulate object-oriented programming by creating structures, and the functions associated with structures are equivalent to methods in other languages. + - Java and C# are object-oriented programming languages, and blocks of code (methods) are usually included as part of a class. A static method behaves like a function because it is bound to the class and cannot access specific instance variables. + - C++ and Python support both procedural programming (functions) and object-oriented programming (methods). + +!!! question "Does the figure "Common Types of Space Complexity" reflect the absolute size of the occupied space?" + + No, that figure shows the space complexity, which reflects the growth trend, not the absolute size of the space occupied. + + Suppose you take $n = 8$ , you may find that the values of each curve do not correspond to the function. This is because each curve contains a constant term that is used to compress the range of values to a visually comfortable range. + + In practice, since we usually don't know what the "constant term" complexity of each method is, it is generally impossible to choose the optimal solution for $n = 8$ based on complexity alone. 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0000000000..0e68c42adc --- /dev/null +++ b/docs-en/chapter_computational_complexity/time_complexity.md @@ -0,0 +1,1092 @@ +# Time Complexity + +Runtime can be a visual and accurate reflection of the efficiency of an algorithm. What should we do if we want to accurately predict the runtime of a piece of code? + +1. **Determine the running platform**, including hardware configuration, programming language, system environment, etc., all of which affect the efficiency of the code. +2. **Evaluates the running time** required for various computational operations, e.g., the addition operation `+` takes 1 ns, the multiplication operation `*` takes 10 ns, the print operation `print()` takes 5 ns, and so on. +3. **Counts all the computational operations in the code** and sums the execution times of all the operations to get the runtime. + +For example, in the following code, the input data size is $n$ : + +=== "Python" + + ```python title="" + # Under an operating platform + def algorithm(n: int): + a = 2 # 1 ns + a = a + 1 # 1 ns + a = a * 2 # 10 ns + # Cycle n times + for _ in range(n): # 1 ns + print(0) # 5 ns + ``` + +=== "C++" + + ```cpp title="" + // Under a particular operating platform + void algorithm(int n) { + int a = 2; // 1 ns + a = a + 1; // 1 ns + a = a * 2; // 10 ns + // Loop n times + for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed + cout << 0 << endl; // 5 ns + } + } + ``` + +=== "Java" + + ```java title="" + // Under a particular operating platform + void algorithm(int n) { + int a = 2; // 1 ns + a = a + 1; // 1 ns + a = a * 2; // 10 ns + // Loop n times + for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed + System.out.println(0); // 5 ns + } + } + ``` + +=== "C#" + + ```csharp title="" + // Under a particular operating platform + void Algorithm(int n) { + int a = 2; // 1 ns + a = a + 1; // 1 ns + a = a * 2; // 10 ns + // Loop n times + for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed + Console.WriteLine(0); // 5 ns + } + } + ``` + +=== "Go" + + ```go title="" + // Under a particular operating platform + func algorithm(n int) { + a := 2 // 1 ns + a = a + 1 // 1 ns + a = a * 2 // 10 ns + // Loop n times + for i := 0; i < n; i++ { // 1 ns + fmt.Println(a) // 5 ns + } + } + ``` + +=== "Swift" + + ```swift title="" + // Under a particular operating platform + func algorithm(n: Int) { + var a = 2 // 1 ns + a = a + 1 // 1 ns + a = a * 2 // 10 ns + // Loop n times + for _ in 0 ..< n { // 1 ns + print(0) // 5 ns + } + } + ``` + +=== "JS" + + ```javascript title="" + // Under a particular operating platform + function algorithm(n) { + var a = 2; // 1 ns + a = a + 1; // 1 ns + a = a * 2; // 10 ns + // Loop n times + for(let i = 0; i < n; i++) { // 1 ns , every round i++ is executed + console.log(0); // 5 ns + } + } + ``` + +=== "TS" + + ```typescript title="" + // Under a particular operating platform + function algorithm(n: number): void { + var a: number = 2; // 1 ns + a = a + 1; // 1 ns + a = a * 2; // 10 ns + // Loop n times + for(let i = 0; i < n; i++) { // 1 ns , every round i++ is executed + console.log(0); // 5 ns + } + } + ``` + +=== "Dart" + + ```dart title="" + // Under a particular operating platform + void algorithm(int n) { + int a = 2; // 1 ns + a = a + 1; // 1 ns + a = a * 2; // 10 ns + // Loop n times + for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed + print(0); // 5 ns + } + } + ``` + +=== "Rust" + + ```rust title="" + // Under a particular operating platform + fn algorithm(n: i32) { + let mut a = 2; // 1 ns + a = a + 1; // 1 ns + a = a * 2; // 10 ns + // Loop n times + for _ in 0..n { // 1 ns for each round i++ + println!("{}", 0); // 5 ns + } + } + ``` + +=== "C" + + ```c title="" + // Under a particular operating platform + void algorithm(int n) { + int a = 2; // 1 ns + a = a + 1; // 1 ns + a = a * 2; // 10 ns + // Loop n times + for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed + printf("%d", 0); // 5 ns + } + } + ``` + +=== "Zig" + + ```zig title="" + // Under a particular operating platform + fn algorithm(n: usize) void { + var a: i32 = 2; // 1 ns + a += 1; // 1 ns + a *= 2; // 10 ns + // Loop n times + for (0..n) |_| { // 1 ns + std.debug.print("{}\n", .{0}); // 5 ns + } + } + ``` + +Based on the above method, the algorithm running time can be obtained as $6n + 12$ ns : + +$$ +1 + 1 + 10 + (1 + 5) \times n = 6n + 12 +$$ + +In practice, however, **statistical algorithm runtimes are neither reasonable nor realistic**. First, we do not want to tie the estimation time to the operation platform, because the algorithm needs to run on a variety of different platforms. Second, it is difficult for us to be informed of the runtime of each operation, which makes the prediction process extremely difficult. + +## Trends In Statistical Time Growth + +The time complexity analysis counts not the algorithm running time, **but the tendency of the algorithm running time to increase as the amount of data gets larger**. + +The concept of "time-growing trend" is rather abstract, so let's try to understand it through an example. Suppose the size of the input data is $n$, and given three algorithmic functions `A`, `B` and `C`: + +=== "Python" + + ```python title="" + # Time complexity of algorithm A: constant order + def algorithm_A(n: int): + print(0) + # Time complexity of algorithm B: linear order + def algorithm_B(n: int): + for _ in range(n): + print(0) + # Time complexity of algorithm C: constant order + def algorithm_C(n: int): + for _ in range(1000000): + print(0) + ``` + +=== "C++" + + ```cpp title="" + // Time complexity of algorithm A: constant order + void algorithm_A(int n) { + cout << 0 << endl; + } + // Time complexity of algorithm B: linear order + void algorithm_B(int n) { + for (int i = 0; i < n; i++) { + cout << 0 << endl; + } + } + // Time complexity of algorithm C: constant order + void algorithm_C(int n) { + for (int i = 0; i < 1000000; i++) { + cout << 0 << endl; + } + } + ``` + +=== "Java" + + ```java title="" + // Time complexity of algorithm A: constant order + void algorithm_A(int n) { + System.out.println(0); + } + // Time complexity of algorithm B: linear order + void algorithm_B(int n) { + for (int i = 0; i < n; i++) { + System.out.println(0); + } + } + // Time complexity of algorithm C: constant order + void algorithm_C(int n) { + for (int i = 0; i < 1000000; i++) { + System.out.println(0); + } + } + ``` + +=== "C#" + + ```csharp title="" + // Time complexity of algorithm A: constant order + void AlgorithmA(int n) { + Console.WriteLine(0); + } + // Time complexity of algorithm B: linear order + void AlgorithmB(int n) { + for (int i = 0; i < n; i++) { + Console.WriteLine(0); + } + } + // Time complexity of algorithm C: constant order + void AlgorithmC(int n) { + for (int i = 0; i < 1000000; i++) { + Console.WriteLine(0); + } + } + ``` + +=== "Go" + + ```go title="" + // Time complexity of algorithm A: constant order + func algorithm_A(n int) { + fmt.Println(0) + } + // Time complexity of algorithm B: linear order + func algorithm_B(n int) { + for i := 0; i < n; i++ { + fmt.Println(0) + } + } + // Time complexity of algorithm C: constant order + func algorithm_C(n int) { + for i := 0; i < 1000000; i++ { + fmt.Println(0) + } + } + ``` + +=== "Swift" + + ```swift title="" + // Time complexity of algorithm A: constant order + func algorithmA(n: Int) { + print(0) + } + + // Time complexity of algorithm B: linear order + func algorithmB(n: Int) { + for _ in 0 ..< n { + print(0) + } + } + + // Time complexity of algorithm C: constant order + func algorithmC(n: Int) { + for _ in 0 ..< 1000000 { + print(0) + } + } + ``` + +=== "JS" + + ```javascript title="" + // Time complexity of algorithm A: constant order + function algorithm_A(n) { + console.log(0); + } + // Time complexity of algorithm B: linear order + function algorithm_B(n) { + for (let i = 0; i < n; i++) { + console.log(0); + } + } + // Time complexity of algorithm C: constant order + function algorithm_C(n) { + for (let i = 0; i < 1000000; i++) { + console.log(0); + } + } + + ``` + +=== "TS" + + ```typescript title="" + // Time complexity of algorithm A: constant order + function algorithm_A(n: number): void { + console.log(0); + } + // Time complexity of algorithm B: linear order + function algorithm_B(n: number): void { + for (let i = 0; i < n; i++) { + console.log(0); + } + } + // Time complexity of algorithm C: constant order + function algorithm_C(n: number): void { + for (let i = 0; i < 1000000; i++) { + console.log(0); + } + } + ``` + +=== "Dart" + + ```dart title="" + // Time complexity of algorithm A: constant order + void algorithmA(int n) { + print(0); + } + // Time complexity of algorithm B: linear order + void algorithmB(int n) { + for (int i = 0; i < n; i++) { + print(0); + } + } + // Time complexity of algorithm C: constant order + void algorithmC(int n) { + for (int i = 0; i < 1000000; i++) { + print(0); + } + } + ``` + +=== "Rust" + + ```rust title="" + // Time complexity of algorithm A: constant order + fn algorithm_A(n: i32) { + println!("{}", 0); + } + // Time complexity of algorithm B: linear order + fn algorithm_B(n: i32) { + for _ in 0..n { + println!("{}", 0); + } + } + // Time complexity of algorithm C: constant order + fn algorithm_C(n: i32) { + for _ in 0..1000000 { + println!("{}", 0); + } + } + ``` + +=== "C" + + ```c title="" + // Time complexity of algorithm A: constant order + void algorithm_A(int n) { + printf("%d", 0); + } + // Time complexity of algorithm B: linear order + void algorithm_B(int n) { + for (int i = 0; i < n; i++) { + printf("%d", 0); + } + } + // Time complexity of algorithm C: constant order + void algorithm_C(int n) { + for (int i = 0; i < 1000000; i++) { + printf("%d", 0); + } + } + ``` + +=== "Zig" + + ```zig title="" + // Time complexity of algorithm A: constant order + fn algorithm_A(n: usize) void { + _ = n; + std.debug.print("{}\n", .{0}); + } + // Time complexity of algorithm B: linear order + fn algorithm_B(n: i32) void { + for (0..n) |_| { + std.debug.print("{}\n", .{0}); + } + } + // Time complexity of algorithm C: constant order + fn algorithm_C(n: i32) void { + _ = n; + for (0..1000000) |_| { + std.debug.print("{}\n", .{0}); + } + } + ``` + +The figure below shows the time complexity of the above three algorithmic functions. + +- Algorithm `A` has only $1$ print operations, and the running time of the algorithm does not increase with $n$. We call the time complexity of this algorithm "constant order". +- The print operation in algorithm `B` requires $n$ cycles, and the running time of the algorithm increases linearly with $n$. The time complexity of this algorithm is called "linear order". +- The print operation in algorithm `C` requires $1000000$ loops, which is a long runtime, but it is independent of the size of the input data $n$. Therefore, the time complexity of `C` is the same as that of `A`, which is still of "constant order". + +![Time growth trends for algorithms A, B and C](time_complexity.assets/time_complexity_simple_example.png) + +What are the characteristics of time complexity analysis compared to direct statistical algorithmic running time? + +- The **time complexity can effectively evaluate the efficiency of an algorithm**. For example, the running time of algorithm `B` increases linearly and is slower than algorithm `A` for $n > 1$ and slower than algorithm `C` for $n > 1,000,000$. In fact, as long as the input data size $n$ is large enough, algorithms with "constant order" of complexity will always outperform algorithms with "linear order", which is exactly what the time complexity trend means. +- The **time complexity of the projection method is simpler**. Obviously, neither the running platform nor the type of computational operation is related to the growth trend of the running time of the algorithm. Therefore, in the time complexity analysis, we can simply consider the execution time of all computation operations as the same "unit time", and thus simplify the "statistics of the running time of computation operations" to the "statistics of the number of computation operations", which is the same as the "statistics of the number of computation operations". The difficulty of the estimation is greatly reduced by considering the execution time of all operations as the same "unit time". +- There are also some limitations of **time complexity**. For example, although algorithms `A` and `C` have the same time complexity, the actual running time varies greatly. Similarly, although the time complexity of algorithm `B` is higher than that of `C` , algorithm `B` significantly outperforms algorithm `C` when the size of the input data $n$ is small. In these cases, it is difficult to judge the efficiency of an algorithm based on time complexity alone. Of course, despite the above problems, complexity analysis is still the most effective and commonly used method to judge the efficiency of algorithms. + +## Functions Asymptotic Upper Bounds + +Given a function with input size $n$: + +=== "Python" + + ```python title="" + def algorithm(n: int): + a = 1 # +1 + a = a + 1 # +1 + a = a * 2 # +1 + # Cycle n times + for i in range(n): # +1 + print(0) # +1 + ``` + +=== "C++" + + ```cpp title="" + void algorithm(int n) { + int a = 1; // +1 + a = a + 1; // +1 + a = a * 2; // +1 + // Loop n times + for (int i = 0; i < n; i++) { // +1 (execute i ++ every round) + cout << 0 << endl; // +1 + } + } + ``` + +=== "Java" + + ```java title="" + void algorithm(int n) { + int a = 1; // +1 + a = a + 1; // +1 + a = a * 2; // +1 + // Loop n times + for (int i = 0; i < n; i++) { // +1 (execute i ++ every round) + System.out.println(0); // +1 + } + } + ``` + +=== "C#" + + ```csharp title="" + void Algorithm(int n) { + int a = 1; // +1 + a = a + 1; // +1 + a = a * 2; // +1 + // Loop n times + for (int i = 0; i < n; i++) { // +1 (execute i ++ every round) + Console.WriteLine(0); // +1 + } + } + ``` + +=== "Go" + + ```go title="" + func algorithm(n int) { + a := 1 // +1 + a = a + 1 // +1 + a = a * 2 // +1 + // Loop n times + for i := 0; i < n; i++ { // +1 + fmt.Println(a) // +1 + } + } + ``` + +=== "Swift" + + ```swift title="" + func algorithm(n: Int) { + var a = 1 // +1 + a = a + 1 // +1 + a = a * 2 // +1 + // Loop n times + for _ in 0 ..< n { // +1 + print(0) // +1 + } + } + ``` + +=== "JS" + + ```javascript title="" + function algorithm(n) { + var a = 1; // +1 + a += 1; // +1 + a *= 2; // +1 + // Loop n times + for(let i = 0; i < n; i++){ // +1 (execute i ++ every round) + console.log(0); // +1 + } + } + ``` + +=== "TS" + + ```typescript title="" + function algorithm(n: number): void{ + var a: number = 1; // +1 + a += 1; // +1 + a *= 2; // +1 + // Loop n times + for(let i = 0; i < n; i++){ // +1 (execute i ++ every round) + console.log(0); // +1 + } + } + ``` + +=== "Dart" + + ```dart title="" + void algorithm(int n) { + int a = 1; // +1 + a = a + 1; // +1 + a = a * 2; // +1 + // Loop n times + for (int i = 0; i < n; i++) { // +1 (execute i ++ every round) + print(0); // +1 + } + } + ``` + +=== "Rust" + + ```rust title="" + fn algorithm(n: i32) { + let mut a = 1; // +1 + a = a + 1; // +1 + a = a * 2; // +1 + + // Loop n times + for _ in 0..n { // +1 (execute i ++ every round) + println!("{}", 0); // +1 + } + } + ``` + +=== "C" + + ```c title="" + void algorithm(int n) { + int a = 1; // +1 + a = a + 1; // +1 + a = a * 2; // +1 + // Loop n times + for (int i = 0; i < n; i++) { // +1 (execute i ++ every round) + printf("%d", 0); // +1 + } + } + ``` + +=== "Zig" + + ```zig title="" + fn algorithm(n: usize) void { + var a: i32 = 1; // +1 + a += 1; // +1 + a *= 2; // +1 + // Loop n times + for (0..n) |_| { // +1 (execute i ++ every round) + std.debug.print("{}\n", .{0}); // +1 + } + } + ``` + +Let the number of operations of the algorithm be a function of the size of the input data $n$, denoted as $T(n)$ , then the number of operations of the above function is: + +$$ +T(n) = 3 + 2n +$$ + +$T(n)$ is a primary function, which indicates that the trend of its running time growth is linear, and thus its time complexity is of linear order. + +We denote the time complexity of the linear order as $O(n)$ , and this mathematical notation is called the "big $O$ notation big-$O$ notation", which denotes the "asymptotic upper bound" of the function $T(n)$. + +Time complexity analysis is essentially the computation of asymptotic upper bounds on the "number of operations function $T(n)$", which has a clear mathematical definition. + +!!! abstract "Function asymptotic upper bound" + + If there exists a positive real number $c$ and a real number $n_0$ such that $T(n) \leq c \cdot f(n)$ for all $n > n_0$ , then it can be argued that $f(n)$ gives an asymptotic upper bound on $T(n)$ , denoted as $T(n) = O(f(n))$ . + +As shown in the figure below, computing the asymptotic upper bound is a matter of finding a function $f(n)$ such that $T(n)$ and $f(n)$ are at the same growth level as $n$ tends to infinity, differing only by a multiple of the constant term $c$. + +![asymptotic upper bound of function](time_complexity.assets/asymptotic_upper_bound.png) + +## Method Of Projection + +Asymptotic upper bounds are a bit heavy on math, so don't worry if you feel you don't have a full solution. Because in practice, we only need to master the projection method, and the mathematical meaning can be gradually comprehended. + +By definition, after determining $f(n)$, we can get the time complexity $O(f(n))$. So how to determine the asymptotic upper bound $f(n)$? The overall is divided into two steps: first count the number of operations, and then determine the asymptotic upper bound. + +### The First Step: Counting The Number Of Operations + +For the code, it is sufficient to calculate from top to bottom line by line. However, since the constant term $c \cdot f(n)$ in the above $c \cdot f(n)$ can take any size, **the various coefficients and constant terms in the number of operations $T(n)$ can be ignored**. Based on this principle, the following counting simplification techniques can be summarized. + +1. **Ignore the constant terms in $T(n)$**. Since none of them are related to $n$, they have no effect on the time complexity. +2. **omits all coefficients**. For example, loops $2n$ times, $5n + 1$ times, etc., can be simplified and notated as $n$ times because the coefficients before $n$ have no effect on the time complexity. +3. **Use multiplication** when loops are nested. The total number of operations is equal to the product of the number of operations of the outer and inner levels of the loop, and each level of the loop can still be nested by applying the techniques in points `1.` and `2.` respectively. + +Given a function, we can use the above trick to count the number of operations. + +=== "Python" + + ```python title="" + def algorithm(n: int): + a = 1 # +0 (trick 1) + a = a + n # +0 (trick 1) + # +n (technique 2) + for i in range(5 * n + 1): + print(0) + # +n*n (technique 3) + for i in range(2 * n): + for j in range(n + 1): + print(0) + ``` + +=== "C++" + + ```cpp title="" + void algorithm(int n) { + int a = 1; // +0 (trick 1) + a = a + n; // +0 (trick 1) + // +n (technique 2) + for (int i = 0; i < 5 * n + 1; i++) { + cout << 0 << endl; + } + // +n*n (technique 3) + for (int i = 0; i < 2 * n; i++) { + for (int j = 0; j < n + 1; j++) { + cout << 0 << endl; + } + } + } + ``` + +=== "Java" + + ```java title="" + void algorithm(int n) { + int a = 1; // +0 (trick 1) + a = a + n; // +0 (trick 1) + // +n (technique 2) + for (int i = 0; i < 5 * n + 1; i++) { + System.out.println(0); + } + // +n*n (technique 3) + for (int i = 0; i < 2 * n; i++) { + for (int j = 0; j < n + 1; j++) { + System.out.println(0); + } + } + } + ``` + +=== "C#" + + ```csharp title="" + void Algorithm(int n) { + int a = 1; // +0 (trick 1) + a = a + n; // +0 (trick 1) + // +n (technique 2) + for (int i = 0; i < 5 * n + 1; i++) { + Console.WriteLine(0); + } + // +n*n (technique 3) + for (int i = 0; i < 2 * n; i++) { + for (int j = 0; j < n + 1; j++) { + Console.WriteLine(0); + } + } + } + ``` + +=== "Go" + + ```go title="" + func algorithm(n int) { + a := 1 // +0 (trick 1) + a = a + n // +0 (trick 1) + // +n (technique 2) + for i := 0; i < 5 * n + 1; i++ { + fmt.Println(0) + } + // +n*n (technique 3) + for i := 0; i < 2 * n; i++ { + for j := 0; j < n + 1; j++ { + fmt.Println(0) + } + } + } + ``` + +=== "Swift" + + ```swift title="" + func algorithm(n: Int) { + var a = 1 // +0 (trick 1) + a = a + n // +0 (trick 1) + // +n (technique 2) + for _ in 0 ..< (5 * n + 1) { + print(0) + } + // +n*n (technique 3) + for _ in 0 ..< (2 * n) { + for _ in 0 ..< (n + 1) { + print(0) + } + } + } + ``` + +=== "JS" + + ```javascript title="" + function algorithm(n) { + let a = 1; // +0 (trick 1) + a = a + n; // +0 (trick 1) + // +n (technique 2) + for (let i = 0; i < 5 * n + 1; i++) { + console.log(0); + } + // +n*n (technique 3) + for (let i = 0; i < 2 * n; i++) { + for (let j = 0; j < n + 1; j++) { + console.log(0); + } + } + } + ``` + +=== "TS" + + ```typescript title="" + function algorithm(n: number): void { + let a = 1; // +0 (trick 1) + a = a + n; // +0 (trick 1) + // +n (technique 2) + for (let i = 0; i < 5 * n + 1; i++) { + console.log(0); + } + // +n*n (technique 3) + for (let i = 0; i < 2 * n; i++) { + for (let j = 0; j < n + 1; j++) { + console.log(0); + } + } + } + ``` + +=== "Dart" + + ```dart title="" + void algorithm(int n) { + int a = 1; // +0 (trick 1) + a = a + n; // +0 (trick 1) + // +n (technique 2) + for (int i = 0; i < 5 * n + 1; i++) { + print(0); + } + // +n*n (technique 3) + for (int i = 0; i < 2 * n; i++) { + for (int j = 0; j < n + 1; j++) { + print(0); + } + } + } + ``` + +=== "Rust" + + ```rust title="" + fn algorithm(n: i32) { + let mut a = 1; // +0 (trick 1) + a = a + n; // +0 (trick 1) + + // +n (technique 2) + for i in 0..(5 * n + 1) { + println!("{}", 0); + } + + // +n*n (technique 3) + for i in 0..(2 * n) { + for j in 0..(n + 1) { + println!("{}", 0); + } + } + } + ``` + +=== "C" + + ```c title="" + void algorithm(int n) { + int a = 1; // +0 (trick 1) + a = a + n; // +0 (trick 1) + // +n (technique 2) + for (int i = 0; i < 5 * n + 1; i++) { + printf("%d", 0); + } + // +n*n (technique 3) + for (int i = 0; i < 2 * n; i++) { + for (int j = 0; j < n + 1; j++) { + printf("%d", 0); + } + } + } + ``` + +=== "Zig" + + ```zig title="" + fn algorithm(n: usize) void { + var a: i32 = 1; // +0 (trick 1) + a = a + @as(i32, @intCast(n)); // +0 (trick 1) + + // +n (technique 2) + for(0..(5 * n + 1)) |_| { + std.debug.print("{}\n", .{0}); + } + + // +n*n (technique 3) + for(0..(2 * n)) |_| { + for(0..(n + 1)) |_| { + std.debug.print("{}\n", .{0}); + } + } + } + ``` + +The following equations show the statistical results before and after using the above technique, both of which were introduced with a time complexity of $O(n^2)$ . + +$$ +\begin{aligned} +T(n) & = 2n(n + 1) + (5n + 1) + 2 & \text{complete statistics (-.-|||)} \newline +& = 2n^2 + 7n + 3 \newline +T(n) & = n^2 + n & \text{Lazy Stats (o.O)} +\end{aligned} +$$ + +### Step 2: Judging The Asymptotic Upper Bounds + +**The time complexity is determined by the highest order term in the polynomial $T(n)$**. This is because as $n$ tends to infinity, the highest order term will play a dominant role and the effects of all other terms can be ignored. + +The table below shows some examples, some of which have exaggerated values to emphasize the conclusion that "the coefficients can't touch the order". As $n$ tends to infinity, these constants become irrelevant. + +

Table   Time complexity corresponding to different number of operations

+ +| number of operations $T(n)$ | time complexity $O(f(n))$ | +| --------------------------- | ------------------------- | +| $100000$ | $O(1)$ | +| $3n + 2$ | $O(n)$ | +| $2n^2 + 3n + 2$ | $O(n^2)$ | +| $n^3 + 10000n^2$ | $O(n^3)$ | +| $2^n + 10000n^{10000}$ | $O(2^n)$ | + +## Common Types + +Let the input data size be $n$ , the common types of time complexity are shown in the figure below (in descending order). + +$$ +\begin{aligned} +O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!) \newline +\text{constant order} < \text{logarithmic order} < \text{linear order} < \text{linear logarithmic order} < \text{square order} < \text{exponential order} < \text{multiplication order} +\end{aligned} +$$ + +![Common time complexity types](time_complexity.assets/time_complexity_common_types.png) + +### Constant Order $O(1)$ + +The number of operations of the constant order is independent of the input data size $n$, i.e., it does not change with $n$. + +In the following function, although the number of operations `size` may be large, the time complexity is still $O(1)$ because it is independent of the input data size $n$ : + +```src +[file]{time_complexity}-[class]{}-[func]{constant} +``` + +### Linear Order $O(N)$ + +The number of operations in a linear order grows in linear steps relative to the input data size $n$. Linear orders are usually found in single level loops: + +```src +[file]{time_complexity}-[class]{}-[func]{linear} +``` + +The time complexity of operations such as traversing an array and traversing a linked list is $O(n)$ , where $n$ is the length of the array or linked list: + +```src +[file]{time_complexity}-[class]{}-[func]{array_traversal} +``` + +It is worth noting that **Input data size $n$ needs to be determined specifically** according to the type of input data. For example, in the first example, the variable $n$ is the input data size; in the second example, the array length $n$ is the data size. + +### Squared Order $O(N^2)$ + +The number of operations in the square order grows in square steps with respect to the size of the input data $n$. The squared order is usually found in nested loops, where both the outer and inner levels are $O(n)$ and therefore overall $O(n^2)$: + +```src +[file]{time_complexity}-[class]{}-[func]{quadratic} +``` + +The figure below compares the three time complexities of constant order, linear order and squared order. + +![Time complexity of constant, linear and quadratic orders](time_complexity.assets/time_complexity_constant_linear_quadratic.png) + +Taking bubble sort as an example, the outer loop executes $n - 1$ times, and the inner loop executes $n-1$, $n-2$, $\dots$, $2$, $1$ times, which averages out to $n / 2$ times, resulting in a time complexity of $O((n - 1) n / 2) = O(n^2)$ . + +```src +[file]{time_complexity}-[class]{}-[func]{bubble_sort} +``` + +## Exponential Order $O(2^N)$ + +Cell division in biology is a typical example of exponential growth: the initial state is $1$ cells, after one round of division it becomes $2$, after two rounds of division it becomes $4$, and so on, after $n$ rounds of division there are $2^n$ cells. + +The figure below and the following code simulate the process of cell division with a time complexity of $O(2^n)$ . + +```src +[file]{time_complexity}-[class]{}-[func]{exponential} +``` + +![time complexity of exponential order](time_complexity.assets/time_complexity_exponential.png) + +In practical algorithms, exponential orders are often found in recursion functions. For example, in the following code, it recursively splits in two and stops after $n$ splits: + +```src +[file]{time_complexity}-[class]{}-[func]{exp_recur} +``` + +Exponential order grows very rapidly and is more common in exhaustive methods (brute force search, backtracking, etc.). For problems with large data sizes, exponential order is unacceptable and usually requires the use of algorithms such as dynamic programming or greedy algorithms to solve. + +### Logarithmic Order $O(\Log N)$ + +In contrast to the exponential order, the logarithmic order reflects the "each round is reduced to half" case. Let the input data size be $n$, and since each round is reduced to half, the number of loops is $\log_2 n$, which is the inverse function of $2^n$. + +The figure below and the code below simulate the process of "reducing each round to half" with a time complexity of $O(\log_2 n)$, which is abbreviated as $O(\log n)$. + +```src +[file]{time_complexity}-[class]{}-[func]{logarithmic} +``` + +![time complexity of logarithmic order](time_complexity.assets/time_complexity_logarithmic.png) + +Similar to the exponential order, the logarithmic order is often found in recursion functions. The following code forms a recursion tree of height $\log_2 n$: + +```src +[file]{time_complexity}-[class]{}-[func]{log_recur} +``` + +Logarithmic order is often found in algorithms based on the divide and conquer strategy, which reflects the algorithmic ideas of "dividing one into many" and "simplifying the complexity into simplicity". It grows slowly and is the second most desirable time complexity after constant order. + +!!! tip "What is the base of $O(\log n)$?" + + To be precise, the corresponding time complexity of "one divided into $m$" is $O(\log_m n)$ . And by using the logarithmic permutation formula, we can get equal time complexity with different bases: + + $$ + O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n) + $$ + + That is, the base $m$ can be converted without affecting the complexity. Therefore we usually omit the base $m$ and write the logarithmic order directly as $O(\log n)$. + +### Linear Logarithmic Order $O(N \Log N)$ + +The linear logarithmic order is often found in nested loops, and the time complexity of the two levels of loops is $O(\log n)$ and $O(n)$ respectively. The related code is as follows: + +```src +[file]{time_complexity}-[class]{}-[func]{linear_log_recur} +``` + +The figure below shows how the linear logarithmic order is generated. The total number of operations at each level of the binary tree is $n$ , and the tree has a total of $\log_2 n + 1$ levels, resulting in a time complexity of $O(n\log n)$ . + +![Time complexity of linear logarithmic order](time_complexity.assets/time_complexity_logarithmic_linear.png) + +Mainstream sorting algorithms typically have a time complexity of $O(n \log n)$ , such as quick sort, merge sort, heap sort, etc. + +### The Factorial Order $O(N!)$ + +The factorial order corresponds to the mathematical "permutations problem". Given $n$ elements that do not repeat each other, find all possible permutations of them, the number of permutations being: + +$$ +n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1 +$$ + +Factorials are usually implemented using recursion. As shown in the figure below and in the code below, the first level splits $n$, the second level splits $n - 1$, and so on, until the splitting stops at the $n$th level: + +```src +[file]{time_complexity}-[class]{}-[func]{factorial_recur} +``` + +![Time complexity of the factorial order](time_complexity.assets/time_complexity_factorial.png) + +Note that since there is always $n! > 2^n$ when $n \geq 4$, the factorial order grows faster than the exponential order, and is also unacceptable when $n$ is large. + +## Worst, Best, Average Time Complexity + +**The time efficiency of algorithms is often not fixed, but is related to the distribution of the input data**. Suppose an array `nums` of length $n$ is input, where `nums` consists of numbers from $1$ to $n$, each of which occurs only once; however, the order of the elements is randomly upset, and the goal of the task is to return the index of element $1$. We can draw the following conclusion. + +- When `nums = [? , ? , ... , 1]` , i.e., when the end element is $1$, a complete traversal of the array is required, **to reach the worst time complexity $O(n)$** . +- When `nums = [1, ? , ? , ...]` , i.e., when the first element is $1$ , there is no need to continue traversing the array no matter how long it is, **reaching the optimal time complexity $\Omega(1)$** . + +The "worst time complexity" corresponds to the asymptotic upper bound of the function and is denoted by the large $O$ notation. Correspondingly, the "optimal time complexity" corresponds to the asymptotic lower bound of the function and is denoted in $\Omega$ notation: + +```src +[file]{worst_best_time_complexity}-[class]{}-[func]{find_one} +``` + +It is worth stating that we rarely use the optimal time complexity in practice because it is usually only attainable with a small probability and may be somewhat misleading. **whereas the worst time complexity is more practical because it gives a safe value for efficiency and allows us to use the algorithm with confidence**. + +From the above examples, it can be seen that the worst or best time complexity only occurs in "special data distributions", and the probability of these cases may be very small, which does not truly reflect the efficiency of the algorithm. In contrast, **the average time complexity of can reflect the efficiency of the algorithm under random input data**, which is denoted by the $\Theta$ notation. + +For some algorithms, we can simply derive the average case under a random data distribution. For example, in the above example, since the input array is scrambled, the probability of an element $1$ appearing at any index is equal, so the average number of loops of the algorithm is half of the length of the array $n / 2$ , and the average time complexity is $\Theta(n / 2) = \Theta(n)$ . + +However, for more complex algorithms, calculating the average time complexity is often difficult because it is hard to analyze the overall mathematical expectation given the data distribution. In this case, we usually use the worst time complexity as a criterion for the efficiency of the algorithm. + +!!! question "Why do you rarely see the $\Theta$ symbol?" + + Perhaps because the $O$ symbol is so catchy, we often use it to denote average time complexity. However, this practice is not standardized in the strict sense. In this book and other sources, if you encounter a statement like "average time complexity $O(n)$", please understand it as $\Theta(n)$. diff --git a/docs-en/chapter_data_structure/basic_data_types.md b/docs-en/chapter_data_structure/basic_data_types.md new file mode 100644 index 0000000000..82fdcbaf90 --- /dev/null +++ b/docs-en/chapter_data_structure/basic_data_types.md @@ -0,0 +1,159 @@ +# Basic Data Types + +When we talk about data in computers, we think of various forms such as text, pictures, videos, speech, 3D models, and so on. Although these data are organized in different ways, they all consist of various basic data types. + +**Basic data types are types that the CPU can directly perform arithmetic on and are used directly in algorithms**, and include the following main types. + +- Integer types `byte`, `short`, `int`, `long`. +- Floating point types `float`, `double`, used to represent decimals. +- Character type `char` , used to represent letters, punctuation marks, and even emoticons in various languages. +- Boolean type `bool`, used to represent the "yes" and "no" judgment. + +**Basic data types are stored in the computer in binary form**. One binary bit is $1$ bit. In most modern systems, $1$ bytes are made up of $8$ bits. + +The range of values for a basic data type depends on the amount of space it takes up. The following is an example from Java. + +- The integer type `byte` occupies $1$ byte = $8$ bits and can represent $2^{8}$ numbers. +- The integer type `int` occupies $4$ bytes = $32$ bits and can represent $2^{32}$ numbers. + +The table below lists the space occupation, value ranges and default values of various basic data types. This table does not need to be memorized, a general solution is sufficient, and can be recalled by checking the table when needed. + +

Table   Space Occupancy and Range of Values for Basic Datatypes

+ +| Type | Symbol | Occupancy | Minimum | Maximum | Default | +| -------------- | -------- | ---------------- | ------------------------ | ----------------------- | ----------------------------- | +| integer | `byte` | 1 byte | $-2^7$ ($-128$) | $2^7 - 1$ ($127$) | $0$ | +| | `short` | 2 bytes | $-2^{15}$ | $2^{15} - 1$ | $0$ | +| | `int` | 4 bytes | $-2^{31}$ | $2^{31} - 1$ | $0$ | +| | `long` | 8 bytes | $-2^{63}$ | $2^{63} - 1$ | $0$ | +| Floating Point | `float` | 4 bytes | $1.175 \times 10^{-38}$ | $3.403 \times 10^{38}$ | $0.0 f$ | +| | `double` | 8 bytes | $2.225 \times 10^{-308}$ | $1.798 \times 10^{308}$ | $0.0$ | +| Characters | `char` | 2 bytes / 1 byte | $0$ | $2^{16} - 1$ | $0$ | +| Boolean | `bool` | 1 byte | $\text{false}$ | $\text{true}$ | $\text{false}$ $\text{false}$ | + +For the above table, the following points need to be noted. + +- C and C++ do not explicitly specify the size of the base datatype, which varies by implementation and platform. The table above follows the LP64 [Data Model](https://en.cppreference.com/w/cpp/language/types#Properties), which is used on Unix 64-bit operating systems including Linux and macOS. +- The size of the character `char` is 1 byte in C and C++, and depends on a specific character encoding method in most programming languages, see the "Character Encoding" section for details. +- Even though it only takes 1 bit ($0$ or $1$) to represent a Boolean quantity, it is usually stored as 1 byte in memory. This is because modern computer CPUs usually use 1 byte as the smallest addressable memory unit. + +So what is the connection between basic data types and data structures? As we know, data structure is the way data is organized and stored in a computer. Its subject is "structure" rather than "data". + +If we want to represent "an array of numbers", we naturally think of using an array. This is because the linear structure of arrays allows us to represent the adjacency and order of numbers, but whether the contents are stored as integers `int`, decimals `float`, or characters `char` has nothing to do with the "data structure". + +In other words, **basic data types provide the "content type" of the data, while data structures provide the "organization" of the data**. For example, in the following code, we use the same data structure (array) to store and represent different basic data types, including `int`, `float`, `char`, `bool`, and so on. + +=== "Python" + + ```python title="" + # Initialize arrays with a variety of basic data types + numbers: list[int] = [0] * 5 + decimals: list[float] = [0.0] * 5 + # Python's characters should be treated as strings of length one + characters: list[str] = ['0'] * 5 + bools: list[bool] = [False] * 5 + # Python's lists are free to store all kinds of basic data types and object references. + data = [0, 0.0, 'a', False, ListNode(0)] + ``` + +=== "C++" + + ```cpp title="" + // Initialize arrays with multiple basic data types + int numbers[5]; + float decimals[5]; + char characters[5]; + bool bools[5]; + ``` + +=== "Java" + + ```java title="" + // Initialize arrays with multiple basic data types + int[] numbers = new int[5]; + float[] decimals = new float[5]; + char[] characters = new char[5]; + boolean[] bools = new boolean[5]; + ``` + +=== "C#" + + ```csharp title="" + // Initialize arrays with multiple basic data types + int[] numbers = new int[5]; + float[] decimals = new float[5]; + char[] characters = new char[5]; + bool[] bools = new bool[5]; + ``` + +=== "Go" + + ```go title="" + // Initialize arrays with multiple basic data types + var numbers = [5]int{} + var decimals = [5]float64{} + var characters = [5]byte{} + var bools = [5]bool{} + ``` + +=== "Swift" + + ```swift title="" + // Initialize arrays with multiple basic data types + let numbers = Array(repeating: Int(), count: 5) + let decimals = Array(repeating: Double(), count: 5) + let characters = Array(repeating: Character("a"), count: 5) + let bools = Array(repeating: Bool(), count: 5) + ``` + +=== "JS" + + ```javascript title="" + // JavaScript arrays are free to store a variety of basic data types and objects. + const array = [0, 0.0, 'a', false]; + ``` + +=== "TS" + + ```typescript title="" + // Initialize arrays with multiple basic data types + const numbers: number[] = []; + const characters: string[] = []; + const bools: boolean[] = []; + ``` + +=== "Dart" + + ```dart title="" + // Initialize arrays with multiple basic data types + List numbers = List.filled(5, 0); + List decimals = List.filled(5, 0.0); + List characters = List.filled(5, 'a'); + List bools = List.filled(5, false); + ``` + +=== "Rust" + + ```rust title="" + // Initialize arrays with multiple basic data types + let numbers: Vec = vec![0; 5]; + let decimals: Vec = vec![0.0, 5]; + let characters: Vec = vec!['0'; 5]; + let bools: Vec = vec![false; 5]; + ``` + +=== "C" + + ```c title="" + // Initialize arrays with multiple basic data types + int numbers[10]; + float decimals[10]; + char characters[10]; + bool bools[10]; + ``` + +=== "Zig" + + ```zig title="" + + ``` diff --git a/docs-en/chapter_data_structure/character_encoding.assets/ascii_table.png b/docs-en/chapter_data_structure/character_encoding.assets/ascii_table.png new file mode 100644 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DRs2Z9 literal 0 HcmV?d00001 diff --git a/docs-en/chapter_data_structure/character_encoding.md b/docs-en/chapter_data_structure/character_encoding.md new file mode 100644 index 0000000000..efd7f87b51 --- /dev/null +++ b/docs-en/chapter_data_structure/character_encoding.md @@ -0,0 +1,87 @@ +# Character Encoding * + +In computers, all data is stored as binary numbers, and the character `char` is no exception. In order to represent the characters, we need to establish a set of "character set", the provisions of each character and the one-to-one correspondence between the binary number. With the character set, the computer can look up the table to complete the conversion of binary numbers to characters. + +## Ascii Character Set + +"ASCII" is the earliest character set known as "American Standard Code for Information Interchange". It uses a 7-bit binary number (i.e. the lower 7 bits of a byte) to represent a character, and can represent up to 128 different characters. As shown in the figure below, the ASCII code consists of the upper and lower case letters of the English alphabet, the numbers 0 ~ 9, some punctuation marks, and some control characters (such as line breaks and tabs). + +![ASCII code](character_encoding.assets/ascii_table.png) + +However, **the ASCII code is only capable of representing the English language**. With the globalization of computers, a character set called "EASCII" was created that could represent more languages. It extends the 7-bit ASCII code to 8-bit and is capable of representing 256 different characters. + +Around the world, there are a number of EASCII character sets that are applicable to different regions. The first 128 characters of these character sets are standardized as ASCII, and the last 128 characters are defined differently to meet the needs of different languages. + +## Gbk Character Set + +It was later discovered that **EASCII still could not meet the number of characters required for many languages**. For example, there are about 100,000 Chinese characters, and several thousand of them are used in daily life. In 1980, the General Administration of National Standards of China released the "GB2312" character set, which contains 6,763 Chinese characters and basically meets the needs of computer processing of Chinese characters. + +However, GB2312 cannot handle some rare and traditional characters. The "GBK" character set is an extension of GB2312, which contains 21886 Chinese characters. In the GBK encoding scheme, ASCII characters are represented by one byte and Chinese characters are represented by two bytes. + +## Unicode Character Set + +With the booming development of computers, character sets and coding standards have blossomed, and this has created many problems. On the one hand, these character sets generally define only the characters of a particular language and cannot work properly in a multilingual environment. On the other hand, there are also multiple character set standards for the same language, so if two computers are installed with different encoding standards, the information will be garbled when transmitted. + +Researchers of that era thought:**Wouldn't the problem of cross-linguistic environments and garbled codes be solved if a character set complete enough to include all the languages and symbols of the world was introduced**? Driven by this idea, Unicode, a large and complete character set, was created. + +"Unicode" is the full name of "Unified Character Encoding", which can theoretically accommodate more than one million characters. It is committed to incorporating characters from all over the world into a unified character set, providing a common character set for processing and displaying various languages, and reducing the problem of garbled codes caused by different encoding standards. + +Since its release in 1991, Unicode has been expanding with new languages and characters. As of September 2022, Unicode contains 149,186 characters, including characters, symbols, and even emoticons for various languages. Within the large Unicode character set, commonly used characters take up 2 bytes, and some obscure characters take up 3 or even 4 bytes. + +Unicode is a character set standard that essentially assigns a number (called a "code point") to each character, **but it doesn't dictate how those code points are stored in a computer**. One wonders: how does the system resolve characters when Unicode code points of multiple lengths appear in the same text at the same time? For example, given a code of 2 bytes in length, how does the system recognize whether it is one 2-byte character or two 1-byte characters? + +A straightforward solution to the above problem **is to store all characters as equal length encodings**. As shown in the figure below, each character in "Hello" occupies 1 byte and each character in "algorithm" occupies 2 bytes. We can encode all the characters in "Hello Algo" as 2 bytes by filling the high bit with zero. This way, the system can parse a character every 2 bytes and recover the content of the phrase. + +![Unicode encoding example](character_encoding.assets/unicode_hello_algo.png) + +However, ASCII code has already proved to us that it takes only 1 byte to encode English. If we use the above scheme, the size of the English text will be twice the size of the ASCII encoding, which is a waste of memory space. Therefore, we need a more efficient way to encode Unicode. + +## Utf-8 Encoding + +Currently, UTF-8 has become the most widely used Unicode encoding method internationally. It is a variable-length encoding that uses between 1 and 4 bytes to represent a character, depending on the complexity of the character. 1 byte is needed for ASCII characters, 2 bytes for Latin and Greek alphabets, 3 bytes for commonly used Chinese characters, and 4 bytes for some other rare characters. + +The encoding rules for UTF-8 are not complicated and are divided into the following two cases. + +- For characters 1 byte long, set the highest bit to $0$ and the remaining 7 bits to Unicode code points. It is worth noting that ASCII characters occupy the first 128 code points in the Unicode character set. In other words, **UTF-8 encoding is backward compatible with ASCII**. This means that we can use UTF-8 to solve old ASCII text. +- For characters of length $n$ bytes (where $n > 1$), set the high $n$ bits of the first byte to $1$ and the $n + 1$ bits to $0$; from the second byte onwards, set the high 2 bits of each byte to $10$; and all the rest of the bits are used to fill in the Unicode code point of the character. + +The figure below shows the UTF-8 encoding of "Hello Algo". It is observed that since the highest $n$ bits are set to $1$, the system can read the number of the highest $1$ bits to parse out the length of the character as $n$. + +But why set the high 2 bits of all the rest of the bytes to $10$? In fact, this $10$ serves as a check digit. Assuming that the system starts parsing the text from an incorrect byte, the $10$ in the byte header can help the system to quickly recognize the exception. + +The reason why $10$ is used as a check digit is because under the UTF-8 encoding rules, it is impossible for any character to have the highest two digits as $10$. This conclusion can be proved by the inverse method: suppose the highest two bits of a character are $10$, that means the length of the character is $1$, which corresponds to the ASCII code. The highest bit of ASCII code should be $0$, which contradicts the assumption. + +![UTF-8 encoding example](character_encoding.assets/utf-8_hello_algo.png) + +In addition to UTF-8, there are two other common encodings. + +- **UTF-16 encoding**: 2 or 4 bytes are used to represent a character. All ASCII characters and common non-English characters are represented by 2 bytes; a few characters require 4 bytes. For 2-byte characters, UTF-16 encoding is equivalent to Unicode code points. +- **UTF-32 encoding**: each character uses 4 bytes. This means that UTF-32 will take up more space than UTF-8 and UTF-16, especially for text with a high proportion of ASCII characters. + +From a storage space point of view, using UTF-8 to represent English characters is very efficient because it takes only 1 byte; using UTF-16 to encode some non-English characters (e.g., Chinese) is more efficient because it takes only 2 bytes, whereas UTF-8 may take 3 bytes. + +From a compatibility standpoint, UTF-8 is the most versatile, and many tools and libraries prioritize UTF-8 support. + +## Character Encoding For Programming Languages + +For most of the programming languages in the past, the strings in the running program are encoded in UTF-16 or UTF-32 encoding of equal length. With equal-length encoding, we can treat strings as arrays, which has the following advantages. + +- **Random access**: UTF-16 encoded strings can be easily randomized. UTF-8 is a variable-length encoding, and to find the first $i$ character, we need to traverse from the beginning of the string to the first $i$ character, which takes $O(n)$ time. +- **Character Count**: Similar to random access, calculating the length of a UTF-16 string is an $O(1)$ operation. However, calculating the length of a UTF-8 encoded string requires traversing the entire string. +- **String Operations**: Many string operations (e.g., splitting, concatenation, insertion, deletion, etc.) are easier to perform on UTF-16 encoded strings. Performing these operations on UTF-8 encoded strings usually requires additional computation to ensure that invalid UTF-8 encodings are not produced. + +In fact, the design of character encoding schemes for programming languages is an interesting topic that involves many factors. + +- Java's `String` type uses UTF-16 encoding and takes up 2 bytes per character. This is because when the Java language was first designed, it was assumed that 16 bits would be enough to represent all possible characters. However, this was an incorrect judgment. The Unicode specification was later extended beyond 16 bits, so characters in Java may now be represented by a pair of 16-bit values (called a "proxy pair"). +- JavaScript and TypeScript strings are UTF-16 encoded for similar reasons to Java. When the JavaScript language was first introduced by Netscape in 1995, Unicode was in its relatively early stages, and a 16-bit encoding was sufficient to represent all Unicode characters. +- C# uses UTF-16 encoding, primarily because the .NET platform was designed by Microsoft, and many of Microsoft's technologies, including the Windows operating system, make extensive use of UTF-16 encoding. + +Due to the underestimation of the number of characters in the above programming languages, they have had to resort to "proxy pairs" to represent Unicode characters longer than 16 bits. This is a last resort. On the one hand, a character in a string containing a proxy pair may take up 2 or 4 bytes, thus losing the advantage of equal-length encoding. On the other hand, handling proxy pairs requires additional code, which increases programming complexity and Debugging difficulty. + +For these reasons, some programming languages have proposed a number of different coding schemes. + +- `str` in Python uses Unicode encoding and a flexible string representation where the length of the stored characters depends on the largest Unicode code point in the string. Each character takes 1 byte if the string is entirely ASCII, 2 bytes if there are characters outside the ASCII range but all within the Basic Multilingual Plane (BMP), and 4 bytes if there are characters outside the BMP. +- Go's `string` type is internally encoded in UTF-8. Go also provides the `rune` type, which is used to represent a single Unicode code point. +- The str and String types of the Rust language are internally encoded in UTF-8. Rust also provides the `char` type for representing individual Unicode code points. + +Note that the above discussion is all about how strings are stored in a programming language, **this is a different issue from how strings are stored in a file or transmitted over a network**. 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HcmV?d00001 diff --git a/docs-en/chapter_backtracking/permutations_problem.md b/docs-en/chapter_backtracking/permutations_problem.md new file mode 100644 index 0000000000..c283d1fc38 --- /dev/null +++ b/docs-en/chapter_backtracking/permutations_problem.md @@ -0,0 +1,95 @@ +# Permutations Problem + +The permutations problem is a typical application of the backtracking algorithm. It is defined as finding all possible permutations of the elements of a set (e.g., an array or a string) given that set. + +The table below lists several example data, including the input array and all the corresponding alignments. + +

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