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<div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div>
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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algorithm for the kth smallest element in an array</div>
<style data-mw-deduplicate="TemplateStyles:r1097763485">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Quickselect" title="Special:EditPage/Quickselect">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i>&#160;<a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&amp;q=%22Quickselect%22">"Quickselect"</a>&#160;–&#160;<a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&amp;q=%22Quickselect%22+-wikipedia&amp;tbs=ar:1">news</a>&#160;<b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&amp;q=%22Quickselect%22&amp;tbs=bkt:s&amp;tbm=bks">newspapers</a>&#160;<b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&amp;q=%22Quickselect%22+-wikipedia">books</a>&#160;<b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Quickselect%22">scholar</a>&#160;<b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Quickselect%22&amp;acc=on&amp;wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2013</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove
<style data-mw-deduplicate="TemplateStyles:r1066479718">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}body.skin-minerva .mw-parser-output .infobox-header,body.skin-minerva .mw-parser-output .infobox-subheader,body.skin-minerva .mw-parser-output .infobox-above,body.skin-minerva .mw-parser-output .infobox-title,body.skin-minerva .mw-parser-output .infobox-image,body.skin-minerva .mw-parser-output .infobox-full-data,body.skin-minerva .mw-parser-output .infobox-below{text-align:center}</style><table class="infobox"><caption class="infobox-title">Quickselect</caption><tbody><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Selecting_quickselect_frames.gif" class="mw-file-description" title="Animated visualization of the quickselect algorithm. Selecting the 22st smallest value."><img alt="Animated visualization of the quickselect algorithm. Selecting the 22st smallest value." src="//upload.wikimedia.org/wikipedia/commons/0/04/Selecting_quickselect_frames.gif" decoding="async" width="280" height="214" class="mw-file-element" data-file-width="280" data-file-height="214" /></a></span><div class="infobox-caption">Animated visualization of the quickselect algorithm. Selecting the 22nd smallest value.</div></td></tr><tr><th scope="row" class="infobox-label">Class</th><td class="infobox-data"><a href="/wiki/Selection_algorithm" title="Selection algorithm">Selection algorithm</a></td></tr><tr><th scope="row" class="infobox-label">Data structure</th><td class="infobox-data"><a href="/wiki/Array_data_structure" class="mw-redirect" title="Array data structure">Array</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Best,_worst_and_average_case" title="Best, worst and average case">Worst-case</a> <a href="/wiki/Time_complexity" title="Time complexity">performance</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="O"></span>(<i>n</i><sup>2</sup>)</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Best,_worst_and_average_case" title="Best, worst and average case">Best-case</a> <a href="/wiki/Time_complexity" title="Time complexity">performance</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="O"></span>(<i>n</i>)</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Best,_worst_and_average_case" title="Best, worst and average case">Average</a> <a href="/wiki/Time_complexity" title="Time complexity">performance</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="O"></span>(<i>n</i>)</td></tr><tr><th scope="row" class="infobox-label">Optimal</th><td class="infobox-data">Yes</td></tr></tbody></table>
<p>In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, <b>quickselect</b> is a <a href="/wiki/Selection_algorithm" title="Selection algorithm">selection algorithm</a> to find the <i>k</i>th smallest element in an unordered list, also known as the <i>k</i>th <a href="/wiki/Order_statistics" class="mw-redirect" title="Order statistics">order statistic</a>. Like the related <a href="/wiki/Quicksort" title="Quicksort">quicksort</a> sorting algorithm, it was developed by <a href="/wiki/Tony_Hoare" title="Tony Hoare">Tony Hoare</a>, and thus is also known as <b>Hoare's selection algorithm</b>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">&#91;1&#93;</a></sup> Like quicksort, it is efficient in practice and has good average-case performance, but has poor worst-case performance. Quickselect and its variants are the selection algorithms most often used in efficient real-world implementations.
</p><p>Quickselect uses the same overall approach as quicksort, choosing one element as a pivot and partitioning the data in two based on the pivot, accordingly as less than or greater than the pivot. However, instead of recursing into both sides, as in quicksort, quickselect only recurses into one side – the side with the element it is searching for. This reduces the average complexity from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n\log n)}">
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<annotation encoding="application/x-tex">{\displaystyle O(n\log n)}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2320768fb54880ca4356e61f60eb02a3f9d9f1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.118ex; height:2.843ex;" alt="O(n\log n)"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}">
<semantics>
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<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="O(n)"></span>, with a worst case of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
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<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle O(n^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd9594a16cb898b8f2a2dff9227a385ec183392" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="O(n^{2})"></span>.
</p><p>As with quicksort, quickselect is generally implemented as an <a href="/wiki/In-place_algorithm" title="In-place algorithm">in-place algorithm</a>, and beyond selecting the <span class="texhtml mvar" style="font-style:italic;">k</span>th element, it also partially sorts the data. See <a href="/wiki/Selection_algorithm" title="Selection algorithm">selection algorithm</a> for further discussion of the connection with sorting.
</p>
<meta property="mw:PageProp/toc" />
<h2><span class="mw-headline" id="Algorithm">Algorithm</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quickselect&amp;action=edit&amp;section=1" title="Edit section: Algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<p>In quicksort, there is a subprocedure called <code>partition</code> that can, in linear time, group a list (ranging from indices <code>left</code> to <code>right</code>) into two parts: those less than a certain element, and those greater than or equal to the element. Here is pseudocode that performs a partition about the element <code>list[pivotIndex]</code>:
</p>
<pre><b>function</b> partition(list, left, right, pivotIndex) <b>is</b>
pivotValue&#160;:= list[pivotIndex]
swap list[pivotIndex] and list[right] <i>// Move pivot to end</i>
storeIndex&#160;:= left
<b>for</b> i <b>from</b> left <b>to</b> right − 1 <b>do</b>
<b>if</b> list[i] &lt; pivotValue <b>then</b>
swap list[storeIndex] and list[i]
increment storeIndex
swap list[right] and list[storeIndex] <i>// Move pivot to its final place</i>
<b>return</b> storeIndex
</pre>
<p>This is known as the <a href="/wiki/Quicksort#Lomuto_partition_scheme" title="Quicksort">Lomuto partition scheme</a>, which is simpler but less efficient than <a href="/wiki/Quicksort#Hoare_partition_scheme" title="Quicksort">Hoare's original partition scheme</a>.
</p><p>In quicksort, we recursively sort both branches, leading to best-case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n\log n)}">
<semantics>
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<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
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<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle O(n\log n)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2320768fb54880ca4356e61f60eb02a3f9d9f1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.118ex; height:2.843ex;" alt="O(n\log n)"></span> time. However, when doing selection, we already know which partition our desired element lies in, since the pivot is in its final sorted position, with all those preceding it in an unsorted order and all those following it in an unsorted order. Therefore, a single recursive call locates the desired element in the correct partition, and we build upon this for quickselect:
</p>
<pre><i>// Returns the k-th smallest element of list within left..right inclusive</i>
<i>// (i.e. left &lt;= k &lt;= right).</i>
<b>function</b> select(list, left, right, k) <b>is</b>
<b>if</b> left = right <b>then</b> <i>// If the list contains only one element,</i>
<b>return</b> list[left] <i>// return that element</i>
pivotIndex &#160;:= ... <i>// select a pivotIndex between left and right,</i>
<i>// e.g.,</i> left + floor(rand()&#160;% (right − left + 1))
pivotIndex &#160;:= partition(list, left, right, pivotIndex)
<i>// The pivot is in its final sorted position</i>
<b>if</b> k = pivotIndex <b>then</b>
<b>return</b> list[k]
<b>else if</b> k &lt; pivotIndex <b>then</b>
<b>return</b> select(list, left, pivotIndex − 1, k)
<b>else</b>
<b>return</b> select(list, pivotIndex + 1, right, k)
</pre>
<hr /><p>Note the resemblance to quicksort: just as the minimum-based selection algorithm is a partial selection sort, this is a partial quicksort, generating and partitioning only <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\log n)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>log</mi>
<mo>&#x2061;<!-- ⁡ --></mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle O(\log n)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae0f22048ba6b7c05dbae17b056bfa16e21807d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="O(\log n)"></span> of its <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>O</mi>
<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="O(n)"></span> partitions. This simple procedure has expected linear performance, and, like quicksort, has quite good performance in practice. It is also an <a href="/wiki/In-place_algorithm" title="In-place algorithm">in-place algorithm</a>, requiring only constant memory overhead if <a href="/wiki/Tail_call" title="Tail call">tail call</a> optimization is available, or if eliminating the <a href="/wiki/Tail_recursion" class="mw-redirect" title="Tail recursion">tail recursion</a> with a loop:
</p><pre><b>function</b> select(list, left, right, k) <b>is</b>
<b>loop</b>
<b>if</b> left = right <b>then</b>
<b>return</b> list[left]
pivotIndex&#160;:= ... <i>// select pivotIndex between left and right</i>
pivotIndex&#160;:= partition(list, left, right, pivotIndex)
<b>if</b> k = pivotIndex <b>then</b>
<b>return</b> list[k]
<b>else if</b> k &lt; pivotIndex <b>then</b>
right&#160;:= pivotIndex − 1
<b>else</b>
left&#160;:= pivotIndex + 1
</pre>
<h2><span class="mw-headline" id="Time_complexity">Time complexity</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quickselect&amp;action=edit&amp;section=2" title="Edit section: Time complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<p>Like quicksort, quickselect has good average performance, but is sensitive to the pivot that is chosen. If good pivots are chosen, meaning ones that consistently decrease the search set by a given fraction, then the search set decreases in size exponentially and by induction (or summing the <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a>) one sees that performance is linear, as each step is linear and the overall time is a constant times this (depending on how quickly the search set reduces). However, if bad pivots are consistently chosen, such as decreasing by only a single element each time, then worst-case performance is quadratic: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{2}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle O(n^{2}).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e16485faf5f5237ac4fdf56e76ac7515282114" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.678ex; height:3.176ex;" alt="{\displaystyle O(n^{2}).}"></span> This occurs for example in searching for the maximum element of a set, using the first element as the pivot, and having sorted data. However, for randomly chosen pivots, this worst case is very unlikely: the probability of using more than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Cn}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mi>n</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle Cn}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eb08676206844b06ee1ddd827013317bd52e950" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.161ex; height:2.176ex;" alt="{\displaystyle Cn}"></span> comparisons, for any sufficiently large constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>C</mi>
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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="C"></span>, is superexponentially small as a function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="C"></span>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">&#91;2&#93;</a></sup>
</p>
<h2><span class="mw-headline" id="Variants">Variants</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quickselect&amp;action=edit&amp;section=3" title="Edit section: Variants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<p>The easiest solution is to choose a random pivot, which yields <a href="/wiki/Almost_certain" class="mw-redirect" title="Almost certain">almost certain</a> linear time. Deterministically, one can use median-of-3 pivot strategy (as in the quicksort), which yields linear performance on partially sorted data, as is common in the real world. However, contrived sequences can still cause worst-case complexity; <a href="/wiki/David_Musser" title="David Musser">David Musser</a> describes a "median-of-3 killer" sequence that allows an attack against that strategy, which was one motivation for his <a href="/wiki/Introselect" title="Introselect">introselect</a> algorithm.
</p><p>One can assure linear performance even in the worst case by using a more sophisticated pivot strategy; this is done in the <a href="/wiki/Median_of_medians" title="Median of medians">median of medians</a> algorithm. However, the overhead of computing the pivot is high, and thus this is generally not used in practice. One can combine basic quickselect with median of medians as fallback to get both fast average case performance and linear worst-case performance; this is done in <a href="/wiki/Introselect" title="Introselect">introselect</a>.
</p><p>Finer computations of the average time complexity yield a worst case of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(2+2\log 2+o(1))\leq 3.4n+o(n)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo stretchy="false">(</mo>
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<mn>2</mn>
<mi>log</mi>
<mo>&#x2061;<!-- ⁡ --></mo>
<mn>2</mn>
<mo>+</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
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<mi>n</mi>
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<mi>n</mi>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle n(2+2\log 2+o(1))\leq 3.4n+o(n)}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a08267fadce2d19b80e2d2fe4ef8a3a541c1216a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.854ex; height:2.843ex;" alt="n(2+2\log 2+o(1))\leq 3.4n+o(n)"></span> for random pivots (in the case of the median; other <i>k</i> are faster).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3">&#91;3&#93;</a></sup> The constant can be improved to 3/2 by a more complicated pivot strategy, yielding the <a href="/wiki/Floyd%E2%80%93Rivest_algorithm" title="Floyd–Rivest algorithm">Floyd–Rivest algorithm</a>, which has average complexity of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.5n+O(n^{1/2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1.5</mn>
<mi>n</mi>
<mo>+</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
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<mn>2</mn>
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</msup>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle 1.5n+O(n^{1/2})}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45eddb6b735b6f51499d879f5b00fcfe40e0c484" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.882ex; height:3.343ex;" alt="1.5n+O(n^{1/2})"></span> for median, with other <i>k</i> being faster.
</p>
<h2><span class="mw-headline" id="See_also">See also</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quickselect&amp;action=edit&amp;section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li><a href="/wiki/Floyd%E2%80%93Rivest_algorithm" title="Floyd–Rivest algorithm">Floyd–Rivest algorithm</a></li>
<li><a href="/wiki/Introselect" title="Introselect">Introselect</a></li>
<li><a href="/wiki/Median_of_medians" title="Median of medians">Median of medians</a></li></ul>
<h2><span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quickselect&amp;action=edit&amp;section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<style data-mw-deduplicate="TemplateStyles:r1011085734">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1133582631">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFHoare1961" class="citation journal cs1"><a href="/wiki/Tony_Hoare" title="Tony Hoare">Hoare, C. A. R.</a> (1961). "Algorithm 65: Find". <i><a href="/wiki/Communications_of_the_ACM" title="Communications of the ACM">Comm. ACM</a></i>. <b>4</b> (7): 321–322. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F366622.366647">10.1145/366622.366647</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Comm.+ACM&amp;rft.atitle=Algorithm+65%3A+Find&amp;rft.volume=4&amp;rft.issue=7&amp;rft.pages=321-322&amp;rft.date=1961&amp;rft_id=info%3Adoi%2F10.1145%2F366622.366647&amp;rft.aulast=Hoare&amp;rft.aufirst=C.+A.+R.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuickselect" class="Z3988"></span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFDevroye1984" class="citation journal cs1">Devroye, Luc (1984). <a rel="nofollow" class="external text" href="http://luc.devroye.org/devroye-selection1984.pdf">"Exponential bounds for the running time of a selection algorithm"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Computer and System Sciences</i>. <b>29</b> (1): 1–7. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-0000%2884%2990009-6">10.1016/0022-0000(84)90009-6</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a>&#160;<a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0761047">0761047</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Journal+of+Computer+and+System+Sciences&amp;rft.atitle=Exponential+bounds+for+the+running+time+of+a+selection+algorithm&amp;rft.volume=29&amp;rft.issue=1&amp;rft.pages=1-7&amp;rft.date=1984&amp;rft_id=info%3Adoi%2F10.1016%2F0022-0000%2884%2990009-6&amp;rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D761047%23id-name%3DMR&amp;rft.aulast=Devroye&amp;rft.aufirst=Luc&amp;rft_id=http%3A%2F%2Fluc.devroye.org%2Fdevroye-selection1984.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuickselect" class="Z3988"></span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFDevroye2001" class="citation journal cs1">Devroye, Luc (2001). <a rel="nofollow" class="external text" href="https://luc.devroye.org/wcfind.pdf">"On the probabilistic worst-case time of 'find'<span class="cs1-kern-right"></span>"</a> <span class="cs1-format">(PDF)</span>. <i>Algorithmica</i>. <b>31</b> (3): 291–303. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00453-001-0046-2">10.1007/s00453-001-0046-2</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a>&#160;<a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1855252">1855252</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Algorithmica&amp;rft.atitle=On+the+probabilistic+worst-case+time+of+%27find%27&amp;rft.volume=31&amp;rft.issue=3&amp;rft.pages=291-303&amp;rft.date=2001&amp;rft_id=info%3Adoi%2F10.1007%2Fs00453-001-0046-2&amp;rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1855252%23id-name%3DMR&amp;rft.aulast=Devroye&amp;rft.aufirst=Luc&amp;rft_id=https%3A%2F%2Fluc.devroye.org%2Fwcfind.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuickselect" class="Z3988"></span></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://11011110.github.io/blog/2007/10/09/blum-style-analysis-of.html">Blum-style analysis of Quickselect</a>, <a href="/wiki/David_Eppstein" title="David Eppstein">David Eppstein</a>, October 9, 2007.</span>
</li>
</ol></div></div>
<h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quickselect&amp;action=edit&amp;section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li>"<a rel="nofollow" class="external text" href="https://www.mathworks.com/matlabcentral/fileexchange/68947-qselect">qselect</a>", <i>Quickselect algorithm in Matlab,</i> Manolis Lourakis</li></ul>
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