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. Dinitz'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. 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 in a residual flow network increases monotonically after each augmenting iteration. Then, it shows that each of the edges can be critical at most times for the duration of the algorithm, giving an upper-bound of augmenting iterations. Since each iteration takes time, the total running time of Edmonds-Karp is 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 before such an augmentation and be the flow after. Denote the minimum distance in a residual flow network from nodes as. One can derive a contradiction by showing that, meaning that the shortest path distance between source node and non-sink node did not in fact decrease.

Pseudocode

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

flow := 0 '
repeat
'
q := queue
q.push
pred := array
while 'not empty and pred = null
cur := q.pop
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:
Image:Edmonds-Karp flow example 0.svg|300px|class=skin-invert-image
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.
PathCapacityResulting network
Image:Edmonds-Karp flow example 1.svg|300px|class=skin-invert-image
Image:Edmonds-Karp flow example 2.svg|300px|class=skin-invert-image
Image:Edmonds-Karp flow example 3.svg|300px|class=skin-invert-image
Image:Edmonds-Karp flow example 4.svg|300px|class=skin-invert-image

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