Edmonds–Karp algorithm: Difference between revisions

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{{Short description|Algorithm to compute the maximum flow in a flow network}}

In [[computer science]] and [[graph theory]], 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 <math>O(VE^2)</math>. It is asymptotically slower than the [[relabel-to-front algorithm]], which runs in <math>O(V^3)</math>, but it is often faster in practice for [[sparse graph]]s. The algorithm was first published by a Russian scientist, Dinic, in 1970<ref>{{cite journal | last = E. A. Dinic | title = Algorithm for solution of a problem of maximum flow in a network with power estimation | journal = Soviet Math. Doklady | volume = Vol 11 | issue = | pages = 1277-1280 | publisher = Doklady | date = 1970 | url = | doi = | id = | accessdate = }}</ref>, and independently by [[Jack Edmonds]] and [[Richard Karp]] in 1972<ref>{{cite journal | author = [[Jack Edmonds]] and [[Richard Karp|Richard M. Karp]] | title = Theoretical improvements in algorithmic efficiency for network flow problems | journal = Journal of the [[Association for Computing Machinery|ACM]] | volume = 19 | issue = 2 | pages = 248-264 | publisher = | date = 1972 | url = http://www.akira.ruc.dk/~keld/teaching/algoritmedesign_f03/Artikler/08/Edmonds72.pdf | doi = | id = | accessdate = }}</ref> (discovered earlier). Dinic's algorithm includes additional techniques that reduce the running time to <math>O(V^2 E)</math>.

{{Use American English|date = April 2019}}

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|<math>O(|V||E|^2)</math>]] time. The algorithm was first published by [[Yefim Dinitz]] in 1970,<ref>{{cite journal |first=E. A. |last=Dinic |author-link=Yefim Dinitz |title=Algorithm for solution of a problem of maximum flow in a network with power estimation |journal=Soviet Mathematics - Doklady |volume=11 |pages=1277–1280 |publisher=Doklady |year=1970 }}</ref><ref name="ipv">{{cite book | author = Yefim Dinitz | editor = [[Oded Goldreich]] |editor2=Arnold L. Rosenberg |editor3=Alan L. Selman |editor3-link = Alan Selman| title = Theoretical Computer Science: Essays in Memory of [[Shimon Even]] | chapter = Dinitz' Algorithm: The Original Version and Even's Version | year = 2006 | publisher = Springer | isbn = 978-3-540-32880-3 | pages = 218–240 | chapter-url = https://rangevoting.org/Dinitz_alg.pdf}}</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 |year=1972 |url=http://www.eecs.umich.edu/%7Epettie/matching/Edmonds-Karp-network-flow.pdf |doi=10.1145/321694.321699 |s2cid=6375478 }}</ref> [[Dinic's algorithm|Dinitz's algorithm]] includes additional techniques that reduce the running time to <math>O(|V|^2|E|)</math>.<ref name="ipv" />

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

==Algorithm==

The algorithm is identical to the [[Ford–Fulkerson algorithm]], except that the search order when finding the [[Flow network#Augmenting paths|augmenting path]] is defined. The path found must be a [[Shortest path problem|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 <math>O(|V||E|^2)</math> is found by showing that each augmenting path can be found in <math>O(|E|)</math> time, that every time at least one of the {{mvar|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 <math>|V|</math>. Another property of this algorithm is that the length of the shortest augmenting path increases monotonically.<ref name='clrs'>{{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 |title-link=Introduction to Algorithms }}</ref> A proof outline using these properties is as follows:

The proof first establishes that distance of the shortest path from the source node {{mvar|s}} to any non-sink node {{mvar|v}} in a residual flow network increases monotonically after each augmenting iteration (Lemma 1, proven below). Then, it shows that each of the <math>|E|</math> edges can be critical at most <math>\frac{|V|}{2}</math> times for the duration of the algorithm, giving an upper-bound of <math>O\left( \frac{|V||E|}{2} \right) \in O(|V||E|)</math> augmenting iterations. Since each iteration takes <math>O(|E|)</math> time (bounded by the time for finding the shortest path using Breadth-First-Search), the total running time of Edmonds-Karp is <math>O(|V||E|^2)</math> as required. <ref name='clrs'/>

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 the shortest path which has available capacity. This can be found by a [[breadth-first search]], as we let edges have unit length. The running time of <math>O(VE^2)</math> is found by showing that the length of the augmenting path found never decreases, that for every time one of the <math>E</math> edge becomes saturated the augmenting path must be longer than last time it was saturated, that a path is at most <math>V</math> long, and can be found in <math>O(E)</math> time. There is an accessible proof in <ref>{{cite book | author = [[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]] and [[Clifford Stein]] | title = [[Introduction to Algorithms]] | publisher = MIT Press and McGraw-Hill | year = 2001 | id = ISBN 0-262-53196-8 | edition = second edition | chapter = 26.2 | pages = 660-663 }}</ref>.

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 {{mvar|s}} to {{mvar|v}} to ''decrease''. Let {{mvar|f}} be the flow before such an augmentation and <math>f'</math> be the flow after. Denote the minimum distance in a residual flow network {{tmath|G_f}} from nodes <math>u, v</math> as <math>\delta_f (u, v)</math>. One can derive a contradiction by showing that <math>\delta_f (s, v) \leq \delta _{f'} (s, v)</math>, meaning that the shortest path distance between source node {{mvar|s}} and non-sink node {{mvar|v}} did not in fact decrease. <ref name='clrs'/>

==Sample implementation==

==Pseudocode==

[[Python programming language|Python]] implementation:

'''algorithm''' EdmondsKarp '''is'''

'''input''':

'''def''' edmonds_karp(C, source, sink):

n = len(C) graph ''#(graph[v] Cshould isbe the capacitylist matrixof edges coming out of vertex v in the''

F = [[0] * n '''for' '' ioriginal graph '''inand''' xrange(n)]their corresponding constructed reverse edges''

'' which are used for push-back flow.''

''# residual capacity from u to v is C[u][v] - F[u][v]''

'' 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)

'''whileif''' True:'''not''' (pred[t] = null) '''then'''

path = bfs ''(C,We F,found source,an sink)augmenting path.''

'' See how much flow we can send)'if''' not path:

df := '''break∞'''

'''for''' (e := pred[t]; e ≠ null; e := pred[e.s]) '''do'''

flow = float("infinity")

''# traverse path to find smallest capacity df := '''min'''(df, e.cap - e.flow)

'''for'''(And (u,vupdate edges by that amount) '''in''' path:

flow'''for''' = min(flow,e C:= pred[u][vt]; -e F≠ null; e := pred[u][ve.s]) '''do'''

''# traverse path to update e.flow'' := e.flow + df

e.rev.flow := e.rev.flow - df

'''for''' u,v '''in''' path:

F[u][v]flow +:= flow + df

F[v][u] -= flow

'''return''' sum([F[source][i] '''for''' i '''in''' xrange(n)])

'''until''' pred[t] = null ''(i.e., until no augmenting path was found)''

'''def''' bfs(C, F, source, sink):

'''return''' flow

queue = [source]

paths = {source: []}

'''while''' queue:

u = queue.pop(0)

'''for''' v '''in''' xrange(len(C)):

'''if''' C[u][v] - F[u][v] > 0 '''and''' v '''not''' in paths:

paths[v] = paths[u] + [(u,v)]

'''if''' v == sink:

'''return''' paths[v]

queue.append(v)

'''return''' None

==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]]

Given a network of seven nodes, and capacities as shown below:

In the pairs <math>f/c</math> written on the edges, <math>f</math> is the current flow, and <math>c</math> is the capacity. The residual capacity from <math>u</math> to <math>v</math> is <math>c_f(u,v)=c(u,v)-f(u,v)</math>, the total capacity, minus the flow that is already used. If the net flow from <math>u</math> to <math>v</math> is negative, it ''contributes'' to the residual capacity.

[[Image:ek-flow_0.png]]

{| class="wikitable"

In the pairs <math>f/c</math> written on the edges, <math>f</math> is the current flow, and <math>c</math> is the capacity. The residual capacity from <math>u</math> to <math>v</math> is <math>c_f(u,v)=c(u,v)-f(u,v)</math>, the total capacity, minus the flow you have already used. If the net flow from <math>u</math> to <math>v</math> is negative, it ''contributes'' to the residual capacity.

|-

! Path

! Capacity

! Resulting network

|-

| align="center" | <math>A,D,E,G</math>

| <math>\begin{align}

& \min(c_f(A,D),c_f(D,E),c_f(E,G)) \\

= & \min(3-0,2-0,1-0) \\

= & \min(3,2,1) = 1

\end{align}</math>

| [[Image:Edmonds-Karp flow example 1.svg|300px|class=skin-invert-image]]</td>

|-

| align="center" | <math>A,D,F,G</math>

| <math>\begin{align}

& \min(c_f(A,D),c_f(D,F),c_f(F,G)) \\

= & \min(3-1,6-0,9-0) \\

= & \min(2,6,9) = 2

\end{align}</math>

| [[Image:Edmonds-Karp flow example 2.svg|300px|class=skin-invert-image]]</td>

|-

| align="center" | <math>A,B,C,D,F,G</math>

| <math>\begin{align}

& \min(c_f(A,B),c_f(B,C),c_f(C,D),c_f(D,F),c_f(F,G)) \\

= & \min(3-0,4-0,1-0,6-2,9-2) \\

= & \min(3,4,1,4,7) = 1

\end{align}</math>

| [[Image:Edmonds-Karp flow example 3.svg|300px|class=skin-invert-image]]</td>

|-

| align="center" | <math>A,B,C,E,D,F,G</math>

| <math>\begin{align}

& \min(c_f(A,B),c_f(B,C),c_f(C,E),c_f(E,D),c_f(D,F),c_f(F,G)) \\

= & \min(3-1,4-1,2-0,0-(-1),6-3,9-3) \\

= & \min(2,3,2,1,3,6) = 1

\end{align}</math>

| [[Image:Edmonds-Karp flow example 4.svg|300px|class=skin-invert-image]]</td>

|}

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 [[max flow min cut theorem|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 <math>\{A,B,C,E\}</math> and <math>\{D,F,G\}</math>, with the capacity

<table width="100%">

:<math>c(A,D)+c(C,D)+c(E,G)=3+1+1=5.\ </math>

<tr>

<th>Path</th>

<th>Capacity</th>

<th>Resulting network</th>

</tr>

==Notes==

<tr>

{{reflist|30em}}

<td><math>A,D,E,G</math></td>

<td>

<math>\min(c_f(A,D),c_f(D,E),c_f(E,G)) = </math></br>

<math>\min(3-0,2-0,1-0) = </math></br>

<math>\min(3,2,1) = 1</math></br>

</td>

<td>[[Image:ek-flow_1.png]]</td>

</tr>

<tr>

<td><math>A,D,F,G</math></td>

<td>

<math>\min(c_f(A,D),c_f(D,F),c_f(F,G)) = </math></br>

<math>\min(3-1,6-0,9-0) = </math></br>

<math>\min(2,6,9) = 2</math></br>

</td>

<td>[[Image:ek-flow_2.png]]</td>

</tr>

<tr>

<td><math>A,B,C,D,F,G</math></td>

<td>

<math>\min(c_f(A,B),c_f(B,C),c_f(C,D),c_f(D,F),c_f(F,G)) = </math></br>

<math>\min(3-0,4-0,1-0,6-2,9-2) = </math></br>

<math>\min(3,4,1,4,7) = 1</math></br>

</td>

<td>[[Image:ek-flow_3.png]]</td>

</tr>

<tr>

<td><math>A,B,C,E,D,F,G</math></td>

<td>

<math>\min(c_f(A,B),c_f(B,C),c_f(C,E),c_f(E,D),c_f(D,F),c_f(F,G)) = </math></br>

<math>\min(3-1,4-1,2-0,0--1,6-3,9-3) = </math></br>

<math>\min(2,3,2,1,3,6) = 1</math></br>

</td>

<td>[[Image:ek-flow_4.png]]</td>

</tr>

</table>

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 [[max flow min cut theorem|smallest 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 <math>\{A,B,C,E\}</math> and <math>\{D,F,G\}</math>, with the capacity <math>c(A,D)+c(C,D)+c(E,G)=3+1+1=5</math>.

==References==

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

{{Optimization algorithms|combinatorial}}

<references />

{{DEFAULTSORT:Edmonds-Karp Algorithm}}

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

[[Category:Network flow problem]]

[[Category:NetworkGraph flowalgorithms]]

[[pt:Algoritmo de Edmonds-Karp]]