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href="https://cs.wikipedia.org/wiki/Edmonds%C5%AFv%E2%80%93Karp%C5%AFv_algoritmus" title="Edmondsův–Karpův algoritmus – Czech" lang="cs" hreflang="cs" data-title="Edmondsův–Karpův algoritmus" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Algorithmus_von_Edmonds_und_Karp" title="Algorithmus von Edmonds und Karp – German" lang="de" hreflang="de" data-title="Algorithmus von Edmonds und Karp" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Algoritmo_de_Edmonds-Karp" title="Algoritmo de Edmonds-Karp – Spanish" lang="es" hreflang="es" data-title="Algoritmo de Edmonds-Karp" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%84%DA%AF%D9%88%D8%B1%DB%8C%D8%AA%D9%85_%D8%A7%D8%AF%D9%85%D9%88%D9%86%D8%AF%D8%B2_%DA%A9%D8%A7%D8%B1%D9%BE" title="الگوریتم ادموندز کارپ – Persian" lang="fa" hreflang="fa" data-title="الگوریتم ادموندز کارپ" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Algorithme_d%27Edmonds-Karp" title="Algorithme d'Edmonds-Karp – French" lang="fr" hreflang="fr" data-title="Algorithme d'Edmonds-Karp" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A8%E3%83%89%E3%83%A2%E3%83%B3%E3%82%BA%E3%83%BB%E3%82%AB%E3%83%BC%E3%83%97%E3%81%AE%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0" title="エドモンズ・カープのアルゴリズム – Japanese" lang="ja" hreflang="ja" data-title="エドモンズ・カープのアルゴリズム" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algorytm_Edmondsa-Karpa" title="Algorytm Edmondsa-Karpa – Polish" lang="pl" hreflang="pl" data-title="Algorytm Edmondsa-Karpa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Algoritmo_de_Edmonds-Karp" title="Algoritmo de Edmonds-Karp – Portuguese" lang="pt" hreflang="pt" data-title="Algoritmo de Edmonds-Karp" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Algoritmul_Edmonds-Karp" title="Algoritmul Edmonds-Karp – Romanian" lang="ro" hreflang="ro" data-title="Algoritmul Edmonds-Karp" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC_%D0%AD%D0%B4%D0%BC%D0%BE%D0%BD%D0%B4%D1%81%D0%B0_%E2%80%94_%D0%9A%D0%B0%D1%80%D0%BF%D0%B0" title="Алгоритм Эдмондса — Карпа – Russian" lang="ru" hreflang="ru" data-title="Алгоритм Эдмондса — Карпа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%B4%D0%BC%D0%BE%D0%BD%D0%B4%D1%81%E2%80%93%D0%9A%D0%B0%D1%80%D0%BF%D0%BE%D0%B2_%D0%B0%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Едмондс–Карпов алгоритам – Serbian" lang="sr" hreflang="sr" data-title="Едмондс–Карпов алгоритам" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%82%E0%B8%B1%E0%B9%89%E0%B8%99%E0%B8%95%E0%B8%AD%E0%B8%99%E0%B8%A7%E0%B8%B4%E0%B8%98%E0%B8%B5%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%80%E0%B8%AD%E0%B9%87%E0%B8%94%E0%B8%A1%E0%B8%AD%E0%B8%99%E0%B8%94%E0%B9%8C-%E0%B8%84%E0%B8%B2%E0%B8%9B" title="ขั้นตอนวิธีของเอ็ดมอนด์-คาป – Thai" lang="th" hreflang="th" data-title="ขั้นตอนวิธีของเอ็ดมอนด์-คาป" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC_%D0%95%D0%B4%D0%BC%D0%BE%D0%BD%D0%B4%D1%81%D0%B0_%E2%80%94_%D0%9A%D0%B0%D1%80%D0%BF%D0%B0" title="Алгоритм Едмондса — Карпа – Ukrainian" lang="uk" hreflang="uk" data-title="Алгоритм Едмондса — Карпа" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Thu%E1%BA%ADt_to%C3%A1n_Edmonds%E2%80%93Karp" title="Thuật toán Edmonds–Karp – Vietnamese" lang="vi" hreflang="vi" data-title="Thuật toán Edmonds–Karp" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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href="/wiki/Ford%E2%80%93Fulkerson_algorithm" title="Ford–Fulkerson algorithm">Ford–Fulkerson method</a> for computing the <a href="/wiki/Maximum_flow_problem" title="Maximum flow problem">maximum flow</a> in a <a href="/wiki/Flow_network" title="Flow network">flow network</a> in <a href="/wiki/Big_O_notation" title="Big O notation"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(|V||E|^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(|V||E|^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5bc5dfaecce53a04efd47719ac640aa983e706" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.787ex; height:3.343ex;" alt="{\displaystyle O(|V||E|^{2})}"></span></a> time. The algorithm was first published by <a href="/wiki/Yefim_Dinitz" title="Yefim Dinitz">Yefim Dinitz</a> in 1970,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ipv_2-0" class="reference"><a href="#cite_note-ipv-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and independently published by <a href="/wiki/Jack_Edmonds" title="Jack Edmonds">Jack Edmonds</a> and <a href="/wiki/Richard_Karp" class="mw-redirect" title="Richard Karp">Richard Karp</a> in 1972.<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> <a href="/wiki/Dinic%27s_algorithm" title="Dinic's algorithm">Dinitz's algorithm</a> includes additional techniques that reduce the running time to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(|V|^{2}|E|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(|V|^{2}|E|)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaab3df4a2120846d5c6cc1619ed39628ad8d80d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.787ex; height:3.343ex;" alt="{\displaystyle O(|V|^{2}|E|)}"></span>.<sup id="cite_ref-ipv_2-1" class="reference"><a href="#cite_note-ipv-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style></p><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></span></span></div> <div class="side-box-text plainlist">The Wikibook <i><a href="https://en.wikibooks.org/wiki/Algorithm_implementation" class="extiw" title="wikibooks:Algorithm implementation">Algorithm implementation</a></i> has a page on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Algorithm_implementation/Graphs/Maximum_flow/Edmonds-Karp" class="extiw" title="wikibooks:Algorithm implementation/Graphs/Maximum flow/Edmonds-Karp">Edmonds-Karp</a></b></i></div></div> </div> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edmonds%E2%80%93Karp_algorithm&action=edit&section=1" title="Edit section: Algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The algorithm is identical to the <a href="/wiki/Ford%E2%80%93Fulkerson_algorithm" title="Ford–Fulkerson algorithm">Ford–Fulkerson algorithm</a>, except that the search order when finding the <a href="/wiki/Flow_network#Augmenting_paths" title="Flow network">augmenting path</a> is defined. The path found must be a shortest path that has available capacity. This can be found by a <a href="/wiki/Breadth-first_search" title="Breadth-first search">breadth-first search</a>, where we apply a weight of 1 to each edge. The running time of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(|V||E|^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(|V||E|^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5bc5dfaecce53a04efd47719ac640aa983e706" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.787ex; height:3.343ex;" alt="{\displaystyle O(|V||E|^{2})}"></span> is found by showing that each augmenting path can be found in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(|E|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(|E|)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/976fe7f1e011d0dcdb3d6163754c877aaad5187f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.652ex; height:2.843ex;" alt="{\displaystyle O(|E|)}"></span> time, that every time at least one of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> edges becomes saturated (an edge which has the maximum possible flow), that the distance from the saturated edge to the source along the augmenting path must be longer than last time it was saturated, and that the length is at most <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |V|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ddcffc28643ac01a14dd0fb32c3157859e365a7" 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|}"></span>. Another property of this algorithm is that the length of the shortest augmenting path increases monotonically. There is an accessible proof in <i><a href="/wiki/Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a></i>.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Pseudocode">Pseudocode</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edmonds%E2%80%93Karp_algorithm&action=edit&section=2" title="Edit section: Pseudocode"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <pre><b>algorithm</b> EdmondsKarp <b>is</b> <b>input</b>: graph <i>(graph[v] should be the list of edges coming out of vertex v in the</i> <i> original graph <b>and</b> their corresponding constructed reverse edges</i> <i> which are used for push-back flow.</i> <i> Each edge should have a capacity 'cap', flow, source 's' and sink 't' </i> <i> as parameters, as well as a pointer to the reverse edge 'rev'.)</i> s <i>(Source vertex)</i> t <i>(Sink vertex)</i> <b>output</b>: flow <i>(Value of maximum flow)</i> flow := 0 <i>(Initialize flow to zero)</i> <b>repeat</b> <i>(Run a breadth-first search (bfs) to find the shortest s-t path.</i> <i> We use 'pred' to store the edge taken to get to each vertex,</i> <i> so we can recover the path afterwards)</i> q := <b>queue</b>() q.push(s) pred := <b>array</b>(graph.length) <b>while</b> <b>not</b> empty(q) <b>and</b> pred[t] = null cur := q.pop() <b>for</b> Edge e <b>in</b> graph[cur] <b>do</b> <b>if</b> pred[e.t] = <b>null</b> <b>and</b> e.t ≠ s <b>and</b> e.cap > e.flow <b>then</b> pred[e.t] := e q.push(e.t) <b>if</b> <b>not</b> (pred[t] = null) <b>then</b> <i>(We found an augmenting path.</i> <i> See how much flow we can send)</i> df := <b>∞</b> <b>for</b> (e := pred[t]; e ≠ null; e := pred[e.s]) <b>do</b> df := <b>min</b>(df, e.cap - e.flow) <i>(And update edges by that amount)</i> <b>for</b> (e := pred[t]; e ≠ null; e := pred[e.s]) <b>do</b> e.flow  := e.flow + df e.rev.flow := e.rev.flow - df flow := flow + df <b>until</b> pred[t] = null <i>(i.e., until no augmenting path was found)</i> <b>return</b> flow </pre> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edmonds%E2%80%93Karp_algorithm&action=edit&section=3" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a network of seven nodes, source A, sink G, and capacities as shown below: </p><p><span typeof="mw:File"><a href="/wiki/File:Edmonds-Karp_flow_example_0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Edmonds-Karp_flow_example_0.svg/300px-Edmonds-Karp_flow_example_0.svg.png" decoding="async" width="300" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Edmonds-Karp_flow_example_0.svg/450px-Edmonds-Karp_flow_example_0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Edmonds-Karp_flow_example_0.svg/600px-Edmonds-Karp_flow_example_0.svg.png 2x" data-file-width="750" data-file-height="400" /></a></span> </p><p>In the pairs <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f/c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb831cd04ec8b60f900a75341088eb25ac2f255a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.448ex; height:2.843ex;" alt="{\displaystyle f/c}"></span> written on the edges, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is the current flow, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> is the capacity. The residual capacity from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{f}(u,v)=c(u,v)-f(u,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{f}(u,v)=c(u,v)-f(u,v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2756fd42d49a52b35b33bfe377a74dcbdf1f4b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.269ex; height:3.009ex;" alt="{\displaystyle c_{f}(u,v)=c(u,v)-f(u,v)}"></span>, the total capacity, minus the flow that is already used. If the net flow from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> is negative, it <i>contributes</i> to the residual capacity. </p> <table class="wikitable"> <tbody><tr> <th>Path </th> <th>Capacity </th> <th>Resulting network </th></tr> <tr> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,D,E,G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>D</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,D,E,G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ec78807332449f32f1a6df8d99a2b62a9ea124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.372ex; height:2.509ex;" alt="{\displaystyle A,D,E,G}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\min(c_{f}(A,D),c_{f}(D,E),c_{f}(E,G))\\=&\min(3-0,2-0,1-0)\\=&\min(3,2,1)=1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>−</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\min(c_{f}(A,D),c_{f}(D,E),c_{f}(E,G))\\=&\min(3-0,2-0,1-0)\\=&\min(3,2,1)=1\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/084c31fb2397580de7841dcf4c301ab3893b4b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:36.628ex; height:9.343ex;" alt="{\displaystyle {\begin{aligned}&\min(c_{f}(A,D),c_{f}(D,E),c_{f}(E,G))\\=&\min(3-0,2-0,1-0)\\=&\min(3,2,1)=1\end{aligned}}}"></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Edmonds-Karp_flow_example_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Edmonds-Karp_flow_example_1.svg/300px-Edmonds-Karp_flow_example_1.svg.png" decoding="async" width="300" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Edmonds-Karp_flow_example_1.svg/450px-Edmonds-Karp_flow_example_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Edmonds-Karp_flow_example_1.svg/600px-Edmonds-Karp_flow_example_1.svg.png 2x" data-file-width="750" data-file-height="400" /></a></span></td> </tr> <tr> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,D,F,G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>D</mi> <mo>,</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,D,F,G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33aabff2634e707fab59858933d8118a45af1897" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:2.509ex;" alt="{\displaystyle A,D,F,G}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\min(c_{f}(A,D),c_{f}(D,F),c_{f}(F,G))\\=&\min(3-1,6-0,9-0)\\=&\min(2,6,9)=2\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo>−</mo> <mn>0</mn> <mo>,</mo> <mn>9</mn> <mo>−</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\min(c_{f}(A,D),c_{f}(D,F),c_{f}(F,G))\\=&\min(3-1,6-0,9-0)\\=&\min(2,6,9)=2\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aca1c62b46a99a47b53ae51dfec6bc635f0b66f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:36.558ex; height:9.343ex;" alt="{\displaystyle {\begin{aligned}&\min(c_{f}(A,D),c_{f}(D,F),c_{f}(F,G))\\=&\min(3-1,6-0,9-0)\\=&\min(2,6,9)=2\end{aligned}}}"></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Edmonds-Karp_flow_example_2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Edmonds-Karp_flow_example_2.svg/300px-Edmonds-Karp_flow_example_2.svg.png" decoding="async" width="300" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Edmonds-Karp_flow_example_2.svg/450px-Edmonds-Karp_flow_example_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Edmonds-Karp_flow_example_2.svg/600px-Edmonds-Karp_flow_example_2.svg.png 2x" data-file-width="750" data-file-height="400" /></a></span></td> </tr> <tr> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C,D,F,G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo>,</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C,D,F,G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86e30ae5b6757df20348100af8dc6af64ec2730d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.935ex; height:2.509ex;" alt="{\displaystyle A,B,C,D,F,G}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\min(c_{f}(A,B),c_{f}(B,C),c_{f}(C,D),c_{f}(D,F),c_{f}(F,G))\\=&\min(3-0,4-0,1-0,6-2,9-2)\\=&\min(3,4,1,4,7)=1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>0</mn> <mo>,</mo> <mn>4</mn> <mo>−</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mn>0</mn> <mo>,</mo> <mn>6</mn> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>9</mn> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\min(c_{f}(A,B),c_{f}(B,C),c_{f}(C,D),c_{f}(D,F),c_{f}(F,G))\\=&\min(3-0,4-0,1-0,6-2,9-2)\\=&\min(3,4,1,4,7)=1\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adcba9bad73845e03b73febd70fdc161a32faabe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:55.66ex; height:9.343ex;" alt="{\displaystyle {\begin{aligned}&\min(c_{f}(A,B),c_{f}(B,C),c_{f}(C,D),c_{f}(D,F),c_{f}(F,G))\\=&\min(3-0,4-0,1-0,6-2,9-2)\\=&\min(3,4,1,4,7)=1\end{aligned}}}"></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Edmonds-Karp_flow_example_3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Edmonds-Karp_flow_example_3.svg/300px-Edmonds-Karp_flow_example_3.svg.png" decoding="async" width="300" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Edmonds-Karp_flow_example_3.svg/450px-Edmonds-Karp_flow_example_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Edmonds-Karp_flow_example_3.svg/600px-Edmonds-Karp_flow_example_3.svg.png 2x" data-file-width="750" data-file-height="400" /></a></span></td> </tr> <tr> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C,E,D,F,G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>D</mi> <mo>,</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C,E,D,F,G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8db2259ed0b4e0c71e251b6150284aaf230b5796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.744ex; height:2.509ex;" alt="{\displaystyle A,B,C,E,D,F,G}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\min(c_{f}(A,B),c_{f}(B,C),c_{f}(C,E),c_{f}(E,D),c_{f}(D,F),c_{f}(F,G))\\=&\min(3-1,4-1,2-0,0-(-1),6-3,9-3)\\=&\min(2,3,2,1,3,6)=1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>−</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mn>6</mn> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>9</mn> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\min(c_{f}(A,B),c_{f}(B,C),c_{f}(C,E),c_{f}(E,D),c_{f}(D,F),c_{f}(F,G))\\=&\min(3-1,4-1,2-0,0-(-1),6-3,9-3)\\=&\min(2,3,2,1,3,6)=1\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13c57f980264e14693eb1836b1ceddcdf05a9778" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:65.231ex; height:9.343ex;" alt="{\displaystyle {\begin{aligned}&\min(c_{f}(A,B),c_{f}(B,C),c_{f}(C,E),c_{f}(E,D),c_{f}(D,F),c_{f}(F,G))\\=&\min(3-1,4-1,2-0,0-(-1),6-3,9-3)\\=&\min(2,3,2,1,3,6)=1\end{aligned}}}"></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Edmonds-Karp_flow_example_4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Edmonds-Karp_flow_example_4.svg/300px-Edmonds-Karp_flow_example_4.svg.png" decoding="async" width="300" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Edmonds-Karp_flow_example_4.svg/450px-Edmonds-Karp_flow_example_4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Edmonds-Karp_flow_example_4.svg/600px-Edmonds-Karp_flow_example_4.svg.png 2x" data-file-width="750" data-file-height="400" /></a></span></td> </tr></tbody></table> <p>Notice how the length of the <a href="/wiki/Augmenting_path" class="mw-redirect" title="Augmenting path">augmenting path</a> found by the algorithm (in red) never decreases. The paths found are the shortest possible. The flow found is equal to the capacity across the <a href="/wiki/Max_flow_min_cut_theorem" class="mw-redirect" title="Max flow min cut theorem">minimum cut</a> in the graph separating the source and the sink. There is only one minimal cut in this graph, partitioning the nodes into the sets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A,B,C,E\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>E</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{A,B,C,E\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f28eddde1d9c1beab81b3393dd658d4efa52838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.476ex; height:2.843ex;" alt="{\displaystyle \{A,B,C,E\}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{D,F,G\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>D</mi> <mo>,</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{D,F,G\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b955c650f55ab75803870a686ecb6f0297dcb0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.885ex; height:2.843ex;" alt="{\displaystyle \{D,F,G\}}"></span>, with the capacity </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(A,D)+c(C,D)+c(E,G)=3+1+1=5.\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>5.</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(A,D)+c(C,D)+c(E,G)=3+1+1=5.\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be3d4be34cfb4d1c776eded127a99fcbe92aef48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.946ex; height:2.843ex;" alt="{\displaystyle c(A,D)+c(C,D)+c(E,G)=3+1+1=5.\ }"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edmonds%E2%80%93Karp_algorithm&action=edit&section=4" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDinic1970" class="citation journal cs1"><a href="/wiki/Yefim_Dinitz" title="Yefim Dinitz">Dinic, E. A.</a> (1970). "Algorithm for solution of a problem of maximum flow in a network with power estimation". <i>Soviet Mathematics - Doklady</i>. <b>11</b>. Doklady: 1277–1280.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Soviet+Mathematics+-+Doklady&rft.atitle=Algorithm+for+solution+of+a+problem+of+maximum+flow+in+a+network+with+power+estimation&rft.volume=11&rft.pages=1277-1280&rft.date=1970&rft.aulast=Dinic&rft.aufirst=E.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEdmonds%E2%80%93Karp+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-ipv-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-ipv_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ipv_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYefim_Dinitz2006" class="citation book cs1">Yefim Dinitz (2006). <a rel="nofollow" class="external text" href="https://rangevoting.org/Dinitz_alg.pdf">"Dinitz' Algorithm: The Original Version and Even's Version"</a> <span class="cs1-format">(PDF)</span>. In <a href="/wiki/Oded_Goldreich" title="Oded Goldreich">Oded Goldreich</a>; Arnold L. Rosenberg; <a href="/wiki/Alan_Selman" title="Alan Selman">Alan L. Selman</a> (eds.). <i>Theoretical Computer Science: Essays in Memory of <a href="/wiki/Shimon_Even" title="Shimon Even">Shimon Even</a></i>. Springer. pp. 218–240. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-32880-3" title="Special:BookSources/978-3-540-32880-3"><bdi>978-3-540-32880-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Dinitz%27+Algorithm%3A+The+Original+Version+and+Even%27s+Version&rft.btitle=Theoretical+Computer+Science%3A+Essays+in+Memory+of+Shimon+Even&rft.pages=218-240&rft.pub=Springer&rft.date=2006&rft.isbn=978-3-540-32880-3&rft.au=Yefim+Dinitz&rft_id=https%3A%2F%2Frangevoting.org%2FDinitz_alg.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEdmonds%E2%80%93Karp+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdmondsKarp1972" class="citation journal cs1"><a href="/wiki/Jack_Edmonds" title="Jack Edmonds">Edmonds, Jack</a>; <a href="/wiki/Richard_Karp" class="mw-redirect" title="Richard Karp">Karp, Richard M.</a> (1972). <a rel="nofollow" class="external text" href="http://www.eecs.umich.edu/%7Epettie/matching/Edmonds-Karp-network-flow.pdf">"Theoretical improvements in algorithmic efficiency for network flow problems"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of the ACM</i>. <b>19</b> (2): 248–264. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F321694.321699">10.1145/321694.321699</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6375478">6375478</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+ACM&rft.atitle=Theoretical+improvements+in+algorithmic+efficiency+for+network+flow+problems&rft.volume=19&rft.issue=2&rft.pages=248-264&rft.date=1972&rft_id=info%3Adoi%2F10.1145%2F321694.321699&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6375478%23id-name%3DS2CID&rft.aulast=Edmonds&rft.aufirst=Jack&rft.au=Karp%2C+Richard+M.&rft_id=http%3A%2F%2Fwww.eecs.umich.edu%2F%257Epettie%2Fmatching%2FEdmonds-Karp-network-flow.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEdmonds%E2%80%93Karp+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomas_H._Cormen,_Charles_E._Leiserson,_Ronald_L._Rivest_and_Clifford_Stein2009" class="citation book cs1"><a href="/wiki/Thomas_H._Cormen" title="Thomas H. Cormen">Thomas H. Cormen</a>, <a href="/wiki/Charles_E._Leiserson" title="Charles E. Leiserson">Charles E. Leiserson</a>, <a href="/wiki/Ronald_L._Rivest" class="mw-redirect" title="Ronald L. Rivest">Ronald L. Rivest</a> and <a href="/wiki/Clifford_Stein" title="Clifford Stein">Clifford Stein</a> (2009). "26.2". <a href="/wiki/Introduction_to_Algorithms" title="Introduction to Algorithms"><i>Introduction to Algorithms</i></a> (third ed.). MIT Press. pp. 727–730. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-03384-8" title="Special:BookSources/978-0-262-03384-8"><bdi>978-0-262-03384-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=26.2&rft.btitle=Introduction+to+Algorithms&rft.pages=727-730&rft.edition=third&rft.pub=MIT+Press&rft.date=2009&rft.isbn=978-0-262-03384-8&rft.au=Thomas+H.+Cormen%2C+Charles+E.+Leiserson%2C+Ronald+L.+Rivest+and+Clifford+Stein&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEdmonds%E2%80%93Karp+algorithm" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edmonds%E2%80%93Karp_algorithm&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>Algorithms and Complexity (see pages 63–69). <a rel="nofollow" class="external free" href="https://web.archive.org/web/20061005083406/http://www.cis.upenn.edu/~wilf/AlgComp3.html">https://web.archive.org/web/20061005083406/http://www.cis.upenn.edu/~wilf/AlgComp3.html</a></li></ol> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output 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href="/wiki/Template:Optimization_algorithms" title="Template:Optimization algorithms"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Optimization_algorithms" title="Template talk:Optimization algorithms"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Optimization_algorithms" title="Special:EditPage/Template:Optimization algorithms"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Optimization:_Algorithms,_methods,_and_heuristics" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a>: <a href="/wiki/Optimization_algorithm" class="mw-redirect" title="Optimization algorithm">Algorithms</a>, <a href="/wiki/Iterative_method" title="Iterative method">methods</a>, and <a href="/wiki/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_nonlinear" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/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="/wiki/Golden-section_search" title="Golden-section search">Golden-section search</a></li> <li><a href="/wiki/Powell%27s_method" title="Powell's method">Powell's method</a></li> <li><a href="/wiki/Line_search" title="Line search">Line search</a></li> <li><a href="/wiki/Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead method</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/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="/wiki/Trust_region" title="Trust region">Trust region</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/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="/wiki/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="/wiki/Limited-memory_BFGS" title="Limited-memory BFGS">L-BFGS</a></li> <li><a href="/wiki/Davidon%E2%80%93Fletcher%E2%80%93Powell_formula" title="Davidon–Fletcher–Powell formula">Davidon–Fletcher–Powell</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Nonlinear_conjugate_gradient_method" title="Nonlinear conjugate gradient method">Conjugate gradient</a></li> <li><a href="/wiki/Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton</a></li> <li><a href="/wiki/Gradient_descent" title="Gradient descent">Gradient</a></li> <li><a href="/wiki/Mirror_descent" title="Mirror descent">Mirror</a></li> <li><a href="/wiki/Levenberg%E2%80%93Marquardt_algorithm" title="Levenberg–Marquardt algorithm">Levenberg–Marquardt</a></li> <li><a href="/wiki/Powell%27s_dog_leg_method" title="Powell's dog leg method">Powell's dog leg method</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Newton%27s_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><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Max_paraboloid.svg" class="mw-file-description" title="Optimization computes maxima and minima."><img alt="Graph of a strictly concave quadratic function with unique maximum." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/150px-Max_paraboloid.svg.png" decoding="async" width="150" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/225px-Max_paraboloid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/300px-Max_paraboloid.svg.png 2x" data-file-width="700" data-file-height="560" /></a><figcaption>Optimization computes maxima and minima.</figcaption></figure></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_nonlinear" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/Barrier_function" title="Barrier function">Barrier methods</a></li> <li><a href="/wiki/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="/wiki/Augmented_Lagrangian_method" title="Augmented Lagrangian method">Augmented Lagrangian methods</a></li> <li><a href="/wiki/Sequential_quadratic_programming" title="Sequential quadratic programming">Sequential quadratic programming</a></li> <li><a href="/wiki/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_optimization" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/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="/wiki/Cutting-plane_method" title="Cutting-plane method">Cutting-plane method</a></li> <li><a href="/wiki/Frank%E2%80%93Wolfe_algorithm" title="Frank–Wolfe algorithm">Reduced gradient (Frank–Wolfe)</a></li> <li><a href="/wiki/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="/wiki/Linear_programming" title="Linear programming">Linear</a> and<br /><a href="/wiki/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="/wiki/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="/wiki/Affine_scaling" title="Affine scaling">Affine scaling</a></li> <li><a href="/wiki/Ellipsoid_method" title="Ellipsoid method">Ellipsoid algorithm of Khachiyan</a></li> <li><a href="/wiki/Karmarkar%27s_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="/wiki/Matroid" title="Matroid">Basis-</a><a href="/wiki/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="/wiki/Simplex_algorithm" title="Simplex algorithm">Simplex algorithm of Dantzig</a></li> <li><a href="/wiki/Revised_simplex_method" title="Revised simplex method">Revised simplex algorithm</a></li> <li><a href="/wiki/Criss-cross_algorithm" title="Criss-cross algorithm">Criss-cross algorithm</a></li> <li><a href="/wiki/Lemke%27s_algorithm" title="Lemke's algorithm">Principal pivoting algorithm of Lemke</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="Combinatorial" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/Approximation_algorithm" title="Approximation algorithm">Approximation algorithm</a></li> <li><a href="/wiki/Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li> <li><a href="/wiki/Greedy_algorithm" title="Greedy algorithm">Greedy algorithm</a></li> <li><a href="/wiki/Integer_programming" title="Integer programming">Integer programming</a> <ul><li><a href="/wiki/Branch_and_bound" title="Branch and bound">Branch and bound</a>/<a href="/wiki/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="/wiki/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_tree" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/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="/wiki/Bor%C5%AFvka%27s_algorithm" title="Borůvka's algorithm">Borůvka</a></li> <li><a href="/wiki/Prim%27s_algorithm" title="Prim's algorithm">Prim</a></li> <li><a href="/wiki/Kruskal%27s_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_path" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/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="/wiki/Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford</a> <ul><li><a href="/wiki/Shortest_Path_Faster_Algorithm" class="mw-redirect" title="Shortest Path Faster Algorithm">SPFA</a></li></ul></li> <li><a href="/wiki/Dijkstra%27s_algorithm" title="Dijkstra's algorithm">Dijkstra</a></li> <li><a href="/wiki/Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/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="/wiki/Dinic%27s_algorithm" title="Dinic's algorithm">Dinic</a></li> <li><a class="mw-selflink selflink">Edmonds–Karp</a></li> <li><a href="/wiki/Ford%E2%80%93Fulkerson_algorithm" title="Ford–Fulkerson algorithm">Ford–Fulkerson</a></li> <li><a href="/wiki/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="Metaheuristics" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithm</a></li> <li><a href="/wiki/Hill_climbing" title="Hill climbing">Hill climbing</a></li> <li><a href="/wiki/Local_search_(optimization)" title="Local search (optimization)">Local search</a></li> <li><a href="/wiki/Parallel_metaheuristic" title="Parallel metaheuristic">Parallel metaheuristics</a></li> <li><a href="/wiki/Simulated_annealing" title="Simulated annealing">Simulated annealing</a></li> <li><a href="/wiki/Spiral_optimization_algorithm" title="Spiral optimization algorithm">Spiral optimization algorithm</a></li> <li><a href="/wiki/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 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