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 O(|V||E|^2) time. The algorithm was first published by Yefim Dinitz in 1970, and independently published by Jack Edmonds and Richard Karp in 1972. Dinitz's algorithm.

Algorithm

The algorithm is identical to the Ford–Fulkerson algorithm, except that the search order when finding the augmenting path is defined. The path found must be a shortest path that has available capacity. This can be found by a breadth-first search time, that every time at least one of the . Another property of this algorithm is that the length of the shortest augmenting path increases monotonically. A proof outline using these properties is as follows:

The proof first establishes that distance of the shortest path from the source node to any non-sink node as required.

To prove Lemma 1, one can use proof by contradiction by assuming that there is an augmenting iteration that causes the shortest path distance from to to decrease. Let be the flow after. Denote the minimum distance in a residual flow network , meaning that the shortest path distance between source node and non-sink node did not in fact decrease.

Pseudocode

algorithm EdmondsKarp is
    input:
        graph   (graph[v] should be the list of edges coming out of vertex v in the
                 original graph and their corresponding constructed reverse edges
                 which are used for push-back flow.
                 Each edge should have a capacity 'cap', flow, source 's' and sink 't' 
                 as parameters, as well as a pointer to the reverse edge 'rev'.)
        s       (Source vertex)
        t       (Sink vertex)
    output:
        flow    (Value of maximum flow)
    
    flow := 0   (Initialize flow to zero)
    repeat
        (Run a breadth-first search (bfs) to find the shortest s-t path.
         We use 'pred' to store the edge taken to get to each vertex,
         so we can recover the path afterwards)
        q := queue()
        q.push(s)
        pred := array(graph.length)
        while not empty(q) and pred[t] = null
            cur := q.pop()
            for Edge e in graph[cur] do
                if pred[e.t] = null and e.t ≠ s and e.cap > e.flow then
                    pred[e.t] := e
                    q.push(e.t)

        if not (pred[t] = null) then
            (We found an augmenting path.
             See how much flow we can send) 
            df := ∞
            for (e := pred[t]; e ≠ null; e := pred[e.s]) do
                df := min(df, e.cap - e.flow)
            (And update edges by that amount)
            for (e := pred[t]; e ≠ null; e := pred[e.s]) do
                e.flow  := e.flow + df
                e.rev.flow := e.rev.flow - df
            flow := flow + df

    until pred[t] = null  (i.e., until no augmenting path was found)
    return flow

Example

Given a network of seven nodes, source A, sink G, and capacities as shown below:

is negative, it contributes to the residual capacity.Notice how the length of the augmenting path found by the algorithm (in red) never decreases. The paths found are the shortest possible. The flow found is equal to the capacity across the minimum cut, with the capacity

Notes

References

1. Algorithms and Complexity (see pages 63–69). https://web.archive.org/web/20061005083406/http://www.cis.upenn.edu/~wilf/AlgComp3.html