-
Notifications
You must be signed in to change notification settings - Fork 13
Expand file tree
/
Copy pathinsertion-sort.html
More file actions
95 lines (88 loc) · 30.2 KB
/
insertion-sort.html
File metadata and controls
95 lines (88 loc) · 30.2 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
<!DOCTYPE html>
<html lang="en-US">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<title>Algorithm - Insertion Sort</title>
<meta name="generator" content="VuePress 1.8.2">
<link rel="icon" type="image/png" sizes="32x32" href="/favicon-32x32.png">
<link rel="icon" type="image/png" sizes="16x16" href="/favicon-16x16.png">
<link rel="manifest" href="/site.webmanifest">
<link rel="apple-touch-icon" sizes="180x180" href="/apple-touch-icon.png">
<link rel="mask-icon" href="/safari-pinned-tab.svg" color="#5bbad5">
<meta name="description" content="Haskell Implementation, Algorithm Basics, C# Implementation">
<meta property="og:site_name" content="DevTut">
<meta property="og:title" content="Algorithm - Insertion Sort">
<meta property="og:description" content="Haskell Implementation, Algorithm Basics, C# Implementation">
<meta property="og:type" content="article">
<meta property="og:url" content="/algorithm/insertion-sort.html">
<meta property="og:image" content="/logo.png">
<meta name="twitter:title" content="Algorithm - Insertion Sort">
<meta name="twitter:description" content="Haskell Implementation, Algorithm Basics, C# Implementation">
<meta name="twitter:url" content="/algorithm/insertion-sort.html">
<meta name="twitter:card" content="summary_large_image">
<meta name="twitter:image" content="/logo.png">
<meta name="theme-color" content="#ffffff">
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black">
<meta name="msapplication-TileImage" content="/mstile-150x150.png">
<meta name="msapplication-TileColor" content="#da532c">
<meta name="google-site-verification" content="76_rKXgwMVIjd-axJC_1zPV9OS4mEjvtgjYOWVkAdnQ">
<link rel="preload" href="/assets/css/0.styles.60619e34.css" as="style"><link rel="preload" href="/assets/js/app.1779e102.js" as="script"><link rel="preload" href="/assets/js/3.2cfa8016.js" as="script"><link rel="preload" href="/assets/js/44.a877bcf7.js" as="script">
<link rel="stylesheet" href="/assets/css/0.styles.60619e34.css">
</head>
<body>
<div id="app" data-server-rendered="true"><div class="theme-container"><header class="navbar"><div class="sidebar-button"><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" role="img" viewBox="0 0 448 512" class="icon"><path fill="currentColor" d="M436 124H12c-6.627 0-12-5.373-12-12V80c0-6.627 5.373-12 12-12h424c6.627 0 12 5.373 12 12v32c0 6.627-5.373 12-12 12zm0 160H12c-6.627 0-12-5.373-12-12v-32c0-6.627 5.373-12 12-12h424c6.627 0 12 5.373 12 12v32c0 6.627-5.373 12-12 12zm0 160H12c-6.627 0-12-5.373-12-12v-32c0-6.627 5.373-12 12-12h424c6.627 0 12 5.373 12 12v32c0 6.627-5.373 12-12 12z"></path></svg></div> <a href="/" class="home-link router-link-active"><!----> <span class="site-name">DevTut</span></a> <div class="links"><form id="search-form" role="search" class="algolia-search-wrapper search-box"><input id="algolia-search-input" class="search-query"></form> <nav class="nav-links can-hide"> <a href="https://github.com/devtut/generate" target="_blank" rel="noopener noreferrer" class="repo-link">
GitHub
<span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></nav></div></header> <div class="sidebar-mask"></div> <aside class="sidebar"><nav class="nav-links"> <a href="https://github.com/devtut/generate" target="_blank" rel="noopener noreferrer" class="repo-link">
GitHub
<span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></nav> <ul class="sidebar-links"><li><section class="sidebar-group depth-0"><p class="sidebar-heading open"><span>Algorithm</span> <!----></p> <ul class="sidebar-links sidebar-group-items"><li><a href="/algorithm/" aria-current="page" class="sidebar-link">Disclaimer</a></li><li><a href="/algorithm/getting-started-with-algorithm.html" class="sidebar-link">Getting started with algorithm</a></li><li><a href="/algorithm/algorithm-complexity.html" class="sidebar-link">Algorithm Complexity</a></li><li><a href="/algorithm/big-o-notation.html" class="sidebar-link">Big-O Notation</a></li><li><a href="/algorithm/trees.html" class="sidebar-link">Trees</a></li><li><a href="/algorithm/binary-search-trees.html" class="sidebar-link">Binary Search Trees</a></li><li><a href="/algorithm/check-if-a-tree-is-bst-or-not.html" class="sidebar-link">Check if a tree is BST or not</a></li><li><a href="/algorithm/binary-tree-traversals.html" class="sidebar-link">Binary Tree traversals</a></li><li><a href="/algorithm/lowest-common-ancestor-of-a-binary-tree.html" class="sidebar-link">Lowest common ancestor of a Binary Tree</a></li><li><a href="/algorithm/graph.html" class="sidebar-link">Graph</a></li><li><a href="/algorithm/graph-traversals.html" class="sidebar-link">Graph Traversals</a></li><li><a href="/algorithm/dijkstras-algorithm.html" class="sidebar-link">Dijkstra’s Algorithm</a></li><li><a href="/algorithm/a-pathfinding.html" class="sidebar-link">A* Pathfinding</a></li><li><a href="/algorithm/a-pathfinding-algorithm.html" class="sidebar-link">A* Pathfinding Algorithm</a></li><li><a href="/algorithm/dynamic-programming.html" class="sidebar-link">Dynamic Programming</a></li><li><a href="/algorithm/applications-of-dynamic-programming.html" class="sidebar-link">Applications of Dynamic Programming</a></li><li><a href="/algorithm/kruskal-s-algorithm.html" class="sidebar-link">Kruskal's Algorithm</a></li><li><a href="/algorithm/greedy-algorithms.html" class="sidebar-link">Greedy Algorithms</a></li><li><a href="/algorithm/applications-of-greedy-technique.html" class="sidebar-link">Applications of Greedy technique</a></li><li><a href="/algorithm/prim-s-algorithm.html" class="sidebar-link">Prim's Algorithm</a></li><li><a href="/algorithm/bellman-ford-algorithm.html" class="sidebar-link">Bellman–Ford Algorithm</a></li><li><a href="/algorithm/line-algorithm.html" class="sidebar-link">Line Algorithm</a></li><li><a href="/algorithm/floyd-warshall-algorithm.html" class="sidebar-link">Floyd-Warshall Algorithm</a></li><li><a href="/algorithm/catalan-number-algorithm.html" class="sidebar-link">Catalan Number Algorithm</a></li><li><a href="/algorithm/multithreaded-algorithms.html" class="sidebar-link">Multithreaded Algorithms</a></li><li><a href="/algorithm/knuth-morris-pratt-kmp-algorithm.html" class="sidebar-link">Knuth Morris Pratt (KMP) Algorithm</a></li><li><a href="/algorithm/edit-distance-dynamic-algorithm.html" class="sidebar-link">Edit Distance Dynamic Algorithm</a></li><li><a href="/algorithm/online-algorithms.html" class="sidebar-link">Online algorithms</a></li><li><a href="/algorithm/integer-partition-algorithm.html" class="sidebar-link">Integer Partition Algorithm</a></li><li><a href="/algorithm/maximum-path-sum-algorithm.html" class="sidebar-link">Maximum Path Sum Algorithm</a></li><li><a href="/algorithm/maximum-subarray-algorithm.html" class="sidebar-link">Maximum Subarray Algorithm</a></li><li><a href="/algorithm/sliding-window-algorithm.html" class="sidebar-link">Sliding Window Algorithm</a></li><li><a href="/algorithm/sorting.html" class="sidebar-link">Sorting</a></li><li><a href="/algorithm/bubble-sort.html" class="sidebar-link">Bubble Sort</a></li><li><a href="/algorithm/merge-sort.html" class="sidebar-link">Merge Sort</a></li><li><a href="/algorithm/insertion-sort.html" aria-current="page" class="active sidebar-link">Insertion Sort</a><ul class="sidebar-sub-headers"><li class="sidebar-sub-header"><a href="/algorithm/insertion-sort.html#haskell-implementation" class="sidebar-link">Haskell Implementation</a></li><li class="sidebar-sub-header"><a href="/algorithm/insertion-sort.html#algorithm-basics" class="sidebar-link">Algorithm Basics</a></li><li class="sidebar-sub-header"><a href="/algorithm/insertion-sort.html#c-implementation" class="sidebar-link">C# Implementation</a></li></ul></li><li><a href="/algorithm/bucket-sort.html" class="sidebar-link">Bucket Sort</a></li><li><a href="/algorithm/quicksort.html" class="sidebar-link">Quicksort</a></li><li><a href="/algorithm/counting-sort.html" class="sidebar-link">Counting Sort</a></li><li><a href="/algorithm/heap-sort.html" class="sidebar-link">Heap Sort</a></li><li><a href="/algorithm/cycle-sort.html" class="sidebar-link">Cycle Sort</a></li><li><a href="/algorithm/odd-even-sort.html" class="sidebar-link">Odd-Even Sort</a></li><li><a href="/algorithm/selection-sort.html" class="sidebar-link">Selection Sort</a></li><li><a href="/algorithm/pigeonhole-sort.html" class="sidebar-link">Pigeonhole Sort</a></li><li><a href="/algorithm/radix-sort.html" class="sidebar-link">Radix Sort</a></li><li><a href="/algorithm/shell-sort.html" class="sidebar-link">Shell Sort</a></li><li><a href="/algorithm/pancake-sort.html" class="sidebar-link">Pancake Sort</a></li><li><a href="/algorithm/searching.html" class="sidebar-link">Searching</a></li><li><a href="/algorithm/substring-search.html" class="sidebar-link">Substring Search</a></li><li><a href="/algorithm/breadth-first-search.html" class="sidebar-link">Breadth-First Search</a></li><li><a href="/algorithm/depth-first-search.html" class="sidebar-link">Depth First Search</a></li><li><a href="/algorithm/hash-functions.html" class="sidebar-link">Hash Functions</a></li><li><a href="/algorithm/travelling-salesman.html" class="sidebar-link">Travelling Salesman</a></li><li><a href="/algorithm/shortest-common-supersequence-problem.html" class="sidebar-link">Shortest Common Supersequence Problem</a></li><li><a href="/algorithm/knapsack-problem.html" class="sidebar-link">Knapsack Problem</a></li><li><a href="/algorithm/equation-solving.html" class="sidebar-link">Equation Solving</a></li><li><a href="/algorithm/longest-common-subsequence.html" class="sidebar-link">Longest Common Subsequence</a></li><li><a href="/algorithm/longest-increasing-subsequence.html" class="sidebar-link">Longest Increasing Subsequence</a></li><li><a href="/algorithm/check-two-strings-are-anagrams.html" class="sidebar-link">Check two strings are anagrams</a></li><li><a href="/algorithm/pascal-s-triangle.html" class="sidebar-link">Pascal's Triangle</a></li><li><a href="/algorithm/algo-print-a-m-n-matrix-in-square-wise.html" class="sidebar-link">Algo:- Print a m*n matrix in square wise</a></li><li><a href="/algorithm/matrix-exponentiation.html" class="sidebar-link">Matrix Exponentiation</a></li><li><a href="/algorithm/polynomial-time-bounded-algorithm-for-minimum-vertex-cover.html" class="sidebar-link">polynomial-time bounded algorithm for Minimum Vertex Cover</a></li><li><a href="/algorithm/dynamic-time-warping.html" class="sidebar-link">Dynamic Time Warping</a></li><li><a href="/algorithm/fast-fourier-transform.html" class="sidebar-link">Fast Fourier Transform</a></li><li><a href="/algorithm/pseudocode.html" class="sidebar-link">Pseudocode</a></li><li><a href="/algorithm/contributors.html" class="sidebar-link">The Contributors</a></li></ul></section></li></ul> </aside> <main class="page"> <div class="theme-default-content content__default"><h1 id="insertion-sort"><a href="#insertion-sort" class="header-anchor">#</a> Insertion Sort</h1> <h2 id="haskell-implementation"><a href="#haskell-implementation" class="header-anchor">#</a> Haskell Implementation</h2> <div class="language-cpp extra-class"><pre class="language-cpp"><code>insertSort <span class="token double-colon punctuation">::</span> Ord a <span class="token operator">=</span><span class="token operator">></span> <span class="token punctuation">[</span>a<span class="token punctuation">]</span> <span class="token operator">-></span> <span class="token punctuation">[</span>a<span class="token punctuation">]</span>
insertSort <span class="token punctuation">[</span><span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token punctuation">]</span>
<span class="token function">insertSort</span> <span class="token punctuation">(</span>x<span class="token operator">:</span>xs<span class="token punctuation">)</span> <span class="token operator">=</span> insert <span class="token function">x</span> <span class="token punctuation">(</span>insertSort xs<span class="token punctuation">)</span>
insert <span class="token double-colon punctuation">::</span> Ord a <span class="token operator">=</span><span class="token operator">></span> a<span class="token operator">-></span> <span class="token punctuation">[</span>a<span class="token punctuation">]</span> <span class="token operator">-></span> <span class="token punctuation">[</span>a<span class="token punctuation">]</span>
insert n <span class="token punctuation">[</span><span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token punctuation">[</span>n<span class="token punctuation">]</span>
insert <span class="token function">n</span> <span class="token punctuation">(</span>x<span class="token operator">:</span>xs<span class="token punctuation">)</span> <span class="token operator">|</span> n <span class="token operator"><=</span> x <span class="token operator">=</span> <span class="token punctuation">(</span>n<span class="token operator">:</span>x<span class="token operator">:</span>xs<span class="token punctuation">)</span>
<span class="token operator">|</span> otherwise <span class="token operator">=</span> x<span class="token operator">:</span>insert n xs
</code></pre></div><h2 id="algorithm-basics"><a href="#algorithm-basics" class="header-anchor">#</a> Algorithm Basics</h2> <p>Insertion sort is a very simple, stable, in-place sorting algorithm. It performs well on small sequences but it is much less efficient on large lists.
At every step, the algorithms considers the i-th element of the given sequence, moving it to the left until it is in the correct position.</p> <p><strong>Graphical Illustration</strong></p> <p><a href="http://i.stack.imgur.com/Jn79T.jpg" target="_blank" rel="noopener noreferrer"><img src="http://i.stack.imgur.com/Jn79T.jpg" alt="Insertion Sort"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p><strong>Pseudocode</strong></p> <div class="language-cpp extra-class"><pre class="language-cpp"><code><span class="token keyword">for</span> j <span class="token operator">=</span> <span class="token number">1</span> to <span class="token function">length</span><span class="token punctuation">(</span>A<span class="token punctuation">)</span>
n <span class="token operator">=</span> A<span class="token punctuation">[</span>j<span class="token punctuation">]</span>
i <span class="token operator">=</span> j <span class="token operator">-</span> <span class="token number">1</span>
<span class="token keyword">while</span> j <span class="token operator">></span> <span class="token number">0</span> <span class="token operator">and</span> A<span class="token punctuation">[</span>i<span class="token punctuation">]</span> <span class="token operator">></span> n
A<span class="token punctuation">[</span>i <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">=</span> A<span class="token punctuation">[</span>i<span class="token punctuation">]</span>
i <span class="token operator">=</span> i <span class="token operator">-</span> <span class="token number">1</span>
A<span class="token punctuation">[</span>i <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">=</span> n
</code></pre></div><p><strong>Example</strong></p> <p>Consider the following list of integers:</p> <div class="language-cpp extra-class"><pre class="language-cpp"><code><span class="token punctuation">[</span><span class="token number">5</span><span class="token punctuation">,</span> <span class="token number">2</span><span class="token punctuation">,</span> <span class="token number">4</span><span class="token punctuation">,</span> <span class="token number">6</span><span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">3</span><span class="token punctuation">]</span>
</code></pre></div><p>The algorithm will perform the following steps:</p> <ol><li><code>[5, 2, 4, 6, 1, 3]</code></li> <li><code>[2, 5, 4, 6, 1, 3]</code></li> <li><code>[2, 4, 5, 6, 1, 3]</code></li> <li><code>[2, 4, 5, 6, 1, 3]</code></li> <li><code>[1, 2, 4, 5, 6, 3]</code></li> <li><code>[1, 2, 3, 4, 5, 6]</code></li></ol> <h2 id="c-implementation"><a href="#c-implementation" class="header-anchor">#</a> C# Implementation</h2> <div class="language-cpp extra-class"><pre class="language-cpp"><code><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">InsertionSort</span>
<span class="token punctuation">{</span>
<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">SortInsertion</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> input<span class="token punctuation">,</span> <span class="token keyword">int</span> n<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator"><</span> n<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">int</span> x <span class="token operator">=</span> input<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token keyword">int</span> j <span class="token operator">=</span> i <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span>
<span class="token keyword">while</span> <span class="token punctuation">(</span>j <span class="token operator">>=</span> <span class="token number">0</span> <span class="token operator">&&</span> input<span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">></span> x<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
input<span class="token punctuation">[</span>j <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">=</span> input<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">;</span>
j <span class="token operator">=</span> j <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
input<span class="token punctuation">[</span>j <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">=</span> x<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> <span class="token function">Main</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> input<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token function">SortInsertion</span><span class="token punctuation">(</span>input<span class="token punctuation">,</span> input<span class="token punctuation">.</span>Length<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">return</span> input<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</code></pre></div><p><strong>Auxiliary Space:</strong> <code>O(1)</code><br> <strong>Time Complexity:</strong> <code>O(n)</code></p> <h4 id="remarks"><a href="#remarks" class="header-anchor">#</a> Remarks</h4> <p>When we analyze the performance of the sorting algorithm, we interest primarily on the number of comparison and exchange.</p> <h3 id="average-exchange"><a href="#average-exchange" class="header-anchor">#</a> Average Exchange</h3> <p>Let E<sub>n</sub> be the total average number of exchange to sort array of N element. E<sub>1</sub> = 0 (we do not need any exchange for array with one element). The average number of exchange to sort N element array is the sum of average number of number of exchange to sort N-1 element array with the average exchange to insert the last element into N-1 element array.</p> <p><a href="http://i.stack.imgur.com/0I2Ba.gif" target="_blank" rel="noopener noreferrer"><img src="http://i.stack.imgur.com/0I2Ba.gif" alt="enter image description here"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p>Simplify the summation (arithmetic series)</p> <p><a href="http://i.stack.imgur.com/i6vUV.gif" target="_blank" rel="noopener noreferrer"><img src="http://i.stack.imgur.com/i6vUV.gif" alt="enter image description here"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p>Expands the term</p> <p><a href="http://i.stack.imgur.com/qQGAc.gif" target="_blank" rel="noopener noreferrer"><img src="http://i.stack.imgur.com/qQGAc.gif" alt="enter image description here"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p>Simplify the summation again (arithmetic series)</p> <p><a href="http://i.stack.imgur.com/D4Iye.gif" target="_blank" rel="noopener noreferrer"><img src="http://i.stack.imgur.com/D4Iye.gif" alt="enter image description here"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <h3 id="average-comparison"><a href="#average-comparison" class="header-anchor">#</a> Average Comparison</h3> <p>Let C<sub>n</sub> be the total average number of comparison to sort array of N element. C<sub>1</sub> = 0 (we do not need any comparison on one element array). The average number of comparison to sort N element array is the sum of average number of number of comparison to sort N-1 element array with the average comparison to insert the last element into N-1 element array. If last element is largest element, we need only one comparison, if the last element is the second smallest element, we need N-1 comparison. However, if last element is the smallest element, we do not need N comparison. We still only need N-1 comparison. That is why we remove 1/N in below equation.</p> <p><a href="http://i.stack.imgur.com/CsWkN.gif" target="_blank" rel="noopener noreferrer"><img src="http://i.stack.imgur.com/CsWkN.gif" alt="enter image description here"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p>Simplify the summation (arithmetic series)</p> <p><a href="http://i.stack.imgur.com/V4kOL.gif" target="_blank" rel="noopener noreferrer"><img src="http://i.stack.imgur.com/V4kOL.gif" alt="enter image description here"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p>Expand the term</p> <p><a href="http://i.stack.imgur.com/RtUcF.gif" target="_blank" rel="noopener noreferrer"><img src="http://i.stack.imgur.com/RtUcF.gif" alt="enter image description here"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p>Simplify the summation again (arithmetic series and harmonic number)</p> <p><a href="http://i.stack.imgur.com/b6ViQ.gif" target="_blank" rel="noopener noreferrer"><img src="http://i.stack.imgur.com/b6ViQ.gif" alt="enter image description here"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p></div> <footer class="page-edit"><div class="edit-link"><a href="https://github.com/devtut/generate/edit/master/docs/algorithm/insertion-sort.md" target="_blank" rel="noopener noreferrer">Edit this page on GitHub</a> <span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></div> <!----></footer> <div class="page-nav"><p class="inner"><span class="prev">
←
<a href="/algorithm/merge-sort.html" class="prev">
Merge Sort
</a></span> <span class="next"><a href="/algorithm/bucket-sort.html">
Bucket Sort
</a>
→
</span></p></div> </main></div><div class="global-ui"><!----></div></div>
<script src="/assets/js/app.1779e102.js" defer></script><script src="/assets/js/3.2cfa8016.js" defer></script><script src="/assets/js/44.a877bcf7.js" defer></script>
</body>
</html>