## Edmonds-Karp algorithm

The Edmonds-Karp algorithm has a running time of O(VE²), it is faster than the algorithm of Ford-Fulkerson for dense graphs, i.e. a graph containing a large number of edges (or arcs) depending on the number of vertices.

The Edmonds-Karp algorithm is identical to the Ford-Fulkerson algorithm, except for the search order used to determine an increasing path. The path found must be the shortest path that has a positive capacity. Such a path can be found by a width course, assuming that the arcs all have unit length. Note that the length of the increasing path increases with each iteration.

EdmondsKarpCode from Wikipediaalgorithminput: C [1..n, 1..n](capacity matrix)E [1..n, 1 ..?](lists of neighbors)s(source)t(well)output: f(maximum flow value)F(matrix giving a valid flow of maximum value)f: = 0(the initial flow is zero)F: =array(1..n, 1..n)(the residual capacity from u to v is C[u,v] - F[u,v])foreverm, P: = BreadthFirstSearch (C, E, s, t, F)yewm = 0breakf: = f + m(Backtrack and note the flow)v: = twhilev ≠ s u: = P [v] F [u, v]: = F [u, v] + m F [v, u]: = F [v, u] - m v: = ureturn(f, F)

algorithmBreadthFirstSearchinput: It's foutput: M [t](capacity of the path found)P(parents table)P: =array(1..n)foruin1..n P [u]: = -1 P [s]: = -2(to ensure that the source is not rediscovered)M: =array(1..n)(capacity of the path found to the node)M [s]: = ∞ Q: = tail () Q.push (s)whileQ.size ()> 0 u: = Q.pop ()forvinE [u]yewC [u, v] - F [u, v]> 0andP [v] = -1 P [v]: = u M [v]: = min (M [u], C [u, v] - F [u, v])yewv ≠ t Q.push (v)elsereturnM [t], Preturn0, P