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<h1>Counting Sort Algorithm</h1>
<p class="editor-contents__short-description">In this tutorial, you will learn about the counting sort algorithm and its implementation in Python, Java, C, and C++.</p>
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<span property="dc:title" content="Counting Sort Algorithm" class="rdf-meta element-hidden"></span>
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<p id="definition">Counting sort is <a href="/dsa.html">a sorting algorithm</a> that sorts the elements of an array by counting the number of occurrences of each unique element in the array. The count is stored in an auxiliary array and the sorting is done by mapping the count as an index of the auxiliary array.</p>
<hr><h2 id="working">Working of Counting Sort</h2>
<ol><li>Find out the maximum element (let it be <code>max</code>) from the given array.
<figure><img alt="Counting Sort steps" src="https://www.programiz.com/sites/tutorial2program/files/Counting-sort-0_0.png" title="Given array" width="544" height="109"><figcaption>Given array</figcaption></figure></li>
<li>Initialize an array of length <code>max+1</code> with all elements 0. This array is used for storing the count of the elements in the array.
<figure><img alt="Counting Sort Step" src="https://www.programiz.com/sites/tutorial2program/files/Counting-sort-1.png" title="Count Array" width="544" height="98"><figcaption>Count array</figcaption></figure></li>
<li>Store the count of each element at their respective index in <code>count</code> array<br><br>
For example: if the count of element 3 is 2 then, 2 is stored in the 3rd position of <var>count</var> array. If element "5" is not present in the array, then 0 is stored in 5th position.
<figure><img alt="Counting Sort Step" src="https://www.programiz.com/sites/tutorial2program/files/Counting-sort-2.png" title="Count of each element" width="544" height="100"><figcaption>Count of each element stored</figcaption></figure></li>
<li>Store cumulative sum of the elements of the count array. It helps in placing the elements into the correct index of the sorted array.
<figure><img alt="Counting Sort Step" src="https://www.programiz.com/sites/tutorial2program/files/Counting-sort-3.png" title="Cumulative count" width="544" height="100"><figcaption>Cumulative count</figcaption></figure></li>
<li>Find the index of each element of the original array in the count array. This gives the cumulative count. Place the element at the index calculated as shown in figure below.
<figure><img alt="Counting Sort Steps" src="https://www.programiz.com/sites/tutorial2program/files/Counting-sort-4_1.png" title="Counting Sort" width="592" height="298"><figcaption>Counting sort</figcaption></figure></li>
<li>After placing each element at its correct position, decrease its count by one.</li>
</ol><hr><h2 id="algorithm">Counting Sort Algorithm</h2>
<pre style="max-height: 600px;"><code class="dsa hljs swift">countingSort(array, size)
<span class="hljs-built_in">max</span> <- <span class="hljs-built_in">find</span> largest element <span class="hljs-keyword">in</span> array
initialize <span class="hljs-built_in">count</span> array with all zeros
<span class="hljs-keyword">for</span> j <- <span class="hljs-number">0</span> to size
<span class="hljs-built_in">find</span> the total <span class="hljs-built_in">count</span> of each unique element and
store the <span class="hljs-built_in">count</span> at jth index <span class="hljs-keyword">in</span> <span class="hljs-built_in">count</span> array
<span class="hljs-keyword">for</span> i <- <span class="hljs-number">1</span> to <span class="hljs-built_in">max</span>
<span class="hljs-built_in">find</span> the cumulative sum and store it <span class="hljs-keyword">in</span> <span class="hljs-built_in">count</span> array itself
<span class="hljs-keyword">for</span> j <- size down to <span class="hljs-number">1</span>
restore the elements to array
decrease <span class="hljs-built_in">count</span> of each element restored by <span class="hljs-number">1</span></code></pre>
<hr><h2 id="code">Counting Sort Code in Python, Java, and C/C++</h2>
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<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"># Counting sort in Python programming</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">countingSort</span><span class="hljs-params">(array)</span>:</span>
size = len(array)
output = [<span class="hljs-number">0</span>] * size
<span class="hljs-comment"># Initialize count array</span>
count = [<span class="hljs-number">0</span>] * <span class="hljs-number">10</span>
<span class="hljs-comment"># Store the count of each elements in count array</span>
<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(<span class="hljs-number">0</span>, size):
count[array[i]] += <span class="hljs-number">1</span>
<span class="hljs-comment"># Store the cummulative count</span>
<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(<span class="hljs-number">1</span>, <span class="hljs-number">10</span>):
count[i] += count[i - <span class="hljs-number">1</span>]
<span class="hljs-comment"># Find the index of each element of the original array in count array</span>
<span class="hljs-comment"># place the elements in output array</span>
i = size - <span class="hljs-number">1</span>
<span class="hljs-keyword">while</span> i >= <span class="hljs-number">0</span>:
output[count[array[i]] - <span class="hljs-number">1</span>] = array[i]
count[array[i]] -= <span class="hljs-number">1</span>
i -= <span class="hljs-number">1</span>
<span class="hljs-comment"># Copy the sorted elements into original array</span>
<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(<span class="hljs-number">0</span>, size):
array[i] = output[i]
data = [<span class="hljs-number">4</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">8</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>, <span class="hljs-number">1</span>]
countingSort(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>
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<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">// Counting sort in Java programming</span>
<span class="hljs-keyword">import</span> java.util.Arrays;
<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">CountingSort</span> </span>{
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">countSort</span><span class="hljs-params">(<span class="hljs-keyword">int</span> array[], <span class="hljs-keyword">int</span> size)</span> </span>{
<span class="hljs-keyword">int</span>[] output = <span class="hljs-keyword">new</span> <span class="hljs-keyword">int</span>[size + <span class="hljs-number">1</span>];
<span class="hljs-comment">// Find the largest element of the array</span>
<span class="hljs-keyword">int</span> max = array[<span class="hljs-number">0</span>];
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">1</span>; i < size; i++) {
<span class="hljs-keyword">if</span> (array[i] > max)
max = array[i];
}
<span class="hljs-keyword">int</span>[] count = <span class="hljs-keyword">new</span> <span class="hljs-keyword">int</span>[max + <span class="hljs-number">1</span>];
<span class="hljs-comment">// Initialize count array with all zeros.</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < max; ++i) {
count[i] = <span class="hljs-number">0</span>;
}
<span class="hljs-comment">// Store the count of each element</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < size; i++) {
count[array[i]]++;
}
<span class="hljs-comment">// Store the cummulative count of each array</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">1</span>; i <= max; i++) {
count[i] += count[i - <span class="hljs-number">1</span>];
}
<span class="hljs-comment">// Find the index of each element of the original array in count array, and</span>
<span class="hljs-comment">// place the elements in output array</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = size - <span class="hljs-number">1</span>; i >= <span class="hljs-number">0</span>; i--) {
output[count[array[i]] - <span class="hljs-number">1</span>] = array[i];
count[array[i]]--;
}
<span class="hljs-comment">// Copy the sorted elements into original array</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < size; i++) {
array[i] = output[i];
}
}
<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">4</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">8</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>, <span class="hljs-number">1</span> };
<span class="hljs-keyword">int</span> size = data.length;
CountingSort cs = <span class="hljs-keyword">new</span> CountingSort();
cs.countSort(data, size);
System.out.println(<span class="hljs-string">"Sorted Array in Ascending Order: "</span>);
System.out.println(Arrays.toString(data));
}
}</code></pre></div>
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<div class="code-editor__area" id="c-code">
<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">// Counting sort in C programming</span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><stdio.h></span></span>
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">countingSort</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">int</span> output[<span class="hljs-number">10</span>];
<span class="hljs-comment">// Find the largest element of the array</span>
<span class="hljs-keyword">int</span> max = <span class="hljs-built_in">array</span>[<span class="hljs-number">0</span>];
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">1</span>; i < size; i++) {
<span class="hljs-keyword">if</span> (<span class="hljs-built_in">array</span>[i] > max)
max = <span class="hljs-built_in">array</span>[i];
}
<span class="hljs-comment">// The size of count must be at least (max+1) but</span>
<span class="hljs-comment">// we cannot declare it as int count(max+1) in C as</span>
<span class="hljs-comment">// it does not support dynamic memory allocation.</span>
<span class="hljs-comment">// So, its size is provided statically.</span>
<span class="hljs-keyword">int</span> count[<span class="hljs-number">10</span>];
<span class="hljs-comment">// Initialize count array with all zeros.</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i <= max; ++i) {
count[i] = <span class="hljs-number">0</span>;
}
<span class="hljs-comment">// Store the count of each element</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < size; i++) {
count[<span class="hljs-built_in">array</span>[i]]++;
}
<span class="hljs-comment">// Store the cummulative count of each array</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">1</span>; i <= max; i++) {
count[i] += count[i - <span class="hljs-number">1</span>];
}
<span class="hljs-comment">// Find the index of each element of the original array in count array, and</span>
<span class="hljs-comment">// place the elements in output array</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = size - <span class="hljs-number">1</span>; i >= <span class="hljs-number">0</span>; i--) {
output[count[<span class="hljs-built_in">array</span>[i]] - <span class="hljs-number">1</span>] = <span class="hljs-built_in">array</span>[i];
count[<span class="hljs-built_in">array</span>[i]]--;
}
<span class="hljs-comment">// Copy the sorted elements into original 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">array</span>[i] = output[i];
}
}
<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-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> <span class="hljs-built_in">array</span>[] = {<span class="hljs-number">4</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">8</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>, <span class="hljs-number">1</span>};
<span class="hljs-keyword">int</span> n = <span class="hljs-keyword">sizeof</span>(<span class="hljs-built_in">array</span>) / <span class="hljs-keyword">sizeof</span>(<span class="hljs-built_in">array</span>[<span class="hljs-number">0</span>]);
countingSort(<span class="hljs-built_in">array</span>, n);
printArray(<span class="hljs-built_in">array</span>, n);
}</code></pre></div>
</div>
<div class="code-editor__area" id="cpp-code">
<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">// Counting sort in C++ programming</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-function"><span class="hljs-keyword">void</span> <span class="hljs-title">countSort</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-comment">// The size of count must be at least the (max+1) but</span>
<span class="hljs-comment">// we cannot assign declare it as int count(max+1) in C++ as</span>
<span class="hljs-comment">// it does not support dynamic memory allocation.</span>
<span class="hljs-comment">// So, its size is provided statically.</span>
<span class="hljs-keyword">int</span> output[<span class="hljs-number">10</span>];
<span class="hljs-keyword">int</span> count[<span class="hljs-number">10</span>];
<span class="hljs-keyword">int</span> max = <span class="hljs-built_in">array</span>[<span class="hljs-number">0</span>];
<span class="hljs-comment">// Find the largest element of the array</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">1</span>; i < size; i++) {
<span class="hljs-keyword">if</span> (<span class="hljs-built_in">array</span>[i] > max)
max = <span class="hljs-built_in">array</span>[i];
}
<span class="hljs-comment">// Initialize count array with all zeros.</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i <= max; ++i) {
count[i] = <span class="hljs-number">0</span>;
}
<span class="hljs-comment">// Store the count of each element</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < size; i++) {
count[<span class="hljs-built_in">array</span>[i]]++;
}
<span class="hljs-comment">// Store the cummulative count of each array</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">1</span>; i <= max; i++) {
count[i] += count[i - <span class="hljs-number">1</span>];
}
<span class="hljs-comment">// Find the index of each element of the original array in count array, and</span>
<span class="hljs-comment">// place the elements in output array</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = size - <span class="hljs-number">1</span>; i >= <span class="hljs-number">0</span>; i--) {
output[count[<span class="hljs-built_in">array</span>[i]] - <span class="hljs-number">1</span>] = <span class="hljs-built_in">array</span>[i];
count[<span class="hljs-built_in">array</span>[i]]--;
}
<span class="hljs-comment">// Copy the sorted elements into original 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">array</span>[i] = output[i];
}
}
<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">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-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> <span class="hljs-built_in">array</span>[] = {<span class="hljs-number">4</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>, <span class="hljs-number">8</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>, <span class="hljs-number">1</span>};
<span class="hljs-keyword">int</span> n = <span class="hljs-keyword">sizeof</span>(<span class="hljs-built_in">array</span>) / <span class="hljs-keyword">sizeof</span>(<span class="hljs-built_in">array</span>[<span class="hljs-number">0</span>]);
countSort(<span class="hljs-built_in">array</span>, n);
printArray(<span class="hljs-built_in">array</span>, n);
}</code></pre></div>
</div>
</div>
<hr><h2 id="complexity">Complexity</h2>
<div class="table-responsive">
<table border="0"><tbody><tr><th><strong>Time Complexity</strong></th>
<td> </td>
</tr><tr><td>Best</td>
<td>O(n+k)</td>
</tr><tr><td>Worst</td>
<td>O(n+k)</td>
</tr><tr><td>Average</td>
<td>O(n+k)</td>
</tr><tr><th><strong>Space Complexity</strong></th>
<td>O(max)</td>
</tr><tr><th><strong>Stability</strong></th>
<td>Yes</td>
</tr></tbody></table></div>
<hr><p><strong>Time Complexities</strong></p>
<p>There are mainly four main loops. (Finding the greatest value can be done outside the function.)</p>
<div class="table-responsive">
<table border="0"><tbody><tr><th>for-loop</th>
<th>time of counting</th>
</tr><tr><td>1st</td>
<td>O(max)</td>
</tr><tr><td>2nd</td>
<td>O(size)</td>
</tr><tr><td>3rd</td>
<td>O(max)</td>
</tr><tr><td>4th</td>
<td>O(size)</td>
</tr></tbody></table></div>
<p>Overall complexity = <code>O(max)+O(size)+O(max)+O(size)</code> = <code>O(max+size)</code></p>
<ul><li><strong>Worst Case Complexity:</strong> <code>O(n+k)</code></li>
<li><strong>Best Case Complexity:</strong> <code>O(n+k)</code></li>
<li><strong>Average Case Complexity:</strong> <code>O(n+k)</code></li>
</ul><p>In all the above cases, the complexity is the same because no matter how the elements are placed in the array, the algorithm goes through <code>n+k</code> times.</p>
<p></p><div class="clearfix"></div><p>There is no comparison between any elements, so it is better than comparison based sorting techniques. But, it is bad if the integers are very large because the array of that size should be made.</p>
<p><strong>Space Complexity</strong></p>
<p>The space complexity of Counting Sort is <code>O(max)</code>. Larger the range of elements, larger is the space complexity.</p>
<hr><h2 id="application">Counting Sort Applications</h2>
<p>Counting sort is used when:</p>
<ul><li>there are smaller integers with multiple counts.</li>
<li>linear complexity is the need.</li>
</ul><hr><h2>Similar Sorting Algorithms</h2>
<ol><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/bucket-sort.html">Bucket Sort</a></li>
<li><a href="/dsa/radix-sort.html">Radix Sort</a></li>
</ol></div>
</div>
<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents
<button class="btn btn--clear align-items-center">
<svg class="programiz-icon"><use xlink:href="/sites/all/themes/programiz/assets/feather-sprite.svg#x"></use></svg></button></h3><div class="tutorial-toc__links"><ul><li><a href="#definition">Definition</a></li>
<li><a href="#working">Working of Counting Sort</a></li>
<li><a href="#algorithm">Counting Sort Algorithm</a></li>
<li><a href="#code">Counting Sort Code in Python, Java, and C/C++</a></li>
<li><a href="#complexity">Complexity</a></li>
<li><a href="#application">Applications</a></li>
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