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fix: updated Traditional Chinese algorithms section
anees4500 Nov 9, 2025
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fix: updated Traditional Chinese translation section
anees4500 Nov 9, 2025
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2 changes: 2 additions & 0 deletions README.md
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# JavaScript Algorithms and Data Structures



> 🇺🇦 UKRAINE [IS BEING ATTACKED](https://war.ukraine.ua/) BY RUSSIAN ARMY. CIVILIANS ARE GETTING KILLED. RESIDENTIAL AREAS ARE GETTING BOMBED.
> - Help Ukraine via:
> - [Serhiy Prytula Charity Foundation](https://prytulafoundation.org/en/)
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215 changes: 121 additions & 94 deletions README.zh-TW.md
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# JavaScript 演算法與資料結構

> 🇺🇦 烏克蘭正遭受攻擊 (https://war.ukraine.ua/) — 由俄羅斯軍隊發動。平民遭到傷害，住宅區被轟炸。
> - 捐助烏克蘭：
> - [Serhiy Prytula 慈善基金會](https://prytulafoundation.org/en/)
> - [Come Back Alive 慈善基金會](https://savelife.in.ua/en/donate-en/)
> - [烏克蘭國家銀行（募款專戶）](https://bank.gov.ua/en/news/all/natsionalniy-bank-vidkriv-spetsrahunok-dlya-zboru-koshtiv-na-potrebi-armiyi)
> - 更多資訊請見 [war.ukraine.ua](https://war.ukraine.ua/) 與 [烏克蘭外交部 (MFA)](https://twitter.com/MFA_Ukraine)

<hr/>

[![CI](https://github.com/trekhleb/javascript-algorithms/workflows/CI/badge.svg)](https://github.com/trekhleb/javascript-algorithms/actions?query=workflow%3ACI+branch%3Amaster)
[![codecov](https://codecov.io/gh/trekhleb/javascript-algorithms/branch/master/graph/badge.svg)](https://codecov.io/gh/trekhleb/javascript-algorithms)

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_Read this in other languages:_
[_English_](https://github.com/trekhleb/javascript-algorithms/),
[_繁體中文_](README.zh-TW.md),
[_简体中文_](README.zh-CN.md),
[_한국어_](README.ko-KR.md),
[_日本語_](README.ja-JP.md),
Expand Down Expand Up @@ -43,108 +53,125 @@ _Read this in other languages:_
* [紅黑樹](src/data-structures/tree/red-black-tree)
* [圖](src/data-structures/graph) (有向跟無向皆包含)
* [互斥集](src/data-structures/disjoint-set)
* [布隆過濾器](src/data-structures/bloom-filter)
* [LRU 快取](src/data-structures/lru-cache/) - 最近最少使用（Least Recently Used）快取
## 演算法

## 演算法
演算法是針對一類問題提供明確解決方案的規範。
它是一組精確定義的規則，用來描述一系列操作步驟。

演算法是一個如何解決一類問題的非模糊規格。演算法是一個具有精確地定義了一系列運作的規則的集合
`B` - 初學者，`A` - 進階

### 演算法議題分類
### 按主題分類的演算法

* **數學類**
* [階層](src/algorithms/math/factorial)
* [費伯納西數列](src/algorithms/math/fibonacci)
* [Primality Test](src/algorithms/math/primality-test) (排除法)
* [歐幾里得算法](src/algorithms/math/euclidean-algorithm) - 計算最大公因數 (GCD)
* [最小公倍數](src/algorithms/math/least-common-multiple) (LCM)
* [整數拆分](src/algorithms/math/integer-partition)
* **集合**
* [笛卡爾積](src/algorithms/sets/cartesian-product) - 多個集合的乘積
* [冪集合](src/algorithms/sets/power-set) - 所有集合的子集合
* [排列](src/algorithms/sets/permutations) (有/無重複)
* [组合](src/algorithms/sets/combinations) (有/無重複)
* [洗牌算法](src/algorithms/sets/fisher-yates) - 隨機置換一有限序列
* [最長共同子序列](src/algorithms/sets/longest-common-subsequence) (LCS)
* [最長遞增子序列](src/algorithms/sets/longest-increasing-subsequence)
* [Shortest Common Supersequence](src/algorithms/sets/shortest-common-supersequence) (SCS)
* [背包問題](src/algorithms/sets/knapsack-problem) - "0/1" and "Unbound" ones
* [最大子序列問題](src/algorithms/sets/maximum-subarray) - 暴力法以及動態編程的(Kadane's)版本
* **字串**
* [萊文斯坦距離](src/algorithms/string/levenshtein-distance) - 兩序列間的最小編輯距離
* [漢明距離](src/algorithms/string/hamming-distance) - number of positions at which the symbols are different
* [KMP 演算法](src/algorithms/string/knuth-morris-pratt) - 子字串搜尋
* [Rabin Karp 演算法](src/algorithms/string/rabin-karp) - 子字串搜尋
* [最長共通子序列](src/algorithms/string/longest-common-substring)
* **搜尋**
* [二元搜尋](src/algorithms/search/binary-search)
* **排序**
* [氣泡排序](src/algorithms/sorting/bubble-sort)
* [選擇排序](src/algorithms/sorting/selection-sort)
* [插入排序](src/algorithms/sorting/insertion-sort)
* [堆排序](src/algorithms/sorting/heap-sort)
* [合併排序](src/algorithms/sorting/merge-sort)
* [快速排序](src/algorithms/sorting/quick-sort)
* [希爾排序](src/algorithms/sorting/shell-sort)
* `B` [位元運算](src/algorithms/math/bits) - 設定/取得/更新/清除位元、乘除以二、取負數等。
* `B` [二進位浮點數](src/algorithms/math/binary-floating-point) - 浮點數的二進位表示。
* `B` [階乘](src/algorithms/math/factorial)
* `B` [費波那契數列](src/algorithms/math/fibonacci) - 經典與閉合形式。
* `B` [質因數分解](src/algorithms/math/prime-factors) - 找出質因數並使用 Hardy-Ramanujan 定理計算。
* `B` [質數測試](src/algorithms/math/primality-test) (試除法)
* `B` [歐幾里得演算法](src/algorithms/math/euclidean-algorithm) - 計算最大公因數 (GCD)
* `B` [最小公倍數](src/algorithms/math/least-common-multiple) (LCM)
* `B` [埃拉托斯特尼篩法](src/algorithms/math/sieve-of-eratosthenes) - 找出所有小於給定值的質數
* `B` [二的冪檢查](src/algorithms/math/is-power-of-two)
* `B` [帕斯卡三角形](src/algorithms/math/pascal-triangle)
* `B` [複數運算](src/algorithms/math/complex-number)
* `B` [弧度與角度](src/algorithms/math/radian)
* `B` [快速冪算法](src/algorithms/math/fast-powering)
* `B` [霍納法則](src/algorithms/math/horner-method)
* `B` [矩陣](src/algorithms/math/matrix)
* `B` [歐幾里得距離](src/algorithms/math/euclidean-distance)
* `A` [整數拆分](src/algorithms/math/integer-partition)
* `A` [平方根](src/algorithms/math/square-root)
* `A` [劉徽圓周率算法](src/algorithms/math/liu-hui)
* `A` [離散傅立葉變換](src/algorithms/math/fourier-transform)
* **集合類**
* `B` [笛卡爾積](src/algorithms/sets/cartesian-product)
* `B` [費雪耶茨洗牌法](src/algorithms/sets/fisher-yates)
* `A` [冪集](src/algorithms/sets/power-set)
* `A` [排列](src/algorithms/sets/permutations)
* `A` [組合](src/algorithms/sets/combinations)
* `A` [最長公共子序列](src/algorithms/sets/longest-common-subsequence)
* `A` [最長遞增子序列](src/algorithms/sets/longest-increasing-subsequence)
* `A` [最短公共超序列](src/algorithms/sets/shortest-common-supersequence)
* `A` [背包問題](src/algorithms/sets/knapsack-problem)
* `A` [最大子序列問題](src/algorithms/sets/maximum-subarray)
* `A` [組合總和](src/algorithms/sets/combination-sum)
* **字串類**
* `B` [漢明距離](src/algorithms/string/hamming-distance)
* `B` [回文檢查](src/algorithms/string/palindrome)
* `A` [萊文斯坦距離](src/algorithms/string/levenshtein-distance)
* `A` [KMP 演算法](src/algorithms/string/knuth-morris-pratt)
* `A` [Z 演算法](src/algorithms/string/z-algorithm)
* `A` [Rabin Karp 演算法](src/algorithms/string/rabin-karp)
* `A` [最長公共子字串](src/algorithms/string/longest-common-substring)
* `A` [正則表達式匹配](src/algorithms/string/regular-expression-matching)
* **搜尋類**
* `B` [線性搜尋](src/algorithms/search/linear-search)
* `B` [跳躍搜尋](src/algorithms/search/jump-search)
* `B` [二元搜尋](src/algorithms/search/binary-search)
* `B` [內插搜尋](src/algorithms/search/interpolation-search)
* **排序類**
* `B` [氣泡排序](src/algorithms/sorting/bubble-sort)
* `B` [選擇排序](src/algorithms/sorting/selection-sort)
* `B` [插入排序](src/algorithms/sorting/insertion-sort)
* `B` [堆排序](src/algorithms/sorting/heap-sort)
* `B` [合併排序](src/algorithms/sorting/merge-sort)
* `B` [快速排序](src/algorithms/sorting/quick-sort)
* `B` [希爾排序](src/algorithms/sorting/shell-sort)
* `B` [計數排序](src/algorithms/sorting/counting-sort)
* `B` [基數排序](src/algorithms/sorting/radix-sort)
* `B` [桶排序](src/algorithms/sorting/bucket-sort)
* **鏈表**
* `B` [正向遍歷](src/algorithms/linked-list/traversal)
* `B` [反向遍歷](src/algorithms/linked-list/reverse-traversal)
* **樹**
* [深度優先搜尋](src/algorithms/tree/depth-first-search) (DFS)
* [廣度優先搜尋](src/algorithms/tree/breadth-first-search) (BFS)
* `B` [深度優先搜尋](src/algorithms/tree/depth-first-search) (DFS)
* `B` [廣度優先搜尋](src/algorithms/tree/breadth-first-search) (BFS)
* **圖**
* [深度優先搜尋](src/algorithms/graph/depth-first-search) (DFS)
* [廣度優先搜尋](src/algorithms/graph/breadth-first-search) (BFS)
* [Dijkstra 演算法](src/algorithms/graph/dijkstra) - 找到所有圖頂點的最短路徑
* [Bellman-Ford 演算法](src/algorithms/graph/bellman-ford) - 找到所有圖頂點的最短路徑
* [Detect Cycle](src/algorithms/graph/detect-cycle) - for both directed and undirected graphs (DFS and Disjoint Set based versions)
* [Prim’s 演算法](src/algorithms/graph/prim) - finding Minimum Spanning Tree (MST) for weighted undirected graph
* [Kruskal’s 演算法](src/algorithms/graph/kruskal) - finding Minimum Spanning Tree (MST) for weighted undirected graph
* [拓撲排序](src/algorithms/graph/topological-sorting) - DFS method
* [關節點](src/algorithms/graph/articulation-points) - Tarjan's algorithm (DFS based)
* [橋](src/algorithms/graph/bridges) - DFS based algorithm
* [尤拉路徑及尤拉環](src/algorithms/graph/eulerian-path) - Fleury's algorithm - Visit every edge exactly once
* [漢彌爾頓環](src/algorithms/graph/hamiltonian-cycle) - Visit every vertex exactly once
* [強連通組件](src/algorithms/graph/strongly-connected-components) - Kosaraju's algorithm
* [旅行推銷員問題](src/algorithms/graph/travelling-salesman) - shortest possible route that visits each city and returns to the origin city
* [Floyd-Warshall algorithm](src/algorithms/graph/floyd-warshall) - 一次循环可以找出所有頂點之间的最短路徑
* `B` [深度優先搜尋](src/algorithms/graph/depth-first-search) (DFS)
* `B` [廣度優先搜尋](src/algorithms/graph/breadth-first-search) (BFS)
* `B` [Kruskal 演算法](src/algorithms/graph/kruskal)
* `A` [Dijkstra 演算法](src/algorithms/graph/dijkstra)
* `A` [Bellman-Ford 演算法](src/algorithms/graph/bellman-ford)
* `A` [Floyd-Warshall 演算法](src/algorithms/graph/floyd-warshall)
* `A` [環檢測](src/algorithms/graph/detect-cycle)
* `A` [Prim 演算法](src/algorithms/graph/prim)
* `A` [拓撲排序](src/algorithms/graph/topological-sorting)
* `A` [關節點](src/algorithms/graph/articulation-points)
* `A` [橋](src/algorithms/graph/bridges)
* `A` [歐拉路徑與歐拉回路](src/algorithms/graph/eulerian-path)
* `A` [哈密頓迴圈](src/algorithms/graph/hamiltonian-cycle)
* `A` [強連通分量](src/algorithms/graph/strongly-connected-components)
* `A` [旅行推銷員問題](src/algorithms/graph/travelling-salesman)
* **密碼學**
* `B` [多項式雜湊](src/algorithms/cryptography/polynomial-hash)
* `B` [鐵軌密碼](src/algorithms/cryptography/rail-fence-cipher)
* `B` [凱撒密碼](src/algorithms/cryptography/caesar-cipher)
* `B` [Hill 密碼](src/algorithms/cryptography/hill-cipher)
* **機器學習**
* `B` [NanoNeuron](https://github.com/trekhleb/nano-neuron)
* `B` [k-最近鄰演算法 (k-NN)](src/algorithms/ml/knn)
* `B` [k-平均演算法 (k-Means)](src/algorithms/ml/k-means)
* **影像處理**
* `B` [縫隙雕刻 (Seam Carving)](src/algorithms/image-processing/seam-carving)
* **統計學**
* `B` [加權隨機](src/algorithms/statistics/weighted-random)
* **進化演算法**
* `A` [遺傳演算法](https://github.com/trekhleb/self-parking-car-evolution)
* **未分類**
* [河內塔](src/algorithms/uncategorized/hanoi-tower)
* [N-皇后問題](src/algorithms/uncategorized/n-queens)
* [騎士走棋盤](src/algorithms/uncategorized/knight-tour)

### 演算法範型

演算法的範型是一個泛用方法或設計一類底層演算法的方式。它是一個比演算法的概念更高階的抽象化，就像是演算法是比電腦程式更高階的抽象化。

* **暴力法** - 尋遍所有的可能解然後選取最好的解
* [最大子序列](src/algorithms/sets/maximum-subarray)
* [旅行推銷員問題](src/algorithms/graph/travelling-salesman) - shortest possible route that visits each city and returns to the origin city
* **貪婪法** - choose the best option at the current time, without any consideration for the future
* [未定背包問題](src/algorithms/sets/knapsack-problem)
* [Dijkstra 演算法](src/algorithms/graph/dijkstra) - 找到所有圖頂點的最短路徑
* [Prim’s 演算法](src/algorithms/graph/prim) - finding Minimum Spanning Tree (MST) for weighted undirected graph
* [Kruskal’s 演算法](src/algorithms/graph/kruskal) - finding Minimum Spanning Tree (MST) for weighted undirected graph
* **分治法** - divide the problem into smaller parts and then solve those parts
* [二元搜尋](src/algorithms/search/binary-search)
* [河內塔](src/algorithms/uncategorized/hanoi-tower)
* [歐幾里得演算法](src/algorithms/math/euclidean-algorithm) - calculate the Greatest Common Divisor (GCD)
* [排列](src/algorithms/sets/permutations) (有/無重複)
* [组合](src/algorithms/sets/combinations) (有/無重複)
* [合併排序](src/algorithms/sorting/merge-sort)
* [快速排序](src/algorithms/sorting/quick-sort)
* [樹深度優先搜尋](src/algorithms/tree/depth-first-search) (DFS)
* [圖深度優先搜尋](src/algorithms/graph/depth-first-search) (DFS)
* **動態編程** - build up to a solution using previously found sub-solutions
* [費伯納西數列](src/algorithms/math/fibonacci)
* [萊溫斯坦距離](src/algorithms/string/levenshtein-distance) - minimum edit distance between two sequences
* [最長共同子序列](src/algorithms/sets/longest-common-subsequence) (LCS)
* [最長共同子字串](src/algorithms/string/longest-common-substring)
* [最長遞增子序列](src/algorithms/sets/longest-increasing-subsequence)
* [最短共同子序列](src/algorithms/sets/shortest-common-supersequence)
* [0/1背包問題](src/algorithms/sets/knapsack-problem)
* [整數拆分](src/algorithms/math/integer-partition)
* [最大子序列](src/algorithms/sets/maximum-subarray)
* [Bellman-Ford 演算法](src/algorithms/graph/bellman-ford) - finding shortest path to all graph vertices
* **回溯法** - 用類似暴力法來嘗試產生所有可能解，但每次只在能滿足所有測試條件，且只有繼續產生子序列方案來產生的解決方案。否則回溯並尋找不同路徑的解決方案。
* [漢彌爾頓迴路](src/algorithms/graph/hamiltonian-cycle) - Visit every vertex exactly once
* [N-皇后問題](src/algorithms/uncategorized/n-queens)
* [騎士走棋盤](src/algorithms/uncategorized/knight-tour)
* `B` [河內塔](src/algorithms/uncategorized/hanoi-tower)
* `B` [矩陣旋轉](src/algorithms/uncategorized/square-matrix-rotation)
* `B` [跳躍遊戲](src/algorithms/uncategorized/jump-game)
* `B` [唯一路徑](src/algorithms/uncategorized/unique-paths)
* `B` [雨水收集](src/algorithms/uncategorized/rain-terraces)
* `B` [遞迴階梯](src/algorithms/uncategorized/recursive-staircase)
* `B` [最佳買賣股票時機](src/algorithms/uncategorized/best-time-to-buy-sell-stocks)
* `B` [有效括號](src/algorithms/stack/valid-parentheses)
* `A` [N 皇后問題](src/algorithms/uncategorized/n-queens)
* `A` [騎士巡邏](src/algorithms/uncategorized/knight-tour)

* **Branch & Bound**

## 如何使用本知識庫
Expand Down