# Difference between revisions of "Edmonds–Karp algorithm"

(wlink big O notation during first use in this article) |
(→Pseudocode: replaced push/pop with offer/poll which is more commonly used for FIFO queues. (push/pop usually implies LIFO, which makes this a depth-first search)) |
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− | In [[computer science | + | In [[computer science]], the '''Edmonds–Karp algorithm''' is an implementation of the [[Ford–Fulkerson algorithm|Ford–Fulkerson method]] for computing the [[maximum flow problem|maximum flow]] in a [[flow network]] in ''[[big O notation|O]]''(''V'' ''E''<sup>2</sup>) time. The algorithm was first published by Yefim (Chaim) Dinic in 1970<ref>{{cite journal |first=E. A. |last=Dinic |title=Algorithm for solution of a problem of maximum flow in a network with power estimation |journal=Soviet Math. Doklady |volume=11 |issue= |pages=1277–1280 |publisher=Doklady |year=1970 |url= |doi= |id= |accessdate= }}</ref> and independently published by [[Jack Edmonds]] and [[Richard Karp]] in 1972.<ref>{{cite journal |last1=Edmonds |first1=Jack |author1-link=Jack Edmonds |last2=Karp |first2=Richard M. |author2-link=Richard Karp |title=Theoretical improvements in algorithmic efficiency for network flow problems |journal=Journal of the ACM |volume=19 |issue=2 |pages=248–264 |publisher=[[Association for Computing Machinery]] |year=1972 |url= |doi=10.1145/321694.321699 |id= |accessdate= }}</ref> [[Dinic's algorithm]] includes additional techniques that reduce the running time to ''O''(''V''<sup>2</sup>''E''). |

==Algorithm== | ==Algorithm== | ||

− | The algorithm is identical to the [[ | + | 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]], as we let edges have unit length. The running time of ''O''(''V'' ''E''<sup>2</sup>) is found by showing that each augmenting path can be found in ''O''(''E'') time, that every time at least one of the ''E'' edges becomes saturated (an edge which has the maximum possible flow), 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 ''V''. 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]]''.<ref>{{cite book |author=[[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]] and [[Clifford Stein]] |title=[[Introduction to Algorithms]] |publisher=MIT Press | year = 2009 |isbn=978-0-262-03384-8 |edition=third |chapter=26.2 |pages=727–730 }}</ref> |

==Pseudocode== | ==Pseudocode== | ||

{{Wikibooks|Algorithm implementation|Graphs/Maximum flow/Edmonds-Karp|Edmonds-Karp}} | {{Wikibooks|Algorithm implementation|Graphs/Maximum flow/Edmonds-Karp|Edmonds-Karp}} | ||

− | :''For a more high level description, see [[ | + | :''For a more high level description, see [[Ford–Fulkerson algorithm]]. |

'''algorithm''' EdmondsKarp | '''algorithm''' EdmondsKarp | ||

Line 34: | Line 34: | ||

v := u | v := u | ||

'''return''' (f, F) | '''return''' (f, F) | ||

− | + | ||

'''algorithm''' BreadthFirstSearch | '''algorithm''' BreadthFirstSearch | ||

'''input''': | '''input''': | ||

Line 44: | Line 44: | ||

'''for''' u '''in''' 1..n | '''for''' u '''in''' 1..n | ||

P[u] := -1 | P[u] := -1 | ||

− | P[s] := -2 ''(make sure source is not rediscovered)'' | + | P[s] := -2 ''(make sure source is not rediscovered)'' |

M := '''array'''(1..n) ''(Capacity of found path to node)'' | M := '''array'''(1..n) ''(Capacity of found path to node)'' | ||

M[s] := ∞ | M[s] := ∞ | ||

Q := queue() | Q := queue() | ||

− | Q. | + | Q.offer(s) |

'''while''' Q.size() > 0 | '''while''' Q.size() > 0 | ||

− | u := Q. | + | u := Q.poll() |

'''for''' v '''in''' E[u] | '''for''' v '''in''' E[u] | ||

''(If there is available capacity, and v is not seen before in search)'' | ''(If there is available capacity, and v is not seen before in search)'' | ||

Line 57: | Line 57: | ||

M[v] := min(M[u], C[u,v] - F[u,v]) | M[v] := min(M[u], C[u,v] - F[u,v]) | ||

'''if''' v ≠ t | '''if''' v ≠ t | ||

− | Q. | + | Q.offer(v) |

'''else''' | '''else''' | ||

'''return''' M[t], P | '''return''' M[t], P | ||

'''return''' 0, P | '''return''' 0, P | ||

+ | |||

+ | |||

+ | :''EdmondsKarp pseudo code using Adjacency nodes. | ||

+ | |||

+ | '''algorithm''' EdmondsKarp | ||

+ | '''input''': | ||

+ | graph ''(Graph with list of Adjacency nodes with capacities,flow,reverse and destinations)'' | ||

+ | s ''(Source)'' | ||

+ | t ''(Sink)'' | ||

+ | '''output''': | ||

+ | flow ''(Value of maximum flow)'' | ||

+ | flow := 0 ''(Initial flow to zero)'' | ||

+ | q := '''array'''(1..n) ''(Initialize q to graph length)'' | ||

+ | '''while''' true | ||

+ | qt := 0 ''(Variable to iterate over all the corresponding edges for a source)'' | ||

+ | q[qt+1] := s ''(initialize source array)'' | ||

+ | pred := '''array'''(q.length) ''(Initialize predecessor List with the graph length)'' | ||

+ | '''for''' qh=0;qh < qt && pred[t] := null | ||

+ | cur := q[qh] | ||

+ | for (graph[cur]) ''(Iterate over list of Edges)'' | ||

+ | Edge[] e := graph[cur] ''(Each edge should be associated with Capacity)'' | ||

+ | if pred[e.t] == null && e.cap > e.f | ||

+ | pred[e.t] := e | ||

+ | q[qt++] : = e.t | ||

+ | if pred[t] = null | ||

+ | break | ||

+ | int df := MAX VALUE ''(Initialize to max integer value)'' | ||

+ | for u = t; u != s; u = pred[u].s | ||

+ | df := min(df, pred[u].cap - pred[u].f) | ||

+ | for u = t; u != s; u = pred[u].s | ||

+ | pred[u].f := pred[u].f + df | ||

+ | pEdge := '''array'''(PredEdge) | ||

+ | pEdge := graph[pred[u].t] | ||

+ | pEdge[pred[u].rev].f := pEdge[pred[u].rev].f - df; | ||

+ | flow := flow + df | ||

+ | '''return''' flow | ||

==Example== | ==Example== | ||

Line 112: | Line 148: | ||

==References== | ==References== | ||

− | # Algorithms and Complexity (see pages | + | # Algorithms and Complexity (see pages 63–69). http://www.cis.upenn.edu/~wilf/AlgComp3.html |

{{DEFAULTSORT:Edmonds-Karp Algorithm}} | {{DEFAULTSORT:Edmonds-Karp Algorithm}} | ||

[[Category:Network flow]] | [[Category:Network flow]] | ||

[[Category:Graph algorithms]] | [[Category:Graph algorithms]] |

## Latest revision as of 02:11, 23 October 2014

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 (Chaim) Dinic in 1970^{[1]} and independently published by Jack Edmonds and Richard Karp in 1972.^{[2]} Dinic's algorithm includes additional techniques that reduce the running time to *O*(*V*^{2}*E*).

## Contents

## 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, as we let edges have unit length. The running time of *O*(*V* *E*^{2}) is found by showing that each augmenting path can be found in *O*(*E*) time, that every time at least one of the *E* edges becomes saturated (an edge which has the maximum possible flow), 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 *V*. 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*.^{[3]}

## Pseudocode

*For a more high level description, see Ford–Fulkerson algorithm.*

algorithmEdmondsKarpinput: C[1..n, 1..n](Capacity matrix)E[1..n, 1..?](Neighbour lists)s(Source)t(Sink)output: f(Value of maximum flow)F(A matrix giving a legal flow with the maximum value)f := 0(Initial flow is zero)F :=array(1..n, 1..n)(Residual capacity from u to v is C[u,v] - F[u,v])foreverm, P := BreadthFirstSearch(C, E, s, t, F)ifm = 0breakf := f + m(Backtrack search, and write 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: C, E, s, t, Foutput: M[t](Capacity of path found)P(Parent table)P :=array(1..n)foruin1..n P[u] := -1 P[s] := -2(make sure source is not rediscovered)M :=array(1..n)(Capacity of found path to node)M[s] := ∞ Q := queue() Q.offer(s)whileQ.size() > 0 u := Q.poll()forvinE[u](If there is available capacity, and v is not seen before in search)ifC[u,v] - F[u,v] > 0andP[v] = -1 P[v] := u M[v] := min(M[u], C[u,v] - F[u,v])ifv ≠ t Q.offer(v)elsereturnM[t], Preturn0, P

*EdmondsKarp pseudo code using Adjacency nodes.*

algorithmEdmondsKarpinput: graph(Graph with list of Adjacency nodes with capacities,flow,reverse and destinations)s(Source)t(Sink)output: flow(Value of maximum flow)flow := 0(Initial flow to zero)q :=array(1..n)(Initialize q to graph length)whiletrue qt := 0(Variable to iterate over all the corresponding edges for a source)q[qt+1] := s(initialize source array)pred :=array(q.length)(Initialize predecessor List with the graph length)forqh=0;qh < qt && pred[t] := null cur := q[qh] for (graph[cur])(Iterate over list of Edges)Edge[] e := graph[cur](Each edge should be associated with Capacity)if pred[e.t] == null && e.cap > e.f pred[e.t] := e q[qt++] : = e.t if pred[t] = null break int df := MAX VALUE(Initialize to max integer value)for u = t; u != s; u = pred[u].s df := min(df, pred[u].cap - pred[u].f) for u = t; u != s; u = pred[u].s pred[u].f := pred[u].f + df pEdge :=array(PredEdge) pEdge := graph[pred[u].t] pEdge[pred[u].rev].f := pEdge[pred[u].rev].f - df; flow := flow + dfreturnflow

## 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.

Capacity | Path |
---|---|

Resulting network | |

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 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

## Notes

## References

- Algorithms and Complexity (see pages 63–69). http://www.cis.upenn.edu/~wilf/AlgComp3.html