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The algorithm was discovered by John Hopcroft and Richard Karp and independently by Alexander Karzanov . [3] As in previous methods for matching such as the Hungarian algorithm and the work of Edmonds (1965), the Hopcroft–Karp algorithm repeatedly increases the size of a partial matching by finding augmenting paths. These paths are sequences ...
A better performance of O(√ V E) for general graphs, matching the performance of the Hopcroft–Karp algorithm on bipartite graphs, can be achieved with the much more complicated algorithm of Micali and Vazirani. [5] The same bound was achieved by an algorithm by Blum [6] and an algorithm by Gabow and Tarjan. [7]
Richard Manning Karp (born January 3, 1935) is an American computer scientist and computational theorist at the University of California, Berkeley.He is most notable for his research in the theory of algorithms, for which he received a Turing Award in 1985, The Benjamin Franklin Medal in Computer and Cognitive Science in 2004, and the Kyoto Prize in 2008.
The problem is solved by the Hopcroft-Karp algorithm in time O(√ V E) time, and there are more efficient randomized algorithms, approximation algorithms, and algorithms for special classes of graphs such as bipartite planar graphs, as described in the main article.
John Edward Hopcroft (born October 7, 1939) is an American theoretical computer scientist. His textbooks on theory of computation (also known as the Cinderella book ) and data structures are regarded as standards in their fields.
The resulting algorithm is also known as Hopcroft–Karp algorithm. More generally, this bound holds for any unit network — a network in which each vertex, except for source and sink, either has a single entering edge of capacity one, or a single outgoing edge of capacity one, and all other capacities are arbitrary integers. [3]
He is the inventor of preflow-push based algorithms for the maximum flow problem, and the co-inventor of the Hopcroft–Karp–Karzanov algorithm for maximum matching in bipartite graphs. [2] He is a chief researcher at the Federal Research Center "Computer Science and Control" (Institute for System Analysis) of the Russian Academy of Sciences. [3]
It is easy to check that a matching of a directed graph composes of a set of vertex-disjoint simple paths and cycles. The maximum matching of a directed network can be efficiently calculated by working in the bipartite representation using the classical Hopcroft–Karp algorithm, which runs in O(E √ N) time in the worst case