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<!DOCTYPE html>
<html>
<head>
<title>Big O Reference</title>
<meta name="description" content="Big O Reference for common algorithms and data structures with comparisons">
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<h1 class="uk-heading-large">Big O Reference</h1>
<p class="uk-text-large">Big O Reference of common algorithms and data structures</p>
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<li class="uk-active"><a href="#introduction">Introduction</a></li>
<li class="uk-nav-header">Algorithms</li>
<li class=""><a href="#sorting">Sorting</a></li>
<li class=""><a href="#searching">Searching</a></li>
<li class="uk-nav-header">Data Structures</li>
<li class=""><a href="#data-structures">Common Data Structures</a></li>
<li class=""><a href="#heaps">Heaps</a></li>
<li class=""><a href="#graphs">Graphs</a></li>
</ul>
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<h2 id="introduction" class="no-print">Introduction</h2>
<p class="no-print">This site summarizes the complexities in terms of space and time (Big O) of the most important algorithms and operations in common data structures. It is <a href="http://github.com/josem/bigoref">100% open-source</a> so you can edit the content with a pull request, with no ads and has a friendly <a href="javascript:if(window.print)window.print()">printable version</a>. If you have any comment please create an issue on Github. To know more about the project and see the credits take a look at <a href="http://github.com/josem/bigoref">the readme file</a>.</p>
<hr class="uk-grid-divider">
<h2 id="sorting">Sorting</h2>
<p class="no-print">Very important and useful type of algorithms for many common problems nowadays, Finder, iTunes, Excel and many more programs use some kind of sorting algorithm.</p>
<div class="uk-overflow-container">
<table class="uk-table uk-text-nowrap uk-table-striped">
<thead>
<tr>
<th>Algorithm</th>
<th>Time</th>
<th></th>
<th></th>
<th>Auxiliary Space</th>
</tr>
<tr>
<th></th>
<th>Best</th>
<th>Average</th>
<th>Worst</th>
<th>Worst</th>
</tr>
</thead>
<tbody>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Bubble_sort" target="_blank">Bubble sort</a></td>
<td><code class="bigo-great">O(n)</code></td>
<td><code class="bigo-bad">O(n<sup>2</sup>)</code></td>
<td><code class="bigo-bad">O(n<sup>2</sup>)</code></td>
<td><code class="bigo-great">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Insertion_sort" target="_blank">Insertion sort</a></td>
<td><code class="bigo-great">O(n)</code></td>
<td><code class="bigo-bad">O(n<sup>2</sup>)</code></td>
<td><code class="bigo-bad">O(n<sup>2</sup>)</code></td>
<td><code class="bigo-great">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Selection_sort" target="_blank">Selection sort</a></td>
<td><code class="bigo-bad">O(n<sup>2</sup>)</code></td>
<td><code class="bigo-bad">O(n<sup>2</sup>)</code></td>
<td><code class="bigo-bad">O(n<sup>2</sup>)</code></td>
<td><code class="bigo-great">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Merge_sort" target="_blank">Mergesort</a></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-bad">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Quicksort" target="_blank">Quicksort</a></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-bad">O(n<sup>2</sup>)</code></td>
<td><code class="bigo-ok">O(log(n))</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Heapsort" target="_blank">Heapsort</a></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-great">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Shellsort">Shell Sort</a><a href="http://www.cs.wcupa.edu/rkline/ds/shell-comparison.html"><code>(h=2^k-1)</code></a></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n<sup>1.25</sup>)</code> conjectured</td>
<td><code class="bigo-ok">O(n<sup>(3/2)</sup>)</code></td>
<td><code class="bigo-great">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Bucket_sort" target="_blank">Bucket sort</a></td>
<td><code class="bigo-great">O(n+k)</code></td>
<td><code class="bigo-great">O(n+k)</code></td>
<td><code class="bigo-bad">O(n<sup>2</sup>)</code></td>
<td><code class="bigo-bad">O(nk)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Radix_sort" target="_blank">Radix sort</a></td>
<td><code class="bigo-great">O(nk)</code></td>
<td><code class="bigo-great">O(nk)</code></td>
<td><code class="bigo-great">O(nk)</code></td>
<td><code class="bigo-ok">O(n+k)</code></td>
</tr>
</tbody>
</table>
</div>
<hr class="uk-grid-divider"><div class="page-break"></div>
<h2 style="page-break-before: always;" id="searching">Searching</h2>
<p class="no-print">Searching efficiently is crucial for many operations in different parts of the software we use on a daily basis.</p>
<div class="uk-overflow-container">
<table class="uk-table uk-text-nowrap uk-table-striped">
<thead>
<tr>
<th>Algorithm</th>
<th>Data Structure</th>
<th>Time</th>
<th></th>
<th>Space</th>
</tr>
<tr>
<th></th>
<th></th>
<th>Average</th>
<th>Worst</th>
<th>Worst</th>
</tr>
</thead>
<tbody>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Depth-first_search" target="_blank">Depth First Search (DFS)</a></td>
<td>Graph of |V| vertices and |E| edges</td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-great">O(|E| + |V|)</code></td>
<td><code class="bigo-great">O(|V|)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Breadth-first_search" target="_blank">Breadth First Search (BFS)</a></td>
<td>Graph of |V| vertices and |E| edges</td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-great">O(|E| + |V|)</code></td>
<td><code class="bigo-great">O(|V|)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Binary_search_algorithm" target="_blank">Binary Search</a></td>
<td>Sorted array of n elements</td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Brute-force_search" target="_blank">Linear (Brute force)</a></td>
<td>Array of n elements</td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-great">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm" target="_blank">Shortest path by Dijkstra,<br> Min-heap as priority queue</a></td>
<td>Graph of |V| vertices and |E| edges</td>
<td><code class="bigo-ok">O((|V|+|E|) log|V|)</code></td>
<td><code class="bigo-ok">O((|V|+|E|) log|V|)</code></td>
<td><code class="bigo-ok">O(|V|)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm" target="_blank">Shortest path by Dijkstra,<br> unsorted array as priority queue</a></td>
<td>Graph of |V| vertices and |E| edges</td>
<td><code class="bigo-ok">O(|V|<sup>2</sup>)</code></td>
<td><code class="bigo-ok">O(|V|<sup>2</sup>)</code></td>
<td><code class="bigo-ok">O(|V|)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm" target="_blank">Shortest path by Bellman-Ford</a></td>
<td>Graph of |V| vertices and |E| edges</td>
<td><code class="bigo-ok">O(|V||E|)</code></td>
<td><code class="bigo-ok">O(|V||E|)</code></td>
<td><code class="bigo-ok">O(|V|)</code></td>
</tr>
</tbody>
</table>
</div>
<hr class="uk-grid-divider"><div class="page-break"></div>
<h2 style="page-break-before: always;" id="data-structures">Common Data Structures</h2>
<p class="no-print">Data structures are the key in some of the most used algorithms and a good data structure is sometimes the way to achieve a very efficient and elegant solution.</p>
<div class="uk-overflow-container">
<table class="uk-table uk-text-nowrap uk-table-striped ">
<thead>
<tr>
<th>Data Structure</th>
<th>Time</th>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
</tr>
<tr>
<th></th>
<th>Average</th>
<th></th>
<th></th>
<th></th>
<th>Worst</th>
<th></th>
<th></th>
<th></th>
</tr>
<tr>
<th></th>
<th>Indexing</th>
<th>Search</th>
<th>Insertion</th>
<th>Deletion</th>
<th>Indexing</th>
<th>Search</th>
<th>Insertion</th>
<th>Deletion</th>
</tr>
</thead>
<tbody>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Array_data_structure" target="_blank">Basic Array</a></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-ok">-</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Dynamic_array" target="_blank">Dynamic Array</a></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Singly_linked_list#Singly_linked_lists" target="_blank">Single-Linked List</a></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Doubly_linked_list" target="_blank">Double-Linked List</a></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Skip_list" target="_blank">Skip List*</a></td>
<td><code class="bigo-ok">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Hash_table" target="_blank">Hash Table</a></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Binary_search_tree" target="_blank">Binary Search Tree</a></td>
<td><code class="bigo-ok">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Ternary_search_tree">Ternary Search Tree</a></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
</tr>
<tr>
<td><a href="https://en.wikipedia.org/wiki/Cartesian_tree" target="_blank">Cartesian Tree</a></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/B_tree" target="_blank">B-Tree</a></td>
<td><code class="bigo-ok">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-ok">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Red-black_tree" target="_blank">Red-Black Tree</a></td>
<td><code class="bigo-ok">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-ok">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
</tr>
<tr>
<td><a href="https://en.wikipedia.org/wiki/Splay_tree" target="_blank">Splay Tree</a></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/AVL_tree" target="_blank">AVL Tree</a></td>
<td><code class="bigo-ok">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-ok">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
<td><code class="bigo-great">O(log(n))</code></td>
</tr>
</tbody>
</table>
</div>
<p class="no-print"><span class="uk-badge">Note</span> * All have space complexity of O(n) except for Skip that has O(n log(n))</p>
<hr class="uk-grid-divider"><div class="page-break"></div>
<h2 style="page-break-before: always;" id="heaps">Heaps</h2>
<div class="uk-overflow-container">
<table class="uk-table uk-text-nowrap uk-table-striped ">
<thead>
<tr>
<th>Heaps</th>
<th>Time</th>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
</tr>
<tr>
<th></th>
<th>Heapify</th>
<th>Find Max</th>
<th>Extract Max</th>
<th>Increase Key</th>
<th>Insert</th>
<th>Delete</th>
<th>Merge</th>
</tr>
</thead>
<tbody>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Linked_list" target="_blank">Linked List</a></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-bad">O(m+n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Linked_list" target="_blank">Linked List (unsorted)</a></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-bad">O(n)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Binary_heap" target="_blank">Binary Heap</a></td>
<td><code class="bigo-ok">O(n)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-bad">O(m+n)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Binomial_heap" target="_blank">Binomial Heap</a></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
<td><code class="bigo-ok">O(n log(n))</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Fibonacci_heap" target="_blank">Fibonnaci Heap</a></td>
<td><code class="bigo-ok">-</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-ok">O(log(n))*</code></td>
<td><code class="bigo-great">O(1)*</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-ok">O(log(n))*</code></td>
<td><code class="bigo-great">O(1)</code></td>
</tr>
</tbody>
</table>
</div>
<p class="no-print"><span class="uk-badge">Note</span> * Amortized time</p>
<hr class="uk-grid-divider"><div class="page-break"></div>
<h2 style="page-break-before: always;" id="graphs">Graphs</h2>
<div class="uk-overflow-container">
<table class="uk-table uk-text-nowrap uk-table-striped ">
<thead>
<tr>
<th>Node / Edge Management</th>
<th>Storage</th>
<th>Add Vertex</th>
<th>Add Edge</th>
<th>Remove Vertex</th>
<th>Remove Edge</th>
<th>Query</th>
</tr>
</thead>
<tbody>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Adjacency_list" target="_blank">Adjacency List</a></td>
<td><code class="bigo-ok">O(|V|+|E|)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-ok">O(|V|+|E|)</code></td>
<td><code class="bigo-ok">O(|E|)</code></td>
<td><code class="bigo-ok">O(|V|)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Incidence_list" target="_blank">Incidence List</a></td>
<td><code class="bigo-ok">O(|V|+|E|)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-ok">O(|E|)</code></td>
<td><code class="bigo-ok">O(|E|)</code></td>
<td><code class="bigo-ok">O(|E|)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Adjacency_matrix" target="_blank">Adjacency Matrix</a></td>
<td><code class="bigo-bad">O(|V|<sup>2</sup>)</code></td>
<td><code class="bigo-bad">O(|V|<sup>2</sup>)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-bad">O(|V|<sup>2</sup>)</code></td>
<td><code class="bigo-great">O(1)</code></td>
<td><code class="bigo-great">O(1)</code></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Incidence_matrix" target="_blank">Incidence Matrix</a></td>
<td><code class="bigo-bad">O(|V|⋅|E|)</code></td>
<td><code class="bigo-bad">O(|V|⋅|E|)</code></td>
<td><code class="bigo-bad">O(|V|⋅|E|)</code></td>
<td><code class="bigo-bad">O(|V|⋅|E|)</code></td>
<td><code class="bigo-bad">O(|V|⋅|E|)</code></td>
<td><code class="bigo-ok">O(|E|)</code></td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</body>
</html>