Hopcroft–Karp algorithm: Difference between revisions

Revision as of 17:03, 10 November 2007

Revision as of 17:04, 10 November 2007

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The '''Hopcroft–Karp algorithm''' finds maximum cardinality [[matching]]s in [[bipartite graph]]s in <math>O(\sqrt{V} E)</math> time (another estimation is ''O(V ^5/2)'' <ref>Lipski W., Kombinatoryka dla Programistow, [Combinatorics for Programmers], Wydawnictwa Naukowo-Techniczne, Warzawa, 1982.</ref>). It is an adaptation of the [[Edmonds-Karp algorithm]] for maximum flow, since bipartite matching is equivalent to finding the maximum (integer) flow if the vertices in each partition are considered sources (respectively sinks) of capacity 1. The [[Max-flow min-cut theorem|minimal cut]] associated with the flow is equivalent to the [[König's theorem (graph theory)|minimal vertex cover]] associated with the matching.

== Algorithm ==