- Edmonds-Karp algorithm
In

computer science andgraph theory , the**Edmonds-Karp algorithm**is an implementation of the Ford-Fulkerson method for computing the maximum flow in aflow network in $mathcal\{O\}(|V|\; cdot\; |E|^2)$. It is asymptotically slower than therelabel-to-front algorithm , which runs in $mathcal\{O\}(|V|^3)$, but it is often faster in practice forsparse graph s. The algorithm was first published by a Russian scientist, Dinic, in 1970 [*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 =*] , and independently byJack Edmonds andRichard Karp in 1972 [*cite journal | author =*] (discovered earlier). Dinic's algorithm includes additional techniques that reduce the running time to $mathcal\{O\}(|V|^2\; cdot\; |E|)$.Jack Edmonds and Richard M. Karp | title = Theoretical improvements in algorithmic efficiency for network flow problems | journal = Journal of the 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 =**Algorithm**The algorithm is identical to the

Ford-Fulkerson algorithm ,except that the search order when finding theaugmenting path isdefined. The path found must be the shortest path which hasavailable capacity. This can be found by abreadth-first search , as we let edges have unit length. The running time of$mathcal\{O\}(|V|\; cdot\; |E|^2)$ is found by showing that each augmenting pathcan be found in $mathcal\{O\}(|E|)$ time, that every time at least one of the$E$ edge becomes saturated, that the distance from thesaturated edge to the source along the augmenting path must belonger than last time it was saturated, and that the distance isat most $V$ long. Another property of this algorithm isthat the length of the shortest augmenting path increasesmonotonically. There is an accessible proof in [*cite book*] .

author =Thomas H. Cormen ,Charles E. Leiserson ,Ronald L. Rivest andClifford 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**Pseudocode**:"For a more high level description, see

Ford-Fulkerson algorithm .**algorithm**EdmondsKarp**input**: 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] )"**forever**m, P := BreadthFirstSearch(C, E, s, t)**if**m = 0**break**f := f + m "(Backtrack search, and write flow)" v := t**while**v ≠ s u := P [v] F [u,v] := F [u,v] + m F [v,u] := F [v,u] - m v := u**return**(f, F)**algorithm**BreadthFirstSearch**input**: C, E, s, t**output**: M [t] "(Capacity of path found)" P "(Parent table)" P :=**array**(1..n)**for**u**in**1..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.push(s)**while**Q.size() > 0 u := Q.pop()**for**v**in**E [u] "(If there is available capacity, and v is not seen before in search)"**if**C [u,v] - F [u,v] > 0**and**P [v] = -1 P [v] := u M [v] := min(M [u] , C [u,v] - F [u,v] )**if**v ≠ t Q.push(v)**else****return**M [t] , P**return**0, P**Example**Given a network of seven nodes, source A, sink G, and capacities as shown below:

In the pairs $f/c$ written on the edges, $f$ is the current flow, and $c$ is the capacity. The residual capacity from $u$ to $v$ is $c\_f(u,v)=c(u,v)-f(u,v)$, the total capacity, minus the flow that is already used. If the net flow from $u$ to $v$ 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 in the graph separating the source and the sink. There is only one minimal cut in this graph, partitioning the nodes into the sets $\{A,B,C,E\}$ and $\{D,F,G\}$, with the capacity $c(A,D)+c(C,D)+c(E,G)=3+1+1=5$.**Java Implementation**public FlowGraph(String fileName) { .. // Read "size" value, "capacity [size] [size] " matrix, // as well as "source" and "sink" node indexes (0-based) // from an input text file. maxFlow(); }

private void maxFlow() // Edmonds-Karp algorithm with O(V³E) complexity { flow = new double [size] [size] ; res_capacity = new double [size] [size] ; parent = new int [size] ; min_capacity = new double [size] ; color = new int [size] ; queue = new int [size] ; for (int i = 0; i < size; i++) for (int j = 0; j < size; j++) res_capacity [i] [j] = capacity [i] [j] ; while (BFS(source)) { max_flow += min_capacity [sink] ; int v = sink, u; while (v != source) { u = parent [v] ; flow [u] [v] += min_capacity [sink] ; flow [v] [u] -= min_capacity [sink] ; res_capacity [u] [v] -= min_capacity [sink] ; res_capacity [v] [u] += min_capacity [sink] ; v = u; } } }

private boolean BFS(int source) // Breadth First Search in O(V²) { for (int i = 0; i < size; i++) { color [i] = WHITE; min_capacity [i] = Double.MAX_VALUE; } first = last = 0; queue [last++] = source; color [source] = GRAY; while (first != last) // While "queue" not empty.. { int v = queue [first++] ; for (int u = 0; u < size; u++) if (color [u] = WHITE && res_capacity [v] [u] > 0) { min_capacity [u] = Math.min(min_capacity [v] , res_capacity [v] [u] ); parent [u] = v; color [u] = GRAY; if (u = sink) return true; queue [last++] = u; } } return false; }

public void toFile(String fileName) { .. // Write the results ("flow" matrix and "max_flow" value) to output file. // To be called in the "main()" method.

**References**

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

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