Edmonds–Karp 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 big O notation| time. The algorithm was first published by Yefim Dinitz in 1970 and independently published by Jack Edmonds and Richard Karp in 1972. Dinic's algorithm includes additional techniques that reduce the running time to.

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, where we apply a weight of 1 to each edge. The running time of is found by showing that each augmenting path can be found in time, that every time at least one of the edges becomes saturated, that the distance from the saturated edge to the source along the augmenting path must be longer than last time it was saturated, and that the length is at most. Another property of this algorithm is that the length of the shortest augmenting path increases monotonically. There is an accessible proof in Introduction to Algorithms.

Pseudocode

algorithm EdmondsKarp is
input:
graph '
s '
t '
output:
flow '

flow := 0 '
repeat
'
q := queue
q.push
pred := array
while not empty
cur := q.pull
for Edge e in graph do
if pred = null and e.t ≠ s and e.cap > e.flow then
pred := e
q.push

if not then
'
df := ∞
for do
df := min
'
for do
e.flow := e.flow + df
e.rev.flow := e.rev.flow - df
flow := flow + df

until pred = null '
return''' flow

Example

Given a network of seven nodes, source A, sink G, and capacities as shown below:
In the pairs written on the edges, is the current flow, and is the capacity. The residual capacity from to is, the total capacity, minus the flow that is already used. If the net flow from to is negative, it contributes to the residual capacity.
CapacityPathResulting network

Notice how the length of the augmenting path found by the algorithm never decreases. The paths found are the shortest possible. The flow found is equal to the capacity across the minimum cut in the graph separating the source and the sink. There is only one minimal cut in this graph, partitioning the nodes into the sets and, with the capacity