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<title>Floyd–Warshall algorithm</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Floyd–Warshall algorithm</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Floyd's algorithm" redirects here. For cycle detection, see <a href="Floyd's_cycle-finding_algorithm" class="mw-redirect" title="Floyd's cycle-finding algorithm">Floyd's cycle-finding algorithm</a>. For computer graphics, see <a href="Floyd%E2%80%93Steinberg_dithering" title="Floyd–Steinberg dithering">Floyd–Steinberg dithering</a>.</div>

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</style><table class="infobox"><caption class="infobox-title">Floyd–Warshall algorithm</caption><tbody><tr><th scope="row" class="infobox-label">Class</th><td class="infobox-data"><a href="All-pairs_shortest_path_problem" class="mw-redirect" title="All-pairs shortest path problem">All-pairs shortest path problem</a> (for weighted graphs)</td></tr><tr><th scope="row" class="infobox-label">Data structure</th><td class="infobox-data"><a href="Graph_(data_structure)" class="mw-redirect" title="Graph (data structure)">Graph</a></td></tr><tr><th scope="row" class="infobox-label"><a href="Best%2C_worst_and_average_case" title="Best, worst and average case">Worst-case</a> <a href="Time_complexity" title="Time complexity">performance</a></th><td class="infobox-data"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|V|^{3})}">

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<annotation encoding="application/x-tex">{\displaystyle \Theta (|V|^{3})}</annotation>

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</math></span><img src="./b89800bb4f210f469a91b23b045aa481e5fd176e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.753ex; height:3.343ex;" alt="{\displaystyle \Theta (|V|^{3})}" loading="lazy"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="Best%2C_worst_and_average_case" title="Best, worst and average case">Best-case</a> <a href="Time_complexity" title="Time complexity">performance</a></th><td class="infobox-data"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|V|^{3})}">

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<annotation encoding="application/x-tex">{\displaystyle \Theta (|V|^{3})}</annotation>

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</math></span><img src="./b89800bb4f210f469a91b23b045aa481e5fd176e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.753ex; height:3.343ex;" alt="{\displaystyle \Theta (|V|^{3})}" loading="lazy"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="Best%2C_worst_and_average_case" title="Best, worst and average case">Average</a> <a href="Time_complexity" title="Time complexity">performance</a></th><td class="infobox-data"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|V|^{3})}">

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<annotation encoding="application/x-tex">{\displaystyle \Theta (|V|^{3})}</annotation>

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</math></span><img src="./b89800bb4f210f469a91b23b045aa481e5fd176e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.753ex; height:3.343ex;" alt="{\displaystyle \Theta (|V|^{3})}" loading="lazy"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="Best%2C_worst_and_average_case" title="Best, worst and average case">Worst-case</a> <a href="Space_complexity" title="Space complexity">space complexity</a></th><td class="infobox-data"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|V|^{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \Theta (|V|^{2})}</annotation>

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</math></span><img src="./5dbb38dcf2365c1dbf786871f4014dd1a2762df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.753ex; height:3.343ex;" alt="{\displaystyle \Theta (|V|^{2})}" loading="lazy"></span></td></tr></tbody></table>

<p>In <a href="Computer_science" title="Computer science">computer science</a>, the <b>Floyd–Warshall algorithm</b> (also known as <b>Floyd's algorithm</b>, the <b>Roy–Warshall algorithm</b>, the <b>Roy–Floyd algorithm</b>, or the <b>WFI algorithm</b>) is an <a href="Algorithm" title="Algorithm">algorithm</a> for finding <a href="Shortest_path_problem" title="Shortest path problem">shortest paths</a> in a directed <a href="Weighted_graph" class="mw-redirect" title="Weighted graph">weighted graph</a> with positive or negative edge weights (but with no negative cycles).<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A single execution of the algorithm will find the lengths (summed weights) of shortest paths between all pairs of vertices. Although it does not return details of the paths themselves, it is possible to reconstruct the paths with simple modifications to the algorithm. Versions of the algorithm can also be used for finding the <a href="Transitive_closure" title="Transitive closure">transitive closure</a> of a relation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>, or (in connection with the <a href="Schulze_method" title="Schulze method">Schulze voting system</a>) <a href="Widest_path_problem" title="Widest path problem">widest paths</a> between all pairs of vertices in a weighted graph.

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<div class="mw-heading mw-heading2"><h2 id="History_and_naming">History and naming</h2></div>

<p>The Floyd–Warshall algorithm is an example of <a href="Dynamic_programming" title="Dynamic programming">dynamic programming</a>, and was published in its currently recognized form by <a href="Robert_W._Floyd" title="Robert W. Floyd">Robert Floyd</a> in 1962.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> However, it is essentially the same as algorithms previously published by <a href="Bernard_Roy" title="Bernard Roy">Bernard Roy</a> in 1959<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and also by <a href="Stephen_Warshall" title="Stephen Warshall">Stephen Warshall</a> in 1962<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> for finding the transitive closure of a graph,<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> and is closely related to <a href="Kleene's_algorithm" title="Kleene's algorithm">Kleene's algorithm</a> (published in 1956) for converting a <a href="Deterministic_finite_automaton" title="Deterministic finite automaton">deterministic finite automaton</a> into a <a href="Regular_expression" title="Regular expression">regular expression</a>, with the difference being the use of a min-plus <a href="Semiring" title="Semiring">semiring</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> The modern formulation of the algorithm as three nested for-loops was first described by Peter Ingerman, also in 1962.<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>

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<div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2></div>

<p>The Floyd–Warshall algorithm compares many possible paths through the graph between each pair of vertices. It is guaranteed to find all shortest paths and is able to do this with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|V|^{3})}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \Theta (|V|^{3})}</annotation>

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</math></span><img src="./b89800bb4f210f469a91b23b045aa481e5fd176e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.753ex; height:3.343ex;" alt="{\displaystyle \Theta (|V|^{3})}" loading="lazy"></span> comparisons in a graph,<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_9-0" class="reference"><a href="#cite_note-:1-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> even though there may be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|V|^{2})}">

</mrow>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|V|^{2})}</annotation>

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</math></span><img src="./5dbb38dcf2365c1dbf786871f4014dd1a2762df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.753ex; height:3.343ex;" alt="{\displaystyle \Theta (|V|^{2})}" loading="lazy"></span> edges in the graph. It does so by incrementally improving an estimate on the shortest path between two vertices, until the estimate is optimal.

</p><p>Consider a graph <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> with vertices <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> numbered 1 through <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span>. Further consider a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {shortestPath} (i,j,k)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,k)}</annotation>

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</math></span><img src="./da0a72bd3bd16adcff4b3ef0baa061643a38393c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.832ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,k)}" loading="lazy"></span> that returns the length of the shortest possible path (if one exists) from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

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</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> using vertices only from the set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,k\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,k\}}</annotation>

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</math></span><img src="./e3836453b6a15e5c6ca10a0408fe82de44322fb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.073ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,k\}}" loading="lazy"></span> as intermediate points along the way. Now, given this function, our goal is to find the length of the shortest path from each <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> to each <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

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</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> using <i>any</i> vertex in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,N\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,N\}}</annotation>

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</math></span><img src="./18d7b4ef5cabdf43044424420855eeaf363f0c09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.926ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,N\}}" loading="lazy"></span>. By definition, this is the value <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {shortestPath} (i,j,N)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,N)}</annotation>

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</math></span><img src="./10b7cee0e900c6d0dfdd2ca265c4e0b396ebd5a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.685ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,N)}" loading="lazy"></span>, which we will find <a href="Recursion" title="Recursion">recursively</a>.

</p><p>Observe that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {shortestPath} (i,j,k)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,k)}</annotation>

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</math></span><img src="./da0a72bd3bd16adcff4b3ef0baa061643a38393c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.832ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,k)}" loading="lazy"></span> must be less than or equal to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {shortestPath} (i,j,k-1)}">

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,k-1)}</annotation>

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</math></span><img src="./d5997ae5bc81aeb05219d13a566255d3953c08c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.835ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,k-1)}" loading="lazy"></span>: we have <i>more</i> flexibility if we are allowed to use the vertex <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>. If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {shortestPath} (i,j,k)}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,k)}</annotation>

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</math></span><img src="./da0a72bd3bd16adcff4b3ef0baa061643a38393c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.832ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,k)}" loading="lazy"></span> is in fact less than <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {shortestPath} (i,j,k-1)}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,k-1)}</annotation>

</semantics>

</math></span><img src="./d5997ae5bc81aeb05219d13a566255d3953c08c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.835ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,k-1)}" loading="lazy"></span>, then there must be a path from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> using the vertices <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,k\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,k\}}</annotation>

</semantics>

</math></span><img src="./e3836453b6a15e5c6ca10a0408fe82de44322fb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.073ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,k\}}" loading="lazy"></span> that is shorter than any such path that does not use the vertex <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>. Since there are no negative cycles this path can be decomposed as:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> that uses the vertices <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,k-1\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,k-1\}}</annotation>

</semantics>

</math></span><img src="./2e31a80ba97c78f8fac9a97a7780d85f90a8619b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.076ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,k-1\}}" loading="lazy"></span>, followed by</dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> that uses the vertices <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,k-1\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,k-1\}}</annotation>

</semantics>

</math></span><img src="./2e31a80ba97c78f8fac9a97a7780d85f90a8619b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.076ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,k-1\}}" loading="lazy"></span>.</dd></dl>

<p>And of course, these must be a <i>shortest</i> such path (or several of them), otherwise we could further decrease the length. In other words, we have arrived at the recursive formula:

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,k)=}</annotation>

</semantics>

</math></span><img src="./f9b75e25063384ccca499c56f9a279abf661ad3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.285ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,k)=}" loading="lazy"></span>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {min} {\Big (}\mathrm {shortestPath} (i,j,k-1),}</annotation>

</semantics>

</math></span><img src="./34ac7c89bbb18df3fd660225fd38997079e5e513.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.745ex; height:4.843ex;" alt="{\displaystyle \mathrm {min} {\Big (}\mathrm {shortestPath} (i,j,k-1),}" loading="lazy"></span>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,k,k-1)+\mathrm {shortestPath} (k,j,k-1){\Big )}}</annotation>

</semantics>

</math></span><img src="./0326d6c14def89269c029da59eba012d0f2edc9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.56ex; height:4.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,k,k-1)+\mathrm {shortestPath} (k,j,k-1){\Big )}}" loading="lazy"></span>.</dd></dl></dd></dl></dd></dl>

<p>The base case is given by

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,0)=w(i,j),}</annotation>

</semantics>

</math></span><img src="./23862c8ea0f4a4f00a00dded1176557e4f33b84c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.797ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,0)=w(i,j),}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w(i,j)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w(i,j)}</annotation>

</semantics>

</math></span><img src="./860cd28acf764c9e3a8955c71ba6ee4e6c831b88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.268ex; height:2.843ex;" alt="{\displaystyle w(i,j)}" loading="lazy"></span> denotes the weight of the edge from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> if one exists and ∞ (infinity) otherwise.

</p><p>These formulas are the heart of the Floyd–Warshall algorithm. The algorithm works by first computing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {shortestPath} (i,j,k)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,k)}</annotation>

</semantics>

</math></span><img src="./da0a72bd3bd16adcff4b3ef0baa061643a38393c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.832ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,k)}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation>

</semantics>

</math></span><img src="./8ef21910f980c6fca2b15bee102a7a0d861ed712.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}" loading="lazy"></span> pairs for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=0}</annotation>

</semantics>

</math></span><img src="./6307c8a99dad7d0bcb712352ae0a748bd99a038b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=1}</annotation>

</semantics>

</math></span><img src="./6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=2}</annotation>

</semantics>

</math></span><img src="./0bd301789e1f25a3da4be297ff637754ebee5f5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=2}" loading="lazy"></span>, and so on. This process continues until <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=N}</annotation>

</semantics>

</math></span><img src="./2e302b20f7f5064ac2ebb7734304fd430ad737a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.373ex; height:2.176ex;" alt="{\displaystyle k=N}" loading="lazy"></span>, and we have found the shortest path for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation>

</semantics>

</math></span><img src="./8ef21910f980c6fca2b15bee102a7a0d861ed712.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}" loading="lazy"></span> pairs using any intermediate vertices. Pseudocode for this basic version follows.

</p>

<div class="mw-heading mw-heading3"><h3 id="Pseudocode">Pseudocode</h3></div>

<pre><b>let</b> dist be a |V| × |V| array of minimum distances initialized to ∞ (infinity)

dist[<i>u</i>][<i>v</i>] = w(<i>u</i>, <i>v</i>) <i>// The weight of the edge (</i>u<i>, </i>v<i>)</i>

<b>for each</b> vertex <i>v</i> <b>do</b>

dist[<i>v</i>][<i>v</i>] = 0

<b>for</b> <i>k</i> <b>from</b> 1 <b>to</b> |V|

<b>for</b> <i>i</i> <b>from</b> 1 <b>to</b> |V|

<b>for</b> <i>j</i> <b>from</b> 1 <b>to</b> |V|

<b>if</b> dist[<i>i</i>][<i>j</i>] > dist[<i>i</i>][<i>k</i>] + dist[<i>k</i>][<i>j</i>]

dist[<i>i</i>][<i>j</i>] = dist[<i>i</i>][<i>k</i>] + dist[<i>k</i>][<i>j</i>]

</pre>

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

<p>The algorithm above is executed on the graph on the left below:

</p><p><span typeof="mw:File"></span>

</p><p>Prior to the first recursion of the outer loop, labeled <span class="texhtml"><i>k</i> = 0</span> above, the only known paths correspond to the single edges in the graph. At <span class="texhtml"><i>k</i> = 1</span>, paths that go through the vertex 1 are found: in particular, the path [2,1,3] is found, replacing the path [2,3] which has fewer edges but is longer (in terms of weight). At <span class="texhtml"><i>k</i> = 2</span>, paths going through the vertices {1,2} are found. The red and blue boxes show how the path [4,2,1,3] is assembled from the two known paths [4,2] and [2,1,3] encountered in previous iterations, with 2 in the intersection. The path [4,2,3] is not considered, because [2,1,3] is the shortest path encountered so far from 2 to 3. At <span class="texhtml"><i>k</i> = 3</span>, paths going through the vertices {1,2,3} are found. Finally, at <span class="texhtml"><i>k</i> = 4</span>, all shortest paths are found.

</p><p>The distance matrix at each iteration of <span class="texhtml mvar" style="font-style:italic;">k</span>, with the updated distances in <b>bold</b>, will be:

</p>

</caption>

</td>

</td></tr>

<tr>

<th>4

</th></tr>

<tr>

</td>

<th>1

</th>

<td>∞

</td></tr>

<tr>

<th>2

</th>

<td>∞

</td></tr>

<tr>

<th>3

</th>

<td>2

</td></tr>

<tr>

<th>4

</th>

<td>0

</td></tr></tbody></table>

</caption>

</td>

</td></tr>

<tr>

<th>4

</th></tr>

<tr>

</td>

<th>1

</th>

<td>∞

</td></tr>

<tr>

<th>2

</th>

<td>∞

</td></tr>

<tr>

<th>3

</th>

<td>2

</td></tr>

<tr>

<th>4

</th>

<td>0

</td></tr></tbody></table>

</caption>

</td>

</td></tr>

<tr>

<th>4

</th></tr>

<tr>

</td>

<th>1

</th>

<td>∞

</td></tr>

<tr>

<th>2

</th>

<td>∞

</td></tr>

<tr>

<th>3

</th>

<td>2

</td></tr>

<tr>

<th>4

</th>

<td>0

</td></tr></tbody></table>

</caption>

</td>

</td></tr>

<tr>

<th>4

</th></tr>

<tr>

</td>

<th>1

</th>

</td></tr>

<tr>

<th>2

</th>

</td></tr>

<tr>

<th>3

</th>

<td>2

</td></tr>

<tr>

<th>4

</th>

<td>0

</td></tr></tbody></table>

</caption>

</td>

</td></tr>

<tr>

<th>4

</th></tr>

<tr>

</td>

<th>1

</th>

<td>0

</td></tr>

<tr>

<th>2

</th>

<td>4

</td></tr>

<tr>

<th>3

</th>

<td>2

</td></tr>

<tr>

<th>4

</th>

<td>0

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Behavior_with_negative_cycles">Behavior with negative cycles</h2></div>

<p>A negative cycle is a cycle whose edges sum to a negative value. There is no shortest path between any pair of vertices <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> which form part of a negative cycle, because path-lengths from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> can be arbitrarily small (negative). For numerically meaningful output, the Floyd–Warshall algorithm assumes that there are no negative cycles. Nevertheless, if there are negative cycles, the Floyd–Warshall algorithm can be used to detect them. The intuition is as follows:

</p>

<ul><li>The Floyd–Warshall algorithm iteratively revises path lengths between all pairs of vertices <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation>

</semantics>

</math></span><img src="./8ef21910f980c6fca2b15bee102a7a0d861ed712.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}" loading="lazy"></span>, including where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=j}</annotation>

</semantics>

</math></span><img src="./706e0928b2bf0f24076b0c90bb20616ff2068343.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle i=j}" loading="lazy"></span>;</li>

<li>Initially, the length of the path <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,i)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (i,i)}</annotation>

</semantics>

</math></span><img src="./899c08ccec9bf245d850a44ba8a5275bee6b3b8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.448ex; height:2.843ex;" alt="{\displaystyle (i,i)}" loading="lazy"></span> is zero;</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [i,k,\ldots ,i]}</annotation>

</semantics>

</math></span><img src="./19877ceaf724fa3a7dcc51b9853630a0f31b33c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.322ex; height:2.843ex;" alt="{\displaystyle [i,k,\ldots ,i]}" loading="lazy"></span> can only improve upon this if it has length less than zero, i.e. denotes a negative cycle;</li>

<li>Thus, after the algorithm, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,i)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (i,i)}</annotation>

</semantics>

</math></span><img src="./899c08ccec9bf245d850a44ba8a5275bee6b3b8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.448ex; height:2.843ex;" alt="{\displaystyle (i,i)}" loading="lazy"></span> will be negative if there exists a negative-length path from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> back to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>.</li></ul>

<p>Hence, to detect negative <a href="Cycle_(graph_theory)" title="Cycle (graph theory)">cycles</a> using the Floyd–Warshall algorithm, one can inspect the diagonal of the path matrix, and the presence of a negative number indicates that the graph contains at least one negative cycle.<sup id="cite_ref-:1_9-1" class="reference"><a href="#cite_note-:1-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> However, when a negative cycle is present, during the execution of the algorithm exponentially large numbers on the order of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (6^{n}\cdot w_{max})}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega (6^{n}\cdot w_{max})}</annotation>

</semantics>

</math></span><img src="./5ed0ab02dc9958f492b34481e7c84022e68608a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.696ex; height:2.843ex;" alt="{\displaystyle \Omega (6^{n}\cdot w_{max})}" loading="lazy"></span> can appear, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{max}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{max}}</annotation>

</semantics>

</math></span><img src="./985bd43745ee322dbfee514b0fb5863dfe102a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.149ex; height:2.009ex;" alt="{\displaystyle w_{max}}" loading="lazy"></span> is the largest absolute value edge weight in the graph. To avoid integer underflow problems, one should check for a negative cycle within the innermost for loop of the algorithm.<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>

</p>

<div class="mw-heading mw-heading2"><h2 id="Path_reconstruction">Path reconstruction</h2></div>

<p>The Floyd–Warshall algorithm typically only provides the lengths of the paths between all pairs of vertices. With simple modifications, it is possible to create a method to reconstruct the actual path between any two endpoint vertices. While one may be inclined to store the actual path from each vertex to each other vertex, this is not necessary, and in fact, is very costly in terms of memory. Instead, we can use the <a href="Shortest-path_tree" title="Shortest-path tree">shortest-path tree</a>, which can be calculated for each node in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|E|)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|E|)}</annotation>

</semantics>

</math></span><img src="./fee8b4657c1eb491bc9d05378522ddc12ddac5fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.687ex; height:2.843ex;" alt="{\displaystyle \Theta (|E|)}" loading="lazy"></span> time using <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|V|)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|V|)}</annotation>

</semantics>

</math></span><img src="./d6859084635373622fc54329d559928d0d564d42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.698ex; height:2.843ex;" alt="{\displaystyle \Theta (|V|)}" loading="lazy"></span> memory, and allows us to efficiently reconstruct a directed path between any two connected vertices.

</p>

<div class="mw-heading mw-heading3"><h3 id="Pseudocode_2">Pseudocode</h3></div>

<p>The array <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">prev[u][v]</code> holds the penultimate vertex on the path from <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">u</code> to <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">v</code> (except in the case of <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">prev[v][v]</code>, where it always contains <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">v</code> even if there is no self-loop on <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">v</code>):<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>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |V|\times |V|}</annotation>

</semantics>

</math></span><img src="./870295636261d8bb4ce017b70c39dde3bd9512fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.002ex; height:2.843ex;" alt="{\displaystyle |V|\times |V|}" loading="lazy"></span> array of minimum distances initialized to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \infty }</annotation>

</semantics>

</math></span><img src="./c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy"></span> (infinity)

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |V|\times |V|}</annotation>

</semantics>

</math></span><img src="./870295636261d8bb4ce017b70c39dde3bd9512fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.002ex; height:2.843ex;" alt="{\displaystyle |V|\times |V|}" loading="lazy"></span> array of vertex indices initialized to <b>null</b>

<b>procedure</b> <i>FloydWarshallWithPathReconstruction</i>() <b>is</b>

dist[u][v] = w(u, v) <i>// The weight of the edge (u, v)</i>

prev[u][v] = u

<b>for each</b> vertex v <b>do</b>

dist[v][v] = 0

prev[v][v] = v

<b>for</b> k <b>from</b> 1 <b>to</b> |V| <b>do</b> <i>// standard Floyd-Warshall implementation</i>

<b>for</b> i <b>from</b> 1 <b>to</b> |V|

<b>for</b> j <b>from</b> 1 <b>to</b> |V|

dist[i][j] = dist[i][k] + dist[k][j]

prev[i][j] = prev[k][j]

</pre>

<pre><b>procedure</b> <i>Path</i>(u, v) <b>is</b>

<b>return</b> []

path = [v]

<b>while</b> <i>u</i> ≠ <i>v</i> <b>do</b>

v = prev[u][v]

path.prepend(v)

<b>return</b> path

</pre>

<div class="mw-heading mw-heading2"><h2 id="Time_complexity">Time complexity</h2></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |V|}</annotation>

</semantics>

</math></span><img src="./9ddcffc28643ac01a14dd0fb32c3157859e365a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.081ex; height:2.843ex;" alt="{\displaystyle |V|}" loading="lazy"></span>, the number of vertices. To find all <span class="mwe-math-element mwe-math-element-inline"><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}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation>

</semantics>

</math></span><img src="./ac9810bbdafe4a6a8061338db0f74e25b7952620.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{2}}" loading="lazy"></span> of

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,k)}</annotation>

</semantics>

</math></span><img src="./da0a72bd3bd16adcff4b3ef0baa061643a38393c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.832ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,k)}" loading="lazy"></span> (for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span>) from those of

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,k-1)}</annotation>

</semantics>

</math></span><img src="./d5997ae5bc81aeb05219d13a566255d3953c08c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.835ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,k-1)}" loading="lazy"></span> requires <a href="Big_theta" class="mw-redirect" title="Big theta"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (n^{2})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (n^{2})}</annotation>

</semantics>

</math></span><img src="./215877752c4f392a7276328d4d37709bf7c3f55d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.066ex; height:3.176ex;" alt="{\displaystyle \Theta (n^{2})}" loading="lazy"></span></a> operations. Since we begin with

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,0)=\mathrm {edgeCost} (i,j)}</annotation>

</semantics>

</math></span><img src="./9d3d4efe797fa90aa43cf20fd20a1562ca4160d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.667ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,0)=\mathrm {edgeCost} (i,j)}" loading="lazy"></span> and compute the sequence of <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> matrices <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {shortestPath} (i,j,1)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,1)}</annotation>

</semantics>

</math></span><img src="./2d579582db8fe3121bb4ee775f60a77e9d3131bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.783ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,1)}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {shortestPath} (i,j,2)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,2)}</annotation>

</semantics>

</math></span><img src="./449ff862e43e958555fc9f8a78f2ce80df0db396.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.783ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,2)}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ldots }</annotation>

</semantics>

</math></span><img src="./3b8619532e44ee1ccae3ab03405a6885260d09ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \ldots }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {shortestPath} (i,j,n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {shortestPath} (i,j,n)}</annotation>

</semantics>

</math></span><img src="./ece65972ee8c39a8be2c1a760ebab5a5159889d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.016ex; height:2.843ex;" alt="{\displaystyle \mathrm {shortestPath} (i,j,n)}" loading="lazy"></span>, each having a cost of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (n^{2})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (n^{2})}</annotation>

</semantics>

</math></span><img src="./215877752c4f392a7276328d4d37709bf7c3f55d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.066ex; height:3.176ex;" alt="{\displaystyle \Theta (n^{2})}" loading="lazy"></span>,

the total <a href="Time_complexity" title="Time complexity">time complexity</a> of the algorithm is <span class="mwe-math-element mwe-math-element-inline"><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\cdot \Theta (n^{2})=\Theta (n^{3})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\cdot \Theta (n^{2})=\Theta (n^{3})}</annotation>

</semantics>

</math></span><img src="./764feb1180d30cfe91fc716ba4011f05f50e4f2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.305ex; height:3.176ex;" alt="{\displaystyle n\cdot \Theta (n^{2})=\Theta (n^{3})}" loading="lazy"></span>.<sup id="cite_ref-:1_9-2" class="reference"><a href="#cite_note-:1-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><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>

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications_and_generalizations">Applications and generalizations</h2></div>

<p>The Floyd–Warshall algorithm can be used to solve the following problems, among others:

</p>

<ul><li>Shortest paths in directed graphs (Floyd's algorithm).</li>

<li><a href="Transitive_closure" title="Transitive closure">Transitive closure</a> of directed graphs (Warshall's algorithm). In Warshall's original formulation of the algorithm, the graph is unweighted and represented by a Boolean <a href="Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a>. Then the addition operation is replaced by <a href="Logical_conjunction" title="Logical conjunction">logical conjunction</a> (AND) and the minimum operation by <a href="Logical_disjunction" title="Logical disjunction">logical disjunction</a> (OR).</li>

<li>Finding a <a href="Regular_expression" title="Regular expression">regular expression</a> denoting the <a href="Regular_language" title="Regular language">regular language</a> accepted by a <a href="Finite_automaton" class="mw-redirect" title="Finite automaton">finite automaton</a> (<a href="Kleene's_algorithm" title="Kleene's algorithm">Kleene's algorithm</a>, a closely related generalization of the Floyd–Warshall algorithm)<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></li>

<li><a href="Invertible_matrix" title="Invertible matrix">Inversion</a> of <a href="Real_number" title="Real number">real</a> <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> (<a href="Gauss%E2%80%93Jordan_elimination" class="mw-redirect" title="Gauss–Jordan elimination">Gauss–Jordan algorithm</a>)<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li>

<li>Optimal routing. In this application one is interested in finding the path with the maximum flow between two vertices. This means that, rather than taking minima as in the pseudocode above, one instead takes maxima. The edge weights represent fixed constraints on flow. Path weights represent bottlenecks; so the addition operation above is replaced by the minimum operation.</li>

<li>Fast computation of <a href="Pathfinder_network" title="Pathfinder network">Pathfinder networks</a>.</li>

<li><a href="Widest_path_problem" title="Widest path problem">Widest paths/Maximum bandwidth paths</a></li>

<li>Computing canonical form of difference bound matrices (DBMs)</li>

<li>Computing the similarity between graphs</li>

<li>Transitive closure in AND/OR/threshold graphs.<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></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Implementations">Implementations</h2></div>

<p>Implementations are available for many <a href="Programming_language" title="Programming language">programming languages</a>.

</p>

<ul><li>For <a href="C%2B%2B" title="C++">C++</a>, in the <a rel="nofollow" class="external text" href="http://www.boost.org/libs/graph/doc/">boost::graph</a> library</li>

<li>For <a href="C_Sharp_(programming_language)" title="C Sharp (programming language)">C#</a>, at <a rel="nofollow" class="external text" href="https://github.com/KeRNeLith/QuikGraph">QuikGraph</a></li>

<li>For <a href="C_Sharp_(programming_language)" title="C Sharp (programming language)">C#</a>, at <a rel="nofollow" class="external text" href="https://www.nuget.org/packages/QuickGraphPCL/3.6.61114.2">QuickGraphPCL</a> (A fork of QuickGraph with better compatibility with projects using Portable Class Libraries.)</li>

<li>For <a href="Java_(programming_language)" title="Java (programming language)">Java</a>, in the <a rel="nofollow" class="external text" href="https://commons.apache.org/sandbox/commons-graph/">Apache Commons Graph</a> library</li>

<li>For <a href="JavaScript" title="JavaScript">JavaScript</a>, in the <a href="Cytoscape" title="Cytoscape">Cytoscape</a> library</li>

<li>For <a href="Julia_(programming_language)" title="Julia (programming language)">Julia</a>, in the <a rel="nofollow" class="external text" href="https://docs.juliahub.com/Graphs/VJ6vx/1.7.0/algorithms/shortestpaths/#Graphs.floyd_warshall_shortest_paths-Union{Tuple{AbstractGraph{U}},%20Tuple{T},%20Tuple{U},%20Tuple{AbstractGraph{U},%20AbstractMatrix{T}}}%20where%20{U%3C:Integer,%20T%3C:Real}">Graphs.jl</a> package</li>

<li>For <a href="MATLAB" title="MATLAB">MATLAB</a>, in the <a rel="nofollow" class="external text" href="http://www.mathworks.com/matlabcentral/fileexchange/10922">Matlab_bgl</a> package</li>

<li>For <a href="Perl" title="Perl">Perl</a>, in the <a rel="nofollow" class="external text" href="https://metacpan.org/module/Graph">Graph</a> module</li>

<li>For <a href="Python_(programming_language)" title="Python (programming language)">Python</a>, in the <a href="SciPy" title="SciPy">SciPy</a> library (module <a rel="nofollow" class="external text" href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.csgraph.floyd_warshall.html#scipy.sparse.csgraph.floyd_warshall">scipy.sparse.csgraph</a>) or <a href="NetworkX" title="NetworkX">NetworkX</a> library</li>

<li>For <a href="R_programming_language" class="mw-redirect" title="R programming language">R</a>, in packages <a rel="nofollow" class="external text" href="https://cran.r-project.org/web/packages/e1071/index.html">e1071</a> and <a rel="nofollow" class="external text" href="https://cran.r-project.org/web/packages/Rfast/index.html">Rfast</a></li>

<li>For <a href="C_(programming_language)" title="C (programming language)">C</a>, a <a href="Pthreads" title="Pthreads">pthreads</a>, <a rel="nofollow" class="external text" href="https://moorejs.github.io/APSP-in-parallel/">parallelized</a>, implementation including a <a href="SQLite" title="SQLite">SQLite</a> interface to the data at <a rel="nofollow" class="external text" href="https://github.com/gdavidbutler/pthreadChannel/blob/master/example/floydWarshall.h">floydWarshall.h</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Comparison_with_other_shortest_path_algorithms">Comparison with other shortest path algorithms</h2></div>

<p>For graphs with non-negative edge weights, <a href="Dijkstra's_algorithm" title="Dijkstra's algorithm">Dijkstra's algorithm</a> can be used to find all shortest paths from a <i>single</i> vertex with running time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|E|+|V|\log |V|)}">

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|E|+|V|\log |V|)}</annotation>

</semantics>

</math></span><img src="./e22162be85d06b346f3b7f7aad9746da0c1019c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.435ex; height:2.843ex;" alt="{\displaystyle \Theta (|E|+|V|\log |V|)}" loading="lazy"></span>. Thus, running Dijkstra starting at <i>each</i> vertex takes time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (|E||V|+|V|^{2}\log |V|)}">

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (|E||V|+|V|^{2}\log |V|)}</annotation>

</semantics>

</math></span><img src="./a62bd0a9e63328e60dcea3f0ec8a585fcff3e68d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.57ex; height:3.343ex;" alt="{\displaystyle \Theta (|E||V|+|V|^{2}\log |V|)}" loading="lazy"></span>. Since <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E|=O(|V|^{2})}">

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |E|=O(|V|^{2})}</annotation>

</semantics>

</math></span><img src="./2721551edc09c244a2619b38791552dfe65928c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.886ex; height:3.343ex;" alt="{\displaystyle |E|=O(|V|^{2})}" loading="lazy"></span>, this yields a worst-case running time of repeated Dijkstra of <span class="mwe-math-element mwe-math-element-inline"><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(|V|^{3})}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(|V|^{3})}</annotation>

</semantics>

</math></span><img src="./04ad8b53a2cea864c6d746895cc7397971f72912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.718ex; height:3.343ex;" alt="{\displaystyle O(|V|^{3})}" loading="lazy"></span>. While this matches the asymptotic worst-case running time of the Floyd-Warshall algorithm, the constants involved matter quite a lot. When a <a href="Dense_graph" title="Dense graph">graph is dense</a> (i.e., <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E|\approx |V|^{2}}">

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |E|\approx |V|^{2}}</annotation>

</semantics>

</math></span><img src="./000d44adc1b18a1e707edbe33c3dba6385bcae5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.303ex; height:3.343ex;" alt="{\displaystyle |E|\approx |V|^{2}}" loading="lazy"></span>), the Floyd-Warshall algorithm tends to perform better in practice. When the graph is sparse (i.e., <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |E|}</annotation>

</semantics>

</math></span><img src="./d8c2b9637808cf805d411190b4ae017dbd4ef8d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.069ex; height:2.843ex;" alt="{\displaystyle |E|}" loading="lazy"></span> is significantly smaller than <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V|^{2}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |V|^{2}}</annotation>

</semantics>

</math></span><img src="./3e2a7aad37527c097101c39b0b7724fe4e1cafb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.135ex; height:3.343ex;" alt="{\displaystyle |V|^{2}}" loading="lazy"></span>), Dijkstra tends to dominate.

</p><p>For sparse graphs with negative edges but no negative cycles, <a href="Johnson's_algorithm" title="Johnson's algorithm">Johnson's algorithm</a> can be used, with the same asymptotic running time as the repeated Dijkstra approach.

</p><p>There are also known algorithms using <a href="Fast_matrix_multiplication" class="mw-redirect" title="Fast matrix multiplication">fast matrix multiplication</a> to speed up all-pairs shortest path computation in dense graphs, but these typically make extra assumptions on the edge weights (such as requiring them to be small integers).<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><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> In addition, because of the high constant factors in their running time, they would only provide a speedup over the Floyd–Warshall algorithm for very large graphs.

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFCormenLeisersonRivest1990" class="citation book cs1"><a href="Thomas_H._Cormen" title="Thomas H. Cormen">Cormen, Thomas H.</a>; <a href="Charles_E._Leiserson" title="Charles E. Leiserson">Leiserson, Charles E.</a>; <a href="Ron_Rivest" title="Ron Rivest">Rivest, Ronald L.</a> (1990). <a href="Introduction_to_Algorithms" title="Introduction to Algorithms"><i>Introduction to Algorithms</i></a> (1st ed.). MIT Press and McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-03141-8</bdi>.</cite> See in particular Section 26.2, "The Floyd–Warshall algorithm", pp. 558–565 and Section 26.4, "A general framework for solving path problems in directed graphs", pp. 570–576.</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"><cite id="CITEREFKenneth_H._Rosen2003" class="citation book cs1">Kenneth H. Rosen (2003). <i>Discrete Mathematics and Its Applications, 5th Edition</i>. Addison Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-119881-3</bdi>.</cite></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"><cite id="CITEREFFloyd1962" class="citation journal cs1"><a href="Robert_W._Floyd" title="Robert W. Floyd">Floyd, Robert W.</a> (June 1962). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F367766.368168">"Algorithm 97: Shortest Path"</a>. <i><a href="Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a></i>. <b>5</b> (6): 345. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F367766.368168">10.1145/367766.368168</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2003382">2003382</a>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFRoy1959" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="Bernard_Roy" title="Bernard Roy">Roy, Bernard</a> (1959). <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k3201c/f222.image">"Transitivité et connexité"</a>. <i><a href="C._R._Acad._Sci._Paris" class="mw-redirect" title="C. R. Acad. Sci. Paris">C. R. Acad. Sci. Paris</a></i> (in French). <b>249</b>: <span class="nowrap">216–</span>218.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFWarshall1962" class="citation journal cs1">Warshall, Stephen (January 1962). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F321105.321107">"A theorem on Boolean matrices"</a>. <i><a href="Journal_of_the_ACM" title="Journal of the ACM">Journal of the ACM</a></i>. <b>9</b> (1): <span class="nowrap">11–</span>12. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F321105.321107">10.1145/321105.321107</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:33763989">33763989</a>.</cite></span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Floyd-Warshall_Algorithm"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Floyd-WarshallAlgorithm.html">"Floyd-Warshall Algorithm"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFKleene1956" class="citation book cs1"><a href="Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene, S. C.</a> (1956). "Representation of events in nerve nets and finite automata". In <a href="Claude_Elwood_Shannon" class="mw-redirect" title="Claude Elwood Shannon">C. E. Shannon</a> and <a href="John_McCarthy_(computer_scientist)" title="John McCarthy (computer scientist)">J. McCarthy</a> (ed.). <i>Automata Studies</i>. Princeton University Press. pp. <span class="nowrap">3–</span>42.</cite></span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFIngerman1962" class="citation journal cs1">Ingerman, Peter Z. (November 1962). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F368996.369016">"Algorithm 141: Path Matrix"</a>. <i><a href="Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a></i>. <b>5</b> (11): 556. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F368996.369016">10.1145/368996.369016</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:29010500">29010500</a>.</cite></span>

</li>

<li id="cite_note-:1-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_9-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_9-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFHochbaum2014" class="citation web cs1"><a href="Dorit_S._Hochbaum" title="Dorit S. Hochbaum">Hochbaum, Dorit</a> (2014). <a rel="nofollow" class="external text" href="http://www.ieor.berkeley.edu/~hochbaum/files/ieor266-2014.pdf">"Section 8.9: Floyd-Warshall algorithm for all pairs shortest paths"</a> <span class="cs1-format">(PDF)</span>. <i>Lecture Notes for IEOR 266: Graph Algorithms and Network Flows</i>. Department of Industrial Engineering and Operations Research, <a href="University_of_California%2C_Berkeley" title="University of California, Berkeley">University of California, Berkeley</a>. pp. <span class="nowrap">41–</span>42.</cite></span>

</li>

<cite id="CITEREFStefan_Hougardy2010" class="citation journal cs1">Stefan Hougardy (April 2010). "The Floyd–Warshall algorithm on graphs with negative cycles". <i>Information Processing Letters</i>. <b>110</b> (<span class="nowrap">8–</span>9): <span class="nowrap">279–</span>281. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ipl.2010.02.001">10.1016/j.ipl.2010.02.001</a>.</cite></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://books.goalkicker.com/AlgorithmsBook/">"Free Algorithms Book"</a>.</cite></span>

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<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFBarasTheodorakopoulos2022" class="citation book cs1">Baras, John; Theodorakopoulos, George (2022). <i>Path Problems in Networks</i>. Springer International Publishing. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783031799839</bdi>.</cite></span>

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<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite id="CITEREFGrossYellen2003" class="citation cs1">Gross, Jonathan L.; Yellen, Jay (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mKkIGIea_BkC&pg=PA65"><i>Handbook of Graph Theory</i></a>. Discrete Mathematics and Its Applications. CRC Press. p. 65. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780203490204</bdi>.</cite>.</span>

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<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><cite id="CITEREFPeñaloza" class="citation web cs1">Peñaloza, Rafael. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091022202353/http://geocities.com/rpenalozan/ps/graphalg2005.ps">"Algebraic Structures for Transitive Closure"</a>. <i>Seminar "Graph Algorithms"</i>. Dresden University of Technology, Department of Computer Science, Institute for Theoretical Computer Science. Archived from <a rel="nofollow" class="external text" href="http://geocities.com/rpenalozan/ps/graphalg2005.ps">the original</a> on 2009-10-22.</cite></span>

</li>

<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFGillies1993" class="citation report cs1">Gillies, Donald (1993). <a rel="nofollow" class="external text" href="http://www.ece.ubc.ca/~gillies/download/Donald_W_Gillies_PhD_1993_Scheduling_With_AND_OR_Precedence.pdf">Scheduling Tasks with AND/OR precedence contraints (PhD Thesis, Appendix B)</a> <span class="cs1-format">(PDF)</span> (Report).</cite></span>

</li>

<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><cite id="CITEREFZwick2002" class="citation cs1"><a href="Uri_Zwick" title="Uri Zwick">Zwick, Uri</a> (May 2002). "All pairs shortest paths using bridging sets and rectangular matrix multiplication". <i><a href="Journal_of_the_ACM" title="Journal of the ACM">Journal of the ACM</a></i>. <b>49</b> (3): <span class="nowrap">289–</span>317. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cs/0008011">cs/0008011</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F567112.567114">10.1145/567112.567114</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1065901">1065901</a>.</cite>.</span>

</li>

<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><cite id="CITEREFChan2010" class="citation cs1"><a href="Timothy_M._Chan" title="Timothy M. Chan">Chan, Timothy M.</a> (January 2010). "More algorithms for all-pairs shortest paths in weighted graphs". <i><a href="SIAM_Journal_on_Computing" title="SIAM Journal on Computing">SIAM Journal on Computing</a></i>. <b>39</b> (5): <span class="nowrap">2075–</span>2089. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.153.6864">10.1.1.153.6864</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F08071990x">10.1137/08071990x</a>.</cite>.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Floyd-Warshall_algorithm" class="extiw external" title="commons:Category:Floyd-Warshall algorithm">Floyd-Warshall algorithm</a></span>.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="http://www.pms.informatik.uni-muenchen.de/lehre/compgeometry/Gosper/shortest_path/shortest_path.html#visualization">Interactive animation of the Floyd–Warshall algorithm</a></li>

<li><a rel="nofollow" class="external text" href="https://algorithms.discrete.ma.tum.de/graph-algorithms/spp-floyd-warshall/index_en.html">Interactive animation of the Floyd–Warshall algorithm (Technical University of Munich)</a></li></ul>

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</style><div id="Graph_and_tree_traversal_algorithms231" style="font-size:114%;margin:0 4em"><a href="Graph_traversal" title="Graph traversal">Graph</a> and <a href="Tree_traversal" title="Tree traversal">tree</a> traversal algorithms</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Graph_traversal" title="Graph traversal">Search</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alpha%E2%80%93beta_pruning" title="Alpha–beta pruning">α–β pruning</a></li>

<li><a href="Best-first_search" title="Best-first search">Best-first search</a></li>

<li><a href="Beam_search" title="Beam search">Beam search</a></li>

<li><a href="Bidirectional_search" title="Bidirectional search">Bidirectional search</a></li>

<li><a href="Breadth-first_search" title="Breadth-first search">Breadth-first search</a>

<ul><li><a href="Lexicographic_breadth-first_search" title="Lexicographic breadth-first search">Lexicographic</a></li>

<li><a href="Parallel_breadth-first_search" title="Parallel breadth-first search">Parallel</a></li></ul></li>

<li><a href="Depth-first_search" title="Depth-first search">Depth-first search</a>

<ul><li><a href="Iterative_deepening_depth-first_search" title="Iterative deepening depth-first search">Iterative deepening</a></li></ul></li>

<li><a href="Fringe_search" title="Fringe search">Fringe search</a></li>

<li><a href="Jump_point_search" title="Jump point search">Jump point search</a></li>

<li><a href="Monte_Carlo_tree_search" title="Monte Carlo tree search">Monte Carlo tree search</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Shortest_path_problem" title="Shortest path problem">Shortest path</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford</a></li>

<li><a href="Dijkstra's_algorithm" title="Dijkstra's algorithm">Dijkstra's</a></li>

<li><a href="Johnson's_algorithm" title="Johnson's algorithm">Johnson's</a></li>

<li><a href="Shortest_path_faster_algorithm" class="mw-redirect" title="Shortest path faster algorithm">Shortest path faster</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Minimum_spanning_tree" title="Minimum spanning tree">Minimum spanning tree</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bor%C5%AFvka's_algorithm" title="Borůvka's algorithm">Borůvka's</a></li>

<li><a href="Kruskal's_algorithm" title="Kruskal's algorithm">Kruskal's</a></li>

<li><a href="Reverse-delete_algorithm" title="Reverse-delete algorithm">Reverse-delete</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="List_of_algorithms#Graph_search" title="List of algorithms">List of graph search algorithms</a></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Optimization:_Algorithms,_methods,_and_heuristics381" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Optimization:_Algorithms,_methods,_and_heuristics381" style="font-size:114%;margin:0 4em"><a href="Mathematical_optimization" title="Mathematical optimization">Optimization</a>: <a href="Optimization_algorithm" class="mw-redirect" title="Optimization algorithm">Algorithms</a>, <a href="Iterative_method" title="Iterative method">methods</a>, and <a href="Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Unconstrained_nonlinear381" style="font-size:114%;margin:0 4em"><a href="Nonlinear_programming" title="Nonlinear programming">Unconstrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Function_(mathematics)" title="Function (mathematics)">Functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Golden-section_search" title="Golden-section search">Golden-section search</a></li>

<li><a href="Powell's_method" title="Powell's method">Powell's method</a></li>

<li><a href="Line_search" title="Line search">Line search</a></li>

<li><a href="Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead method</a></li>

<li><a href="Successive_parabolic_interpolation" title="Successive parabolic interpolation">Successive parabolic interpolation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Gradient" title="Gradient">Gradients</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Local_convergence" title="Local convergence">Convergence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Trust_region" title="Trust region">Trust region</a></li>

<li><a href="Wolfe_conditions" title="Wolfe conditions">Wolfe conditions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quasi-Newton_method" title="Quasi-Newton method">Quasi–Newton</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Berndt%E2%80%93Hall%E2%80%93Hall%E2%80%93Hausman_algorithm" title="Berndt–Hall–Hall–Hausman algorithm">Berndt–Hall–Hall–Hausman</a></li>

<li><a href="Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm" title="Broyden–Fletcher–Goldfarb–Shanno algorithm">Broyden–Fletcher–Goldfarb–Shanno</a> and <a href="Limited-memory_BFGS" title="Limited-memory BFGS">L-BFGS</a></li>

<li><a href="Davidon%E2%80%93Fletcher%E2%80%93Powell_formula" title="Davidon–Fletcher–Powell formula">Davidon–Fletcher–Powell</a></li>

<li><a href="Symmetric_rank-one" title="Symmetric rank-one">Symmetric rank-one (SR1)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Iterative_method" title="Iterative method">Other methods</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonlinear_conjugate_gradient_method" title="Nonlinear conjugate gradient method">Conjugate gradient</a></li>

<li><a href="Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton</a></li>

<li><a href="Gradient_descent" title="Gradient descent">Gradient</a></li>

<li><a href="Mirror_descent" title="Mirror descent">Mirror</a></li>

<li><a href="Levenberg%E2%80%93Marquardt_algorithm" title="Levenberg–Marquardt algorithm">Levenberg–Marquardt</a></li>

<li><a href="Powell's_dog_leg_method" title="Powell's dog leg method">Powell's dog leg method</a></li>

<li><a href="Truncated_Newton_method" title="Truncated Newton method">Truncated Newton</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Hessian_matrix" title="Hessian matrix">Hessians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Newton's_method_in_optimization" title="Newton's method in optimization">Newton's method</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Constrained_nonlinear381" style="font-size:114%;margin:0 4em"><a href="Nonlinear_programming" title="Nonlinear programming">Constrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Barrier_function" title="Barrier function">Barrier methods</a></li>

<li><a href="Penalty_method" title="Penalty method">Penalty methods</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Differentiable</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Augmented_Lagrangian_method" title="Augmented Lagrangian method">Augmented Lagrangian methods</a></li>

<li><a href="Sequential_quadratic_programming" title="Sequential quadratic programming">Sequential quadratic programming</a></li>

<li><a href="Successive_linear_programming" title="Successive linear programming">Successive linear programming</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Convex_optimization381" style="font-size:114%;margin:0 4em"><a href="Convex_optimization" title="Convex optimization">Convex optimization</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Convex_minimization" class="mw-redirect" title="Convex minimization">Convex<br> minimization</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cutting-plane_method" title="Cutting-plane method">Cutting-plane method</a></li>

<li><a href="Frank%E2%80%93Wolfe_algorithm" title="Frank–Wolfe algorithm">Reduced gradient (Frank–Wolfe)</a></li>

<li><a href="Subgradient_method" title="Subgradient method">Subgradient method</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Linear_programming" title="Linear programming">Linear</a> and<br><a href="Quadratic_programming" title="Quadratic programming">quadratic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Linear_programming#Interior_point" title="Linear programming">Interior point</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_scaling" title="Affine scaling">Affine scaling</a></li>

<li><a href="Ellipsoid_method" title="Ellipsoid method">Ellipsoid algorithm of Khachiyan</a></li>

<li><a href="Karmarkar's_algorithm" title="Karmarkar's algorithm">Projective algorithm of Karmarkar</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Matroid" title="Matroid">Basis-</a><a href="Exchange_algorithm" class="mw-redirect" title="Exchange algorithm">exchange</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Simplex_algorithm" title="Simplex algorithm">Simplex algorithm of Dantzig</a></li>

<li><a href="Revised_simplex_method" title="Revised simplex method">Revised simplex algorithm</a></li>

<li><a href="Criss-cross_algorithm" title="Criss-cross algorithm">Criss-cross algorithm</a></li>

<li><a href="Lemke's_algorithm" title="Lemke's algorithm">Principal pivoting algorithm of Lemke</a></li>

<li><a href="Active-set_method" title="Active-set method">Active-set method</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial381" style="font-size:114%;margin:0 4em"><a href="Combinatorial_optimization" title="Combinatorial optimization">Combinatorial</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Paradigms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approximation_algorithm" title="Approximation algorithm">Approximation algorithm</a></li>

<li><a href="Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li>

<li><a href="Greedy_algorithm" title="Greedy algorithm">Greedy algorithm</a></li>

<li><a href="Integer_programming" title="Integer programming">Integer programming</a>

<ul><li><a href="Branch_and_bound" title="Branch and bound">Branch and bound</a>/<a href="Branch_and_cut" title="Branch and cut">cut</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Graph_algorithm" class="mw-redirect" title="Graph algorithm">Graph<br> algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Minimum_spanning_tree52" scope="row" class="navbox-group" style="width:1%"><a href="Minimum_spanning_tree" title="Minimum spanning tree">Minimum<br> spanning tree</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bor%C5%AFvka's_algorithm" title="Borůvka's algorithm">Borůvka</a></li>

<li><a href="Kruskal's_algorithm" title="Kruskal's algorithm">Kruskal</a></li></ul>

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

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Shortest_path39" scope="row" class="navbox-group" style="width:1%"><a href="Shortest_path_problem" title="Shortest path problem">Shortest path</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford</a>

<li><a href="Dijkstra's_algorithm" title="Dijkstra's algorithm">Dijkstra</a></li>

</ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Flow_network" title="Flow network">Network flows</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dinic's_algorithm" title="Dinic's algorithm">Dinic</a></li>

<li><a href="Edmonds%E2%80%93Karp_algorithm" title="Edmonds–Karp algorithm">Edmonds–Karp</a></li>

<li><a href="Ford%E2%80%93Fulkerson_algorithm" title="Ford–Fulkerson algorithm">Ford–Fulkerson</a></li>

<li><a href="Push%E2%80%93relabel_maximum_flow_algorithm" title="Push–relabel maximum flow algorithm">Push–relabel maximum flow</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Metaheuristics381" style="font-size:114%;margin:0 4em"><a href="Metaheuristic" title="Metaheuristic">Metaheuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithm</a></li>

<li><a href="Hill_climbing" title="Hill climbing">Hill climbing</a></li>

<li><a href="Local_search_(optimization)" title="Local search (optimization)">Local search</a></li>

<li><a href="Parallel_metaheuristic" title="Parallel metaheuristic">Parallel metaheuristics</a></li>

<li><a href="Simulated_annealing" title="Simulated annealing">Simulated annealing</a></li>

<li><a href="Spiral_optimization_algorithm" title="Spiral optimization algorithm">Spiral optimization algorithm</a></li>

<li><a href="Tabu_search" title="Tabu search">Tabu search</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li><a href="Comparison_of_optimization_software" title="Comparison of optimization software">Software</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-05-23" href="https://en.wikipedia.org/wiki/?title=Floyd%E2%80%93Warshall_algorithm&oldid=1291762042">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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