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<h1>Shell Sort Algorithm</h1>
<p class="editor-contents__short-description">In this tutorial, you will learn about the shell sort algorithm and its implementation in Python, Java, C, and C++.</p>
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<span property="dc:title" content="Shell Sort Algorithm" class="rdf-meta element-hidden"></span>
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<p id="definition">Shell sort is a generalized version of the <a href="/dsa/insertion-sort.html" title="Insertion Sort">insertion sort algorithm</a>. It first sorts elements that are far apart from each other and successively reduces the interval between the elements to be sorted.</p>
<p>The interval between the elements is reduced based on the sequence used. Some of the optimal sequences that can be used in the shell sort algorithm are:</p>
<ul><li><strong>Shell's original sequence</strong>: <code>N/2 , N/4 , …, 1</code></li>
<li><strong>Knuth's increments</strong>: <code>1, 4, 13, …, (3<sup>k</sup></code> <code>– 1) / 2</code></li>
<li><strong>Sedgewick's increments</strong>: <code>1, 8, 23, 77, 281, 1073, 4193, 16577...4j+1+ 3·2j+ 1</code></li>
<li><strong>Hibbard's increments</strong>: <code>1, 3, 7, 15, 31, 63, 127, 255, 511…</code></li>
<li><strong>Papernov & Stasevich increment</strong>: <code>1, 3, 5, 9, 17, 33, 65,...</code></li>
<li><strong>Pratt</strong>: <code>1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 16, 24, 36, 54, 81....</code></li>
</ul><p class="note-tip"><strong>Note:</strong> The performance of the shell sort depends on the type of sequence used for a given input array.</p>
<hr><h2 id="working">Working of Shell Sort</h2>
<ol><li>Suppose, we need to sort the following array.
<figure><img alt="Shell sort step" src="https://www.programiz.com/sites/tutorial2program/files/shell-sort-0.0.png" title="Input Array" width="500" height="85"><figcaption>Initial array</figcaption></figure></li>
<li>We are using the shell's original sequence <code>(N/2, N/4, ...1</code>) as intervals in our algorithm.<br><br>
In the first loop, if the array size is <code>N = 8</code> then, the elements lying at the interval of <code>N/2 = 4</code> are compared and swapped if they are not in order.
<ol type="a"><li>The 0th element is compared with the <var>4th</var> element.</li>
<li>If the 0th element is greater than the <var>4th</var> one then, the <var>4th</var> element is first stored in <code>temp</code> variable and the <code>0th</code> element (ie. greater element) is stored in the <code>4th</code> position and the element stored in <code>temp</code> is stored in the <code>0th</code> position.
<figure><img alt="Shell Sort step" src="https://www.programiz.com/sites/tutorial2program/files/shell-sort-0.1.png" title="First loop of Shell Sort" width="575" height="262"><figcaption>Rearrange the elements at n/2 interval</figcaption></figure><br>
This process goes on for all the remaining elements.
<figure><img alt="Shell Sort steps" src="https://www.programiz.com/sites/tutorial2program/files/shell-sort-0.2.png" title="First loop of Shell Sort" width="500" height="279"><figcaption>Rearrange all the elements at n/2 interval</figcaption></figure></li>
</ol></li>
<li>In the second loop, an interval of <code>N/4 = 8/4 = 2</code> is taken and again the elements lying at these intervals are sorted.
<figure><img alt="Shell Sort step" src="https://www.programiz.com/sites/tutorial2program/files/shell-sort-0.3.png" title="Second Loop" width="500" height="150"><figcaption>Rearrange the elements at n/4 interval</figcaption></figure><br>
You might get confused at this point.
<figure><img alt="Shell Sort step" src="https://www.programiz.com/sites/tutorial2program/files/shell-sort-0.4.png" title="Shell Sort steps" width="500" height="150"><figcaption>All the elements in the array lying at the current interval are compared.</figcaption></figure><br>
The elements at <var>4th</var> and <code>2nd</code> position are compared. The elements at <var>2nd</var> and <code>0th</code> position are also compared. All the elements in the array lying at the current interval are compared.</li>
<li>The same process goes on for remaining elements.
<figure><img alt="Shell Sort step" src="https://www.programiz.com/sites/tutorial2program/files/shell-sort-step_4.png" title="Shell Sort steps" width="500" height="279"><figcaption>Rearrange all the elements at n/4 interval</figcaption></figure></li>
<li>Finally, when the interval is <code>N/8 = 8/8 =1</code> then the array elements lying at the interval of 1 are sorted. The array is now completely sorted.
<figure><img alt="Shell Sort step" src="https://www.programiz.com/sites/tutorial2program/files/shell-sort-step_5.png" title="Shell Sort steps" width="500" height="603"><figcaption>Rearrange the elements at n/8 interval</figcaption></figure></li>
</ol><hr><h2 id="algorithm">Shell Sort Algorithm</h2>
<pre style="max-height: 600px;"><code class="dsa hljs cpp">shellSort(<span class="hljs-built_in">array</span>, size)
<span class="hljs-keyword">for</span> interval i <- size/<span class="hljs-number">2</span>n down to <span class="hljs-number">1</span>
<span class="hljs-keyword">for</span> each interval <span class="hljs-string">"i"</span> in <span class="hljs-built_in">array</span>
sort all the elements at interval <span class="hljs-string">"i"</span>
end shellSort</code></pre>
<hr><h2 id="code">Shell Sort Code in Python, Java, and C/C++</h2>
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<div id="java1" onclick="changejava()" class="tabbed-editor__node"><a href="#java-code">Java</a></div>
<div id="c1" onclick="changec()" class="tabbed-editor__node"><a href="#c-code">C</a></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="python hljs"><span class="hljs-comment"># Shell sort in python</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">shellSort</span><span class="hljs-params">(array, n)</span>:</span>
<span class="hljs-comment"># Rearrange elements at each n/2, n/4, n/8, ... intervals</span>
interval = n // <span class="hljs-number">2</span>
<span class="hljs-keyword">while</span> interval > <span class="hljs-number">0</span>:
<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(interval, n):
temp = array[i]
j = i
<span class="hljs-keyword">while</span> j >= interval <span class="hljs-keyword">and</span> array[j - interval] > temp:
array[j] = array[j - interval]
j -= interval
array[j] = temp
interval //= <span class="hljs-number">2</span>
data = [<span class="hljs-number">9</span>, <span class="hljs-number">8</span>, <span class="hljs-number">3</span>, <span class="hljs-number">7</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>, <span class="hljs-number">4</span>, <span class="hljs-number">1</span>]
size = len(data)
shellSort(data, size)
<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="code-editor__area" id="java-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="java hljs"><span class="hljs-comment">// Shell sort in Java programming</span>
<span class="hljs-keyword">import</span> java.util.Arrays;
<span class="hljs-comment">// Shell sort</span>
<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">ShellSort</span> </span>{
<span class="hljs-comment">// Rearrange elements at each n/2, n/4, n/8, ... intervals</span>
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">shellSort</span><span class="hljs-params">(<span class="hljs-keyword">int</span> array[], <span class="hljs-keyword">int</span> n)</span> </span>{
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> interval = n / <span class="hljs-number">2</span>; interval > <span class="hljs-number">0</span>; interval /= <span class="hljs-number">2</span>) {
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = interval; i < n; i += <span class="hljs-number">1</span>) {
<span class="hljs-keyword">int</span> temp = array[i];
<span class="hljs-keyword">int</span> j;
<span class="hljs-keyword">for</span> (j = i; j >= interval && array[j - interval] > temp; j -= interval) {
array[j] = array[j - interval];
}
array[j] = temp;
}
}
}
<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">8</span>, <span class="hljs-number">3</span>, <span class="hljs-number">7</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>, <span class="hljs-number">4</span>, <span class="hljs-number">1</span> };
<span class="hljs-keyword">int</span> size = data.length;
ShellSort ss = <span class="hljs-keyword">new</span> ShellSort();
ss.shellSort(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="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">// Shell 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-comment">// Shell sort</span>
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">shellSort</span><span class="hljs-params">(<span class="hljs-keyword">int</span> <span class="hljs-built_in">array</span>[], <span class="hljs-keyword">int</span> n)</span> </span>{
<span class="hljs-comment">// Rearrange elements at each n/2, n/4, n/8, ... intervals</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> interval = n / <span class="hljs-number">2</span>; interval > <span class="hljs-number">0</span>; interval /= <span class="hljs-number">2</span>) {
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = interval; i < n; i += <span class="hljs-number">1</span>) {
<span class="hljs-keyword">int</span> temp = <span class="hljs-built_in">array</span>[i];
<span class="hljs-keyword">int</span> j;
<span class="hljs-keyword">for</span> (j = i; j >= interval && <span class="hljs-built_in">array</span>[j - interval] > temp; j -= interval) {
<span class="hljs-built_in">array</span>[j] = <span class="hljs-built_in">array</span>[j - interval];
}
<span class="hljs-built_in">array</span>[j] = temp;
}
}
}
<span class="hljs-comment">// 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> data[] = {<span class="hljs-number">9</span>, <span class="hljs-number">8</span>, <span class="hljs-number">3</span>, <span class="hljs-number">7</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>, <span class="hljs-number">4</span>, <span class="hljs-number">1</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>]);
shellSort(data, size);
<span class="hljs-built_in">printf</span>(<span class="hljs-string">"Sorted array: \n"</span>);
printArray(data, size);
}</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">// Shell 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-comment">// Shell sort</span>
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">shellSort</span><span class="hljs-params">(<span class="hljs-keyword">int</span> <span class="hljs-built_in">array</span>[], <span class="hljs-keyword">int</span> n)</span> </span>{
<span class="hljs-comment">// Rearrange elements at each n/2, n/4, n/8, ... intervals</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> interval = n / <span class="hljs-number">2</span>; interval > <span class="hljs-number">0</span>; interval /= <span class="hljs-number">2</span>) {
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = interval; i < n; i += <span class="hljs-number">1</span>) {
<span class="hljs-keyword">int</span> temp = <span class="hljs-built_in">array</span>[i];
<span class="hljs-keyword">int</span> j;
<span class="hljs-keyword">for</span> (j = i; j >= interval && <span class="hljs-built_in">array</span>[j - interval] > temp; j -= interval) {
<span class="hljs-built_in">array</span>[j] = <span class="hljs-built_in">array</span>[j - interval];
}
<span class="hljs-built_in">array</span>[j] = temp;
}
}
}
<span class="hljs-comment">// 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">int</span> i;
<span class="hljs-keyword">for</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> data[] = {<span class="hljs-number">9</span>, <span class="hljs-number">8</span>, <span class="hljs-number">3</span>, <span class="hljs-number">7</span>, <span class="hljs-number">5</span>, <span class="hljs-number">6</span>, <span class="hljs-number">4</span>, <span class="hljs-number">1</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>]);
shellSort(data, size);
<span class="hljs-built_in">cout</span> << <span class="hljs-string">"Sorted array: \n"</span>;
printArray(data, size);
}</code></pre></div>
</div>
</div>
<hr><h2 id="complexity">Shell Sort 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(nlog n)</td>
</tr><tr><td>Worst</td>
<td>O(n<sup>2</sup>)</td>
</tr><tr><td>Average</td>
<td>O(nlog n)</td>
</tr><tr><th><strong>Space Complexity</strong></th>
<td>O(1)</td>
</tr><tr><th><strong>Stability</strong></th>
<td>No</td>
</tr></tbody></table></div>
<hr><p>Shell sort is an unstable sorting algorithm because this algorithm does not examine the elements lying in between the intervals.</p>
<h3>Time Complexity</h3>
<ul><li><strong>Worst Case Complexity</strong>: less than or equal to <code>O(n<sup>2</sup>)</code><br>
Worst case complexity for shell sort is always less than or equal to <code>O(n<sup>2</sup>)</code>.<br><br>
According to Poonen Theorem, worst case complexity for shell sort is <code>Θ(Nlog N)<sup>2</sup>/(log log N)<sup>2</sup>)</code> or <code>Θ(Nlog N)<sup>2</sup>/log log N)</code> or <code>Θ(N(log N)<sup>2</sup>)</code> or something in between.</li>
<li><strong>Best Case Complexity</strong>: <code>O(n*log n)</code><br>
When the array is already sorted, the total number of comparisons for each interval (or increment) is equal to the size of the array.</li>
<li><strong>Average Case Complexity</strong>: <code>O(n*log n)</code><br>
It is around <code>O(n<sup>1.25</sup>)</code>.</li>
</ul><p>The complexity depends on the interval chosen. The above complexities differ for different increment sequences chosen. Best increment sequence is unknown.</p>
<h3>Space Complexity:</h3>
<p></p><div class="clearfix"></div><p>The space complexity for shell sort is <code>O(1)</code>.</p>
<hr><h2 id="application">Shell Sort Applications</h2>
<p>Shell sort is used when:</p>
<ul><li>calling a stack is overhead. <code>uClibc</code> library uses this sort.</li>
<li>recursion exceeds a limit. <code>bzip2</code> compressor uses it.</li>
<li>Insertion sort does not perform well when the close elements are far apart. Shell sort helps in reducing the distance between the close elements. Thus, there will be less number of swappings to be performed.</li>
</ul><hr><h2>Other 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/insertion-sort.html">Insertion 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
<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 Shell Sort</a></li>
<li><a href="#algorithm">Shell Sort Algorithm</a></li>
<li><a href="#code">Shell 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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