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The following example shows the first steps of Ford–Fulkerson in a flow network with 4 nodes, source and sink . This example shows the worst-case behaviour of the algorithm. In each step, only a flow of is sent across the network. If breadth-first-search were used instead, only two steps would be needed.
The simplest way to compute a maximum cardinality matching is to follow the Ford–Fulkerson algorithm. This algorithm solves the more general problem of computing the maximum flow. A bipartite graph (X + Y, E) can be converted to a flow network as follows. Add a source vertex s; add an edge from s to each vertex in X.
The maximum flow problem was first formulated in 1954 by T. E. Harris and F. S. Ross as a simplified model of Soviet railway traffic flow. [1] [2] [3]In 1955, Lester R. Ford, Jr. and Delbert R. Fulkerson created the first known algorithm, the Ford–Fulkerson algorithm.
In computer science, the Edmonds–Karp algorithm is an implementation of the Ford–Fulkerson method for computing the maximum flow in a flow network in time. The algorithm was first published by Yefim Dinitz in 1970, [1][2] and independently published by Jack Edmonds and Richard Karp in 1972. [3] Dinitz's algorithm includes additional ...
Dinic's algorithm or Dinitz's algorithm is a strongly polynomial algorithm for computing the maximum flow in a flow network, conceived in 1970 by Israeli (formerly Soviet) computer scientist Yefim Dinitz. [1] The algorithm runs in time and is similar to the Edmonds–Karp algorithm, which runs in time, in that it uses shortest augmenting paths.
Bellman–Ford algorithm. The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph. [1] It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it is capable of handling graphs in which some of the edge weights are ...
D. R. Fulkerson. Delbert Ray Fulkerson (/ ˈfʌlkərsən /; August 14, 1924 – January 10, 1976) was an American mathematician who co-developed the Ford–Fulkerson algorithm, one of the most well-known algorithms to solve the maximum flow problem in networks.
Consider the flow f computed for G by Ford–Fulkerson algorithm. In the residual graph (G f ) obtained for G (after the final flow assignment by Ford–Fulkerson algorithm), define two subsets of vertices as follows: A: the set of vertices reachable from s in G f; A c: the set of remaining vertices i.e. V − A