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{{Short description|Algorithm for maximum cardinality matching}}
{{Infobox algorithm|image = |data = [[Graph (data structure)|Graph]]|time = <math>O(E \sqrt V)</math>|class = Graph algorithm|space = <math>O(V)</math>}}
In [[computer science]], the '''Hopcroft–Karp algorithm''' (sometimes more accurately called the '''Hopcroft–Karp–Karzanov algorithm''')<ref>{{harvtxt|Gabow|2017}}; {{harvtxt|Annamalai|2018}}</ref> is an [[algorithm]] that takes a [[bipartite graph]] as input and produces a [[maximum-cardinality matching]] as output — a set of as many edges as possible with the property that no two edges share an endpoint. It runs in <math>O(|E|\sqrt{|V|})</math> time in the [[worst case]], where <math>E</math> is set of edges in the graph, <math>V</math> is set of vertices of the graph, and it is assumed that <math>|E|=\Omega(|V|)</math>. In the case of [[dense graph]]s the time bound becomes <math>O(|V|^{2.5})</math>, and for sparse [[random graph]]s it runs in time <math>O(|E|\log |V|)</math> with high probability.{{sfnp|Bast|Mehlhorn|Schäfer|Tamaki|2006}} The algorithm was discovered by {{harvs|first1=John|last1=Hopcroft|author1-link=John Hopcroft|first2=Richard|last2=Karp|author2-link=Richard Karp|txt|year=1973}} and independently by {{harvs|first=Alexander|last=Karzanov|authorlink=Alexander V. Karzanov|year=1973|txt}}.{{sfnp|Dinitz|2006}} As in previous methods for matching such as the [[Hungarian algorithm]] and the work of {{harvtxt|Edmonds|1965}}, the Hopcroft–Karp algorithm repeatedly increases the size of a partial matching by finding ''augmenting paths''. These paths are sequences of edges of the graph, which alternate between edges in the matching and edges out of the partial matching, and where the initial and final edge are not in the partial matching. Finding an augmenting path allows us to increment the size of the partial matching, by simply toggling the edges of the augmenting path (putting in the partial matching those that were not, and vice versa). Simpler algorithms for bipartite matching, such as the [[Ford–Fulkerson algorithm]]‚ find one augmenting path per iteration: the Hopcroft-Karp algorithm instead finds a maximal set of shortest augmenting paths, so as to ensure that only <math>O(\sqrt{|V|})</math> iterations are needed instead of <math>O(|V|)</math> iterations. The same performance of <math>O(|E|\sqrt{|V|})</math> can be achieved to find maximum-cardinality matchings in arbitrary graphs, with the more complicated algorithm of Micali and Vazirani.{{sfnp|Peterson|Loui|1988}}
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