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The algorithm runs while there is a vertex with positive excess, i.e. an active vertex in the graph. The push operation increases the flow on a residual edge, and a height function on the vertices controls through which residual edges can flow be pushed. The height function is changed by the relabel operation.
The Ford–Fulkerson method or Ford–Fulkerson algorithm (FFA) is a greedy algorithm that computes the maximum flow in a flow network.It is sometimes called a "method" instead of an "algorithm" as the approach to finding augmenting paths in a residual graph is not fully specified [1] or it is specified in several implementations with different running times. [2]
Residual Graph Initialise the residual graph by setting the preflow to values 0 and initialising the labeling. Initial saturating push is performed across all preflow arcs out of the source, s. Node a is relabeled in order to push its excess flow towards the sink, t.
Figure A illustrates a weighted graph G. Figure B calculates the shortest path P 1 from A to F (A–B–D–F). Figure C illustrates the shortest path tree T rooted at A, and the computed distances from A to every vertex (u). Figure D shows the residual graph G t with the updated cost of each edge and the edges of path P 1 reversed.
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]
The residual capacity of an arc e with respect to a pseudo-flow f is denoted c f, and it is the difference between the arc's capacity and its flow. That is, c f ( e ) = c ( e ) - f ( e ) . From this we can construct a residual network , denoted G f ( V , E f ) , with a capacity function c f which models the amount of available capacity on the ...
The residual capacity is a mapping : ... An augmenting path is an – path in the residual graph . Define to be the length of the shortest path ...
a finite directed graph G = (V, E), where V denotes the finite set of vertices and E ⊆ V×V is the set of directed edges; a source s ∈ V and a sink t ∈ V; a capacity function, which is a mapping : + denoted by c uv or c(u, v) for (u,v) ∈ E. It represents the maximum amount of flow that can pass through an edge.