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<!DOCTYPE html>

<head>

<title>Insertion Sort</title>

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<h3>Page Index</h3>

<ul>

<li><a href="/dsa.html#data-structure-1" title="Data Structures (I)">Data Structures

(I)</a></li>

<li><a href="/dsa.html#data-structure-2" title="Data Structures (II)">Data Structures

(II)</a></li>

<li><a href="/dsa.html#tree-1" title="Tree based DSA (I)">Tree based DSA (I)</a></li>

<li><a href="/dsa.html#tree-2" title="Tree based DSA (II)">Tree based DSA (II)</a></li>

<li><a href="/dsa.html#graph" title="Graph Data Structures and Algorithm">Graph based

DSA</a></li>

<li><a href="/dsa.html#sorting-searching" title="Sorting and Searching">Sorting and

Searching</a></li>

<li><a href="/dsa.html#greedy-algorithm" title="Greedy Algorithms">Greedy Algorithms</a>

</li>

<li><a href="/dsa.html#dynamic-programming" title="Dynamic Programming">Dynamic

Programming</a></li>

<li><a href="/dsa.html#other-algorithms" title="Other Algorithms">Other Algorithms</a>

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</ul>

</div>

<h1>Insertion Sort Algorithm</h1>

<p class="editor-contents__short-description">In this tutorial, you will learn about insertion sort algorithm and its implementation in C, C++, Java and Python. </p>

<p>Insertion sort is <a href="/dsa.html">a sorting algorithm</a> that places an unsorted element at its suitable place in each iteration.</p>

<p>Insertion sort works similarly as we sort cards in our hand in a card game.</p>

<p>We assume that the first card is already sorted then, we select an unsorted card. If the unsorted card is greater than the card in hand, it is placed on the right otherwise, to the left. In the same way, other unsorted cards are taken and put in their right place.</p>

<p>A similar approach is used by insertion sort.</p>

<hr><h2 id="working">Working of Insertion Sort</h2>

<p>Suppose we need to sort the following array.</p>

<figure><img alt="Insertion Sort Steps" src="https://www.programiz.com/sites/tutorial2program/files/Frame%204_0.png" title="Initial array" width="330" height="85"><figcaption>Initial array</figcaption></figure><ol><li>The first element in the array is assumed to be sorted. Take the second element and store it separately in <code>key</code>.<br><br>

Compare <code>key</code> with the first element. If the first element is greater than <code>key</code>, then <var>key</var> is placed in front of the first element.

<figure><img alt="Insertion Sort Steps" src="https://www.programiz.com/sites/tutorial2program/files/Insertion-sort-0_1.png" title="Insertion Sort Step 0" width="400" height="358"><figcaption>If the first element is greater than key, then key is placed in front of the first element.</figcaption></figure></li>

<li>Now, the first two elements are sorted.<br><br>

Take the third element and compare it with the elements on the left of it. Placed it just behind the element smaller than it. If there is no element smaller than it, then place it at the beginning of the array.

<figure><img alt="Insertion Sort Steps" src="https://www.programiz.com/sites/tutorial2program/files/Insertion-sort-1_1.png" title="Insertion Sort Step 1" width="400" height="440"><figcaption>Place 1 at the beginning</figcaption></figure></li>

<li>Similarly, place every unsorted element at its correct position.

<figure><img alt="Insertion Sort Steps" src="https://www.programiz.com/sites/tutorial2program/files/Insertion-sort-2_2.png" title="Insertion Sort Step 2" width="400" height="443"><figcaption>Place 4 behind 1</figcaption></figure><figure><img alt="Insertion Sort Steps" src="https://www.programiz.com/sites/tutorial2program/files/Insertion-sort-3_2.png" title="Insertion Sort Step 3" width="400" height="530"><figcaption>Place 3 behind 1 and the array is sorted</figcaption></figure></li>

</ol><hr><h2 id="algorithm">Insertion Sort Algorithm</h2>

<span class="hljs-attr">insertionSort(array)</span>

<span class="hljs-attr">mark</span> <span class="hljs-string">first element as sorted</span>

<span class="hljs-attr">for</span> <span class="hljs-string">each unsorted element X</span>

<span class="hljs-meta">'extract'</span> <span class="hljs-string">the element X</span>

<span class="hljs-attr">if</span> <span class="hljs-string">current element j > X</span>

<span class="hljs-attr">move</span> <span class="hljs-string">sorted element to the right by 1</span>

<span class="hljs-attr">break</span> <span class="hljs-string">loop and insert X here</span>

<span class="hljs-attr">end</span> <span class="hljs-string">insertionSort</span></code></pre>

<hr><h2 id="code">Insertion Sort in Python, Java, and C/C++</h2>

<div id="py1" onclick="changepy()" class="tabbed-editor__node tabbed-editor__node--active"><a href="#python-code">Python</a></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="python hljs"><span class="hljs-comment"># Insertion sort in Python</span>

<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">insertionSort</span><span class="hljs-params">(array)</span>:</span>

<span class="hljs-keyword">for</span> step <span class="hljs-keyword">in</span> range(<span class="hljs-number">1</span>, len(array)):

key = array[step]

j = step - <span class="hljs-number">1</span>

<span class="hljs-comment"># Compare key with each element on the left of it until an element smaller than it is found</span>

<span class="hljs-comment"># For descending order, change key<array[j] to key>array[j]. </span>

<span class="hljs-keyword">while</span> j >= <span class="hljs-number">0</span> <span class="hljs-keyword">and</span> key < array[j]:

array[j + <span class="hljs-number">1</span>] = array[j]

j = j - <span class="hljs-number">1</span>

<span class="hljs-comment"># Place key at after the element just smaller than it.</span>

array[j + <span class="hljs-number">1</span>] = key

data = [<span class="hljs-number">9</span>, <span class="hljs-number">5</span>, <span class="hljs-number">1</span>, <span class="hljs-number">4</span>, <span class="hljs-number">3</span>]

insertionSort(data)

<span class="hljs-keyword">print</span>(<span class="hljs-string">'Sorted Array in Ascending Order:'</span>)

<span class="hljs-keyword">print</span>(data)</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="java hljs"><span class="hljs-comment">// Insertion sort in Java</span>

<span class="hljs-keyword">import</span> java.util.Arrays;

<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">InsertionSort</span> </span>{

<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">insertionSort</span><span class="hljs-params">(<span class="hljs-keyword">int</span> array[])</span> </span>{

<span class="hljs-keyword">int</span> size = array.length;

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> step = <span class="hljs-number">1</span>; step < size; step++) {

<span class="hljs-keyword">int</span> key = array[step];

<span class="hljs-keyword">int</span> j = step - <span class="hljs-number">1</span>;

<span class="hljs-comment">// Compare key with each element on the left of it until an element smaller than</span>

<span class="hljs-comment">// it is found.</span>

<span class="hljs-comment">// For descending order, change key<array[j] to key>array[j].</span>

<span class="hljs-keyword">while</span> (j >= <span class="hljs-number">0</span> && key < array[j]) {

array[j + <span class="hljs-number">1</span>] = array[j];

--j;

}

<span class="hljs-comment">// Place key at after the element just smaller than it.</span>

array[j + <span class="hljs-number">1</span>] = key;

}

<span class="hljs-comment">// Driver code</span>

<span class="hljs-function"><span class="hljs-keyword">public</span> <span class="hljs-keyword">static</span> <span class="hljs-keyword">void</span> <span class="hljs-title">main</span><span class="hljs-params">(String args[])</span> </span>{

<span class="hljs-keyword">int</span>[] data = { <span class="hljs-number">9</span>, <span class="hljs-number">5</span>, <span class="hljs-number">1</span>, <span class="hljs-number">4</span>, <span class="hljs-number">3</span> };

InsertionSort is = <span class="hljs-keyword">new</span> InsertionSort();

is.insertionSort(data);

System.out.println(<span class="hljs-string">"Sorted Array in Ascending Order: "</span>);

System.out.println(Arrays.toString(data));

}

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="c hljs cpp"><span class="hljs-comment">// Insertion sort in C</span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><stdio.h></span></span>

<span class="hljs-comment">// Function to print an array</span>

<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">printArray</span><span class="hljs-params">(<span class="hljs-keyword">int</span> <span class="hljs-built_in">array</span>[], <span class="hljs-keyword">int</span> size)</span> </span>{

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < size; i++) {

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"%d "</span>, <span class="hljs-built_in">array</span>[i]);

}

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"\n"</span>);

}

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> step = <span class="hljs-number">1</span>; step < size; step++) {

<span class="hljs-keyword">int</span> key = <span class="hljs-built_in">array</span>[step];

<span class="hljs-keyword">int</span> j = step - <span class="hljs-number">1</span>;

<span class="hljs-comment">// Compare key with each element on the left of it until an element smaller than</span>

<span class="hljs-comment">// it is found.</span>

<span class="hljs-comment">// For descending order, change key<array[j] to key>array[j].</span>

<span class="hljs-keyword">while</span> (key < <span class="hljs-built_in">array</span>[j] && j >= <span class="hljs-number">0</span>) {

<span class="hljs-built_in">array</span>[j + <span class="hljs-number">1</span>] = <span class="hljs-built_in">array</span>[j];

--j;

}

<span class="hljs-built_in">array</span>[j + <span class="hljs-number">1</span>] = key;

}

<span class="hljs-comment">// Driver code</span>

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{

<span class="hljs-keyword">int</span> data[] = {<span class="hljs-number">9</span>, <span class="hljs-number">5</span>, <span class="hljs-number">1</span>, <span class="hljs-number">4</span>, <span class="hljs-number">3</span>};

<span class="hljs-keyword">int</span> size = <span class="hljs-keyword">sizeof</span>(data) / <span class="hljs-keyword">sizeof</span>(data[<span class="hljs-number">0</span>]);

insertionSort(data, size);

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"Sorted array in ascending order:\n"</span>);

printArray(data, size);

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="cpp hljs"><span class="hljs-comment">// Insertion sort in C++</span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><iostream></span></span>

<span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> <span class="hljs-built_in">std</span>;

<span class="hljs-comment">// Function to print an array</span>

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < size; i++) {

<span class="hljs-built_in">cout</span> << <span class="hljs-built_in">array</span>[i] << <span class="hljs-string">" "</span>;

}

<span class="hljs-built_in">cout</span> << <span class="hljs-built_in">endl</span>;

}

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> step = <span class="hljs-number">1</span>; step < size; step++) {

<span class="hljs-keyword">int</span> key = <span class="hljs-built_in">array</span>[step];

<span class="hljs-keyword">int</span> j = step - <span class="hljs-number">1</span>;

<span class="hljs-comment">// Compare key with each element on the left of it until an element smaller than</span>

<span class="hljs-comment">// it is found.</span>

<span class="hljs-comment">// For descending order, change key<array[j] to key>array[j].</span>

<span class="hljs-keyword">while</span> (key < <span class="hljs-built_in">array</span>[j] && j >= <span class="hljs-number">0</span>) {

<span class="hljs-built_in">array</span>[j + <span class="hljs-number">1</span>] = <span class="hljs-built_in">array</span>[j];

--j;

}

<span class="hljs-built_in">array</span>[j + <span class="hljs-number">1</span>] = key;

}

<span class="hljs-comment">// Driver code</span>

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{

<span class="hljs-keyword">int</span> size = <span class="hljs-keyword">sizeof</span>(data) / <span class="hljs-keyword">sizeof</span>(data[<span class="hljs-number">0</span>]);

insertionSort(data, size);

<span class="hljs-built_in">cout</span> << <span class="hljs-string">"Sorted array in ascending order:\n"</span>;

printArray(data, size);

}</code></pre></div>

</div>

<hr><h2 id="complexity">Insertion Sort Complexity</h2>

<table border="0"><tbody><tr><th><strong>Time Complexity</strong></th>

</tr><tr><td>Best</td>

</tr><tr><td>Worst</td>

</tr><tr><td>Average</td>

</tr><tr><th><strong>Space Complexity</strong></th>

</tr><tr><th><strong>Stability</strong></th>

</tr></tbody></table></div>

<hr><p></p><div class="clearfix"></div><p><strong>Time Complexities</strong></p>

<ul><li><strong>Worst Case Complexity:</strong> <code>O(n<sup>2</sup>)</code><br>

Suppose, an array is in ascending order, and you want to sort it in descending order. In this case, worst case complexity occurs.<br><br>

Each element has to be compared with each of the other elements so, for every nth element, <code>(n-1)</code> number of comparisons are made.<br><br>

Thus, the total number of comparisons = <code>n*(n-1) ~ n<sup>2</sup></code></li>

<li><strong>Best Case Complexity:</strong> <code>O(n)</code><br>

When the array is already sorted, the outer loop runs for <code>n</code> number of times whereas the inner loop does not run at all. So, there are only <code>n</code> number of comparisons. Thus, complexity is linear.</li>

<li><strong>Average Case Complexity:</strong> <code>O(n<sup>2</sup>)</code><br>

It occurs when the elements of an array are in jumbled order (neither ascending nor descending).</li>

</ul><p><strong>Space Complexity</strong></p>

<p>Space complexity is <code>O(1)</code> because an extra variable <code>key</code> is used.</p>

<hr><h2 id="application">Insertion Sort Applications</h2>

<p>The insertion sort is used when:</p>

<ul><li>the array is has a small number of elements</li>

<li>there are only a few elements left to be sorted</li>

</ul><hr><h2>Similar Sorting Algorithms</h2>

<ol><li><a href="/dsa/bubble-sort.html">Bubble Sort</a></li>

<li><a href="/dsa/quick-sort.html">Quicksort</a></li>

<li><a href="/dsa/merge-sort.html">Merge Sort</a></li>

<li><a href="/dsa/selection-sort.html">Selection Sort</a></li>

</ol></div>

</div>

<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents

<li><a href="#algorithm">Insertion Sort Algorithm</a></li>

<li><a href="#code">Insertion Sort in Python, Java, and C/C++</a></li>

<li><a href="#complexity">Complexity</a></li>

<li><a href="#application">Applications</a></li>

</ul></div></div></div> </div>

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