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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Types_of_edit_distance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_edit_distance"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Types of edit distance</span> </div> </a> <ul id="toc-Types_of_edit_distance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_definition_and_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formal_definition_and_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Formal definition and properties</span> </div> </a> <button aria-controls="toc-Formal_definition_and_properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Formal definition and properties subsection</span> </button> <ul id="toc-Formal_definition_and_properties-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Computation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Computation</span> </div> </a> <button aria-controls="toc-Computation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Computation subsection</span> </button> <ul id="toc-Computation-sublist" class="vector-toc-list"> <li id="toc-Common_algorithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Common_algorithm"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Common algorithm</span> </div> </a> <ul id="toc-Common_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Improved_algorithms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Improved_algorithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Improved algorithms</span> </div> </a> <ul id="toc-Improved_algorithms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Language_edit_distance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Language_edit_distance"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Language edit distance</span> </div> </a> <ul id="toc-Language_edit_distance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" 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class="mw-page-title-main">Edit distance</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <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">Computer science metric of string similarity</div> <p>In <a href="/wiki/Computational_linguistics" title="Computational linguistics">computational linguistics</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a>, <b>edit distance</b> is a <a href="/wiki/String_metric" title="String metric">string metric</a>, i.e. a way of quantifying how dissimilar two <a href="/wiki/String_(computing)" class="mw-redirect" title="String (computing)">strings</a> (e.g., words) are to one another, that is measured by counting the minimum number of operations required to transform one string into the other. Edit distances find applications in <a href="/wiki/Natural_language_processing" title="Natural language processing">natural language processing</a>, where automatic <a href="/wiki/Spell_checker" title="Spell checker">spelling correction</a> can determine candidate corrections for a misspelled word by selecting words from a dictionary that have a low distance to the word in question. In <a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a>, it can be used to quantify the similarity of <a href="/wiki/DNA" title="DNA">DNA</a> sequences, which can be viewed as strings of the letters A, C, G and T. </p><p>Different definitions of an edit distance use different sets of like operations. <a href="/wiki/Levenshtein_distance" title="Levenshtein distance">Levenshtein distance</a> operations are the removal, insertion, or substitution of a character in the string. Being the most common metric, the term <i>Levenshtein distance</i> is often used interchangeably with <i>edit distance</i>.<sup id="cite_ref-navarnarutoro_1-0" class="reference"><a href="#cite_note-navarnarutoro-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Types_of_edit_distance">Types of edit distance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edit_distance&action=edit&section=1" title="Edit section: Types of edit distance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Different types of edit distance allow different sets of string operations. For instance: </p> <table class="wikitable"> <caption> </caption> <tbody><tr> <th>Algorithm </th> <th colspan="4">Operations Allowed </th></tr> <tr> <td> </td> <th>Insertions </th> <th>Deletions </th> <th>Substitutions </th> <th><a href="/wiki/Transposition_(mathematics)" class="mw-redirect" title="Transposition (mathematics)">Transposition</a> </th></tr> <tr> <td><a href="/wiki/Levenshtein_distance" title="Levenshtein distance">Levenshtein Distance</a> </td> <td>✓ </td> <td>✓ </td> <td>✓ </td> <td> </td></tr> <tr> <td><a href="/wiki/Longest_common_subsequence" title="Longest common subsequence">Longest Common Subsequence</a> (LCS) </td> <td>✓ </td> <td>✓ </td> <td> </td> <td> </td></tr> <tr> <td><a href="/wiki/Hamming_distance" title="Hamming distance">Hamming Distance</a> </td> <td> </td> <td> </td> <td>✓ </td> <td> </td></tr> <tr> <td><a href="/wiki/Damerau%E2%80%93Levenshtein_distance" title="Damerau–Levenshtein distance">Damerau–Levenshtein Distance</a> </td> <td>✓ </td> <td>✓ </td> <td>✓ </td> <td>✓ </td></tr> <tr> <td><a href="/wiki/Jaro_distance" class="mw-redirect" title="Jaro distance">Jaro distance</a> </td> <td> </td> <td> </td> <td> </td> <td>✓ </td></tr></tbody></table> <p>Some edit distances are defined as a parameterizable metric calculated with a specific set of allowed edit operations, and each operation is assigned a cost (possibly infinite). This is further generalized by DNA <a href="/wiki/Sequence_alignment" title="Sequence alignment">sequence alignment</a> algorithms such as the <a href="/wiki/Smith%E2%80%93Waterman_algorithm" title="Smith–Waterman algorithm">Smith–Waterman algorithm</a>, which make an operation's cost depend on where it is applied. </p> <div class="mw-heading mw-heading2"><h2 id="Formal_definition_and_properties">Formal definition and properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edit_distance&action=edit&section=2" title="Edit section: Formal definition and properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given two strings <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> on an alphabet <span class="texhtml">Σ</span> (e.g. the set of <a href="/wiki/ASCII" title="ASCII">ASCII</a> characters, the set of <a href="/wiki/Byte" title="Byte">bytes</a> [0..255], etc.), the edit distance <span class="nowrap">d(<span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>)</span> is the minimum-weight series of edit operations that transforms <span class="texhtml mvar" style="font-style:italic;">a</span> into <span class="texhtml mvar" style="font-style:italic;">b</span>. One of the simplest sets of edit operations is that defined by Levenshtein in 1966:<sup id="cite_ref-slp_2-0" class="reference"><a href="#cite_note-slp-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><b>Insertion</b> of a single symbol. If <span class="texhtml mvar" style="font-style:italic;">a</span> = <span class="texhtml mvar" style="font-style:italic;">u</span><span class="texhtml mvar" style="font-style:italic;">v</span>, then inserting the symbol <span class="texhtml mvar" style="font-style:italic;">x</span> produces <span class="texhtml mvar" style="font-style:italic;">u</span><span class="texhtml mvar" style="font-style:italic;">x</span><span class="texhtml mvar" style="font-style:italic;">v</span>. This can also be denoted ε→<span class="texhtml mvar" style="font-style:italic;">x</span>, using ε to denote the empty string.</dd> <dd><b>Deletion</b> of a single symbol changes <span class="texhtml mvar" style="font-style:italic;">u</span><span class="texhtml mvar" style="font-style:italic;">x</span><span class="texhtml mvar" style="font-style:italic;">v</span> to <span class="texhtml mvar" style="font-style:italic;">u</span><span class="texhtml mvar" style="font-style:italic;">v</span> (<span class="texhtml mvar" style="font-style:italic;">x</span>→ε).</dd> <dd><b>Substitution</b> of a single symbol <span class="texhtml mvar" style="font-style:italic;">x</span> for a symbol <span class="texhtml mvar" style="font-style:italic;">y</span> ≠ <span class="texhtml mvar" style="font-style:italic;">x</span> changes <span class="texhtml mvar" style="font-style:italic;">u</span><span class="texhtml mvar" style="font-style:italic;">x</span><span class="texhtml mvar" style="font-style:italic;">v</span> to <span class="texhtml mvar" style="font-style:italic;">u</span><span class="texhtml mvar" style="font-style:italic;">y</span><span class="texhtml mvar" style="font-style:italic;">v</span> (<span class="texhtml mvar" style="font-style:italic;">x</span>→<span class="texhtml mvar" style="font-style:italic;">y</span>).</dd></dl> <p>In Levenshtein's original definition, each of these operations has unit cost (except that substitution of a character by itself has zero cost), so the Levenshtein distance is equal to the minimum <i>number</i> of operations required to transform <span class="texhtml mvar" style="font-style:italic;">a</span> to <span class="texhtml mvar" style="font-style:italic;">b</span>. A more general definition associates non-negative weight functions <span class="texhtml mvar" style="font-style:italic;">w</span><sub>ins</sub>(<span class="texhtml mvar" style="font-style:italic;">x</span>), <span class="texhtml mvar" style="font-style:italic;">w</span><sub>del</sub>(<span class="texhtml mvar" style="font-style:italic;">x</span>) and <span class="texhtml mvar" style="font-style:italic;">w</span><sub>sub</sub>(<span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span>) with the operations.<sup id="cite_ref-slp_2-1" class="reference"><a href="#cite_note-slp-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Additional primitive operations have been suggested. <a href="/wiki/Damerau%E2%80%93Levenshtein_distance" title="Damerau–Levenshtein distance">Damerau–Levenshtein distance</a> counts as a single edit a common mistake: <b>transposition</b> of two adjacent characters, formally characterized by an operation that changes <span class="texhtml mvar" style="font-style:italic;">u</span><span class="texhtml mvar" style="font-style:italic;">x</span><span class="texhtml mvar" style="font-style:italic;">y</span><span class="texhtml mvar" style="font-style:italic;">v</span> into <span class="texhtml mvar" style="font-style:italic;">u</span><span class="texhtml mvar" style="font-style:italic;">y</span><span class="texhtml mvar" style="font-style:italic;">x</span><span class="texhtml mvar" style="font-style:italic;">v</span>.<sup id="cite_ref-ukkonen83_3-0" class="reference"><a href="#cite_note-ukkonen83-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ssm_4-0" class="reference"><a href="#cite_note-ssm-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> For the task of correcting <a href="/wiki/Optical_character_recognition" title="Optical character recognition">OCR</a> output, <b>merge</b> and <b>split</b> operations have been used which replace a single character into a pair of them or vice versa.<sup id="cite_ref-ssm_4-1" class="reference"><a href="#cite_note-ssm-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Other variants of edit distance are obtained by restricting the set of operations. <a href="/wiki/Longest_common_subsequence_problem" class="mw-redirect" title="Longest common subsequence problem">Longest common subsequence (LCS)</a> distance is edit distance with insertion and deletion as the only two edit operations, both at unit cost.<sup id="cite_ref-navarnarutoro_1-1" class="reference"><a href="#cite_note-navarnarutoro-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 37">: 37 </span></sup> Similarly, by only allowing substitutions (again at unit cost), <a href="/wiki/Hamming_distance" title="Hamming distance">Hamming distance</a> is obtained; this must be restricted to equal-length strings.<sup id="cite_ref-navarnarutoro_1-2" class="reference"><a href="#cite_note-navarnarutoro-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Jaro%E2%80%93Winkler_distance" title="Jaro–Winkler distance">Jaro–Winkler distance</a> can be obtained from an edit distance where only transpositions are allowed. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edit_distance&action=edit&section=3" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Levenshtein_distance" title="Levenshtein distance">Levenshtein distance</a> between "kitten" and "sitting" is 3. A minimal edit script that transforms the former into the latter is: </p> <ol><li><b>k</b>itten → <b>s</b>itten (substitute "s" for "k")</li> <li>sitt<b>e</b>n → sitt<b>i</b>n (substitute "i" for "e")</li> <li>sittin → sittin<b>g</b> (insert "g" at the end)</li></ol> <p>LCS distance (insertions and deletions only) gives a different distance and minimal edit script: </p> <ol><li><b>k</b>itten → itten (delete "k" at 0)</li> <li>itten → <b>s</b>itten (insert "s" at 0)</li> <li>sitt<b>e</b>n → sittn (delete "e" at 4)</li> <li>sittn → sitt<b>i</b>n (insert "i" at 4)</li> <li>sittin → sittin<b>g</b> (insert "g" at 6)</li></ol> <p>for a total cost/distance of 5 operations. </p> <div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edit_distance&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Edit distance with non-negative cost satisfies the axioms of a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>, giving rise to a <a href="/wiki/Metric_space" title="Metric space">metric space</a> of strings, when the following conditions are met:<sup id="cite_ref-navarnarutoro_1-3" class="reference"><a href="#cite_note-navarnarutoro-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 37">: 37 </span></sup> </p> <ul><li>Every edit operation has positive cost;</li> <li>for every operation, there is an inverse operation with equal cost.</li></ul> <p>With these properties, the metric axioms are satisfied as follows: </p> <dl><dd><span class="texhtml mvar" style="font-style:italic;">d</span>(<span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>) = 0 if and only if a=b, since each string can be trivially transformed to itself using exactly zero operations.</dd> <dd><span class="texhtml mvar" style="font-style:italic;">d</span>(<span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>) > 0 when <span class="texhtml mvar" style="font-style:italic;">a</span> ≠ <span class="texhtml mvar" style="font-style:italic;">b</span>, since this would require at least one operation at non-zero cost.</dd> <dd><span class="texhtml mvar" style="font-style:italic;">d</span>(<span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>) = <span class="texhtml mvar" style="font-style:italic;">d</span>(<span class="texhtml mvar" style="font-style:italic;">b</span>, <span class="texhtml mvar" style="font-style:italic;">a</span>) by equality of the cost of each operation and its inverse.</dd> <dd>Triangle inequality: <span class="texhtml mvar" style="font-style:italic;">d</span>(<span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">c</span>) ≤ <span class="texhtml mvar" style="font-style:italic;">d</span>(<span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>) + <span class="texhtml mvar" style="font-style:italic;">d</span>(<span class="texhtml mvar" style="font-style:italic;">b</span>, <span class="texhtml mvar" style="font-style:italic;">c</span>).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Levenshtein distance and LCS distance with unit cost satisfy the above conditions, and therefore the metric axioms. Variants of edit distance that are not proper metrics have also been considered in the literature.<sup id="cite_ref-navarnarutoro_1-4" class="reference"><a href="#cite_note-navarnarutoro-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Other useful properties of unit-cost edit distances include: </p> <ul><li>LCS distance is bounded above by the sum of lengths of a pair of strings.<sup id="cite_ref-navarnarutoro_1-5" class="reference"><a href="#cite_note-navarnarutoro-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 37">: 37 </span></sup></li> <li>LCS distance is an upper bound on Levenshtein distance.</li> <li>For strings of the same length, Hamming distance is an upper bound on Levenshtein distance.<sup id="cite_ref-navarnarutoro_1-6" class="reference"><a href="#cite_note-navarnarutoro-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul> <p>Regardless of cost/weights, the following property holds of all edit distances: </p> <ul><li>When <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> share a common prefix, this prefix has no effect on the distance. Formally, when <span class="texhtml mvar" style="font-style:italic;">a</span> = <span class="texhtml mvar" style="font-style:italic;">uv</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> = <span class="texhtml mvar" style="font-style:italic;">uw</span>, then <span class="texhtml mvar" style="font-style:italic;">d</span>(<span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>) = <span class="texhtml mvar" style="font-style:italic;">d</span>(<span class="texhtml mvar" style="font-style:italic;">v</span>, <span class="texhtml mvar" style="font-style:italic;">w</span>).<sup id="cite_ref-ssm_4-2" class="reference"><a href="#cite_note-ssm-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This allows speeding up many computations involving edit distance and edit scripts, since common prefixes and suffixes can be skipped in linear time.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Computation">Computation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edit_distance&action=edit&section=5" title="Edit section: Computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first algorithm for computing minimum edit distance between a pair of strings was published by <a href="/wiki/Frederick_J._Damerau" title="Frederick J. Damerau">Damerau</a> in 1964.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Common_algorithm">Common algorithm</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edit_distance&action=edit&section=6" title="Edit section: Common algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Wagner%E2%80%93Fischer_algorithm" title="Wagner–Fischer algorithm">Wagner–Fischer algorithm</a></div> <p>Using Levenshtein's original operations, the (nonsymmetric) edit distance 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 a=a_{1}\ldots a_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=a_{1}\ldots a_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa05f4c5868023d9fd5f9a2e83c435da612ec502" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.015ex; height:2.009ex;" alt="{\displaystyle a=a_{1}\ldots a_{m}}"></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 b=b_{1}\ldots b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=b_{1}\ldots b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3309d61d2ad02ea8293da4e166450fe0c449b449" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.861ex; height:2.509ex;" alt="{\displaystyle b=b_{1}\ldots b_{n}}"></span> is given by <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 d_{mn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{mn}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65dc535bfabf91fffb04553387ec98324c286c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.87ex; height:2.509ex;" alt="{\displaystyle d_{mn}}"></span>, defined by the <a href="/wiki/Recursive_definition" title="Recursive definition">recurrence</a><sup id="cite_ref-slp_2-2" class="reference"><a href="#cite_note-slp-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 {\begin{aligned}d_{i0}&=\sum _{k=1}^{i}w_{\mathrm {del} }(a_{k}),&&\quad {\text{for}}\;1\leq i\leq m\\d_{0j}&=\sum _{k=1}^{j}w_{\mathrm {ins} }(b_{k}),&&\quad {\text{for}}\;1\leq j\leq n\\d_{ij}&={\begin{cases}d_{i-1,j-1}&{\text{for}}\;a_{i}=b_{j}\\\min {\begin{cases}d_{i-1,j}+w_{\mathrm {del} }(a_{i})\\d_{i,j-1}+w_{\mathrm {ins} }(b_{j})\\d_{i-1,j-1}+w_{\mathrm {sub} }(a_{i},b_{j})\end{cases}}&{\text{for}}\;a_{i}\neq b_{j}\end{cases}}&&\quad {\text{for}}\;1\leq i\leq m,1\leq j\leq n.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" 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<mn>0</mn> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd /> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="thickmathspace" /> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="thickmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≠</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> <mtd /> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="thickmathspace" /> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>m</mi> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}d_{i0}&=\sum _{k=1}^{i}w_{\mathrm {del} }(a_{k}),&&\quad {\text{for}}\;1\leq i\leq m\\d_{0j}&=\sum _{k=1}^{j}w_{\mathrm {ins} }(b_{k}),&&\quad {\text{for}}\;1\leq j\leq n\\d_{ij}&={\begin{cases}d_{i-1,j-1}&{\text{for}}\;a_{i}=b_{j}\\\min {\begin{cases}d_{i-1,j}+w_{\mathrm {del} }(a_{i})\\d_{i,j-1}+w_{\mathrm {ins} }(b_{j})\\d_{i-1,j-1}+w_{\mathrm {sub} }(a_{i},b_{j})\end{cases}}&{\text{for}}\;a_{i}\neq b_{j}\end{cases}}&&\quad {\text{for}}\;1\leq i\leq m,1\leq j\leq n.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b7a79599eeadd727a2d8564abd34dd1a11a5c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:82.808ex; height:27.176ex;" alt="{\displaystyle {\begin{aligned}d_{i0}&=\sum _{k=1}^{i}w_{\mathrm {del} }(a_{k}),&&\quad {\text{for}}\;1\leq i\leq m\\d_{0j}&=\sum _{k=1}^{j}w_{\mathrm {ins} }(b_{k}),&&\quad {\text{for}}\;1\leq j\leq n\\d_{ij}&={\begin{cases}d_{i-1,j-1}&{\text{for}}\;a_{i}=b_{j}\\\min {\begin{cases}d_{i-1,j}+w_{\mathrm {del} }(a_{i})\\d_{i,j-1}+w_{\mathrm {ins} }(b_{j})\\d_{i-1,j-1}+w_{\mathrm {sub} }(a_{i},b_{j})\end{cases}}&{\text{for}}\;a_{i}\neq b_{j}\end{cases}}&&\quad {\text{for}}\;1\leq i\leq m,1\leq j\leq n.\end{aligned}}}"></span></dd></dl> <p>This algorithm can be generalized to handle transpositions by adding another term in the recursive clause's minimization.<sup id="cite_ref-ukkonen83_3-1" class="reference"><a href="#cite_note-ukkonen83-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The straightforward, <a href="/wiki/Recursion_(computer_science)" title="Recursion (computer science)">recursive</a> way of evaluating this recurrence takes <a href="/wiki/Exponential_time" class="mw-redirect" title="Exponential time">exponential time</a>. Therefore, it is usually computed using a <a href="/wiki/Dynamic_programming" title="Dynamic programming">dynamic programming</a> algorithm that is commonly credited to <a href="/wiki/Wagner%E2%80%93Fischer_algorithm" title="Wagner–Fischer algorithm">Wagner and Fischer</a>,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> although it has a history of multiple invention.<sup id="cite_ref-slp_2-3" class="reference"><a href="#cite_note-slp-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ukkonen83_3-2" class="reference"><a href="#cite_note-ukkonen83-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> After completion of the Wagner–Fischer algorithm, a minimal sequence of edit operations can be read off as a backtrace of the operations used during the dynamic programming algorithm starting at <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 d_{mn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{mn}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65dc535bfabf91fffb04553387ec98324c286c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.87ex; height:2.509ex;" alt="{\displaystyle d_{mn}}"></span>. </p><p>This algorithm has a <a href="/wiki/Time_complexity" title="Time complexity">time complexity</a> of Θ(<span class="texhtml mvar" style="font-style:italic;">m</span><span class="texhtml mvar" style="font-style:italic;">n</span>) where <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> are the lengths of the strings. When the full dynamic programming table is constructed, its <a href="/wiki/Space_complexity" title="Space complexity">space complexity</a> is also <span class="nowrap">Θ(<span class="texhtml mvar" style="font-style:italic;">m</span><span class="texhtml mvar" style="font-style:italic;">n</span>)</span>; this can be improved to <span class="nowrap">Θ(min(<span class="texhtml mvar" style="font-style:italic;">m</span>,<span class="texhtml mvar" style="font-style:italic;">n</span>))</span> by observing that at any instant, the algorithm only requires two rows (or two columns) in memory. However, this optimization makes it impossible to read off the minimal series of edit operations.<sup id="cite_ref-ukkonen83_3-3" class="reference"><a href="#cite_note-ukkonen83-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> A linear-space solution to this problem is offered by <a href="/wiki/Hirschberg%27s_algorithm" title="Hirschberg's algorithm">Hirschberg's algorithm</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 634">: 634 </span></sup> A general recursive divide-and-conquer framework for solving such recurrences and extracting an optimal sequence of operations cache-efficiently in space linear in the size of the input is given by Chowdhury, Le, and Ramachandran.<sup id="cite_ref-CLR-08_9-0" class="reference"><a href="#cite_note-CLR-08-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Improved_algorithms">Improved algorithms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edit_distance&action=edit&section=7" title="Edit section: Improved algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Improving on the Wagner–Fisher algorithm described above, <a href="/wiki/Esko_Ukkonen" title="Esko Ukkonen">Ukkonen</a> describes several variants,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> one of which takes two strings and a maximum edit distance <span class="texhtml mvar" style="font-style:italic;">s</span>, and returns <span class="nowrap">min(<span class="texhtml mvar" style="font-style:italic;">s</span>, <span class="texhtml mvar" style="font-style:italic;">d</span>)</span>. It achieves this by only computing and storing a part of the dynamic programming table around its diagonal. This algorithm takes time <span class="nowrap">O(<span class="texhtml mvar" style="font-style:italic;">s</span>×min(<span class="texhtml mvar" style="font-style:italic;">m</span>,<span class="texhtml mvar" style="font-style:italic;">n</span>))</span>, where <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> are the lengths of the strings. Space complexity is <span class="nowrap">O(<span class="texhtml mvar" style="font-style:italic;">s</span><sup>2</sup>)</span> or <span class="nowrap">O(<span class="texhtml mvar" style="font-style:italic;">s</span>)</span>, depending on whether the edit sequence needs to be read off.<sup id="cite_ref-ukkonen83_3-4" class="reference"><a href="#cite_note-ukkonen83-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Further improvements by <a href="/wiki/Gad_Landau" title="Gad Landau">Landau</a>, <a href="/wiki/Eugene_Myers" title="Eugene Myers">Myers</a>, and Schmidt <a rel="nofollow" class="external autonumber" href="https://dblp.org/pers/hd/s/Schmidt:Jeanette_P=">[1]</a> give an <span class="nowrap">O(<span class="texhtml mvar" style="font-style:italic;">s</span><sup>2</sup> + max(<span class="texhtml mvar" style="font-style:italic;">m</span>,<span class="texhtml mvar" style="font-style:italic;">n</span>))</span> time algorithm.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>For a finite alphabet and edit costs which are multiples of each other, the fastest known exact algorithm is of Masek and Paterson<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> having worst case runtime of O(nm/logn). </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edit_distance&action=edit&section=8" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Edit distance finds applications in <a href="/wiki/Computational_biology" title="Computational biology">computational biology</a> and natural language processing, e.g. the correction of spelling mistakes or OCR errors, and <a href="/wiki/Approximate_string_matching" title="Approximate string matching">approximate string matching</a>, where the objective is to find matches for short strings in many longer texts, in situations where a small number of differences is to be expected. </p><p>Various algorithms exist that solve problems beside the computation of distance between a pair of strings, to solve related types of problems. </p> <ul><li><a href="/wiki/Hirschberg%27s_algorithm" title="Hirschberg's algorithm">Hirschberg's algorithm</a> computes the optimal <a href="/wiki/Sequence_alignment" title="Sequence alignment">alignment</a> of two strings, where optimality is defined as minimizing edit distance.</li> <li><a href="/wiki/Approximate_string_matching" title="Approximate string matching">Approximate string matching</a> can be formulated in terms of edit distance. Ukkonen's 1985 algorithm takes a string <span class="texhtml mvar" style="font-style:italic;">p</span>, called the pattern, and a constant <span class="texhtml mvar" style="font-style:italic;">k</span>; it then builds a <a href="/wiki/Deterministic_finite_state_automaton" class="mw-redirect" title="Deterministic finite state automaton">deterministic finite state automaton</a> that finds, in an arbitrary string <span class="texhtml mvar" style="font-style:italic;">s</span>, a substring whose edit distance to <span class="texhtml mvar" style="font-style:italic;">p</span> is at most <span class="texhtml mvar" style="font-style:italic;">k</span><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> (cf. the <a href="/wiki/Aho%E2%80%93Corasick_string_matching_algorithm" class="mw-redirect" title="Aho–Corasick string matching algorithm">Aho–Corasick algorithm</a>, which similarly constructs an automaton to search for any of a number of patterns, but without allowing edit operations). A similar algorithm for approximate string matching is the <a href="/wiki/Bitap_algorithm" title="Bitap algorithm">bitap algorithm</a>, also defined in terms of edit distance.</li> <li><a href="/wiki/Levenshtein_automaton" title="Levenshtein automaton">Levenshtein automata</a> are finite-state machines that recognize a set of strings within bounded edit distance of a fixed reference string.<sup id="cite_ref-ssm_4-3" class="reference"><a href="#cite_note-ssm-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Language_edit_distance">Language edit distance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edit_distance&action=edit&section=9" title="Edit section: Language edit distance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A generalization of the edit distance between strings is the language edit distance between a string and a language, usually a <a href="/wiki/Formal_language" title="Formal language">formal language</a>. Instead of considering the edit distance between one string and another, the language edit distance is the minimum edit distance that can be attained between a fixed string and <i>any</i> string taken from a set of strings. More formally, for any language <i>L</i> and string <i>x</i> over an alphabet <span class="texhtml">Σ</span>, the <i>language edit distance</i> d(<i>L</i>, <i>x</i>) is given by<sup id="cite_ref-:0_14-0" class="reference"><a href="#cite_note-:0-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><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 d(L,x)=\min _{y\in L}d(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <mi>L</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(L,x)=\min _{y\in L}d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5563abfbb022d4904e52c0c0762cb461b1e039b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.877ex; height:4.343ex;" alt="{\displaystyle d(L,x)=\min _{y\in L}d(x,y)}"></span>, where <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 d(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3772957879a8bbf7946bddf5743c508a1d5072c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.544ex; height:2.843ex;" alt="{\displaystyle d(x,y)}"></span> is the string edit distance. When the language <i>L</i> is <a href="/wiki/Context-free_language" title="Context-free language">context free</a>, there is a cubic time dynamic programming algorithm proposed by Aho and Peterson in 1972 which computes the language edit distance.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> For less expressive families of grammars, such as the <a href="/wiki/Regular_grammar" title="Regular grammar">regular grammars</a>, faster algorithms exist for computing the edit distance.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Language edit distance has found many diverse applications, such as RNA folding, error correction, and solutions to the Optimum Stack Generation problem.<sup id="cite_ref-:0_14-1" class="reference"><a href="#cite_note-:0-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edit_distance&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Graph_edit_distance" title="Graph edit distance">Graph edit distance</a></li> <li><a href="/wiki/String-to-string_correction_problem" title="String-to-string correction problem">String-to-string correction problem</a></li> <li><a href="/wiki/String_metric" title="String metric">String metric</a></li> <li><a href="/wiki/Time_Warp_Edit_Distance" title="Time Warp Edit Distance">Time Warp Edit Distance</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edit_distance&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-navarnarutoro-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-navarnarutoro_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-navarnarutoro_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-navarnarutoro_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-navarnarutoro_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-navarnarutoro_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-navarnarutoro_1-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-navarnarutoro_1-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.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.id-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.id-lock-limited a,.mw-parser-output .id-lock-registration.id-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.id-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 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