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EdmondsKarp algorithm1
Edmonds
€
Karp algorithm
In computer science and graph theory, the
Edmonds
€
Karp algorithm
is an implementation of the Ford
€
Fulkersonmethod for computing the maximum flow in a flow network in
O
(
V
E
2
) time. It is asymptotically slower than therelabel-to-front algorithm, which runs in
O
(
V
3
) time, but it is often faster in practice for sparse graphs. The algorithmwas first published by Yefim (Chaim) Dinic in 1970 and independently published by Jack Edmonds and RichardKarp in 1972. Dinic's algorithm includes additional techniques that reduce the running time to
O
(
V
2
E
).
Algorithm
The algorithm is identical to the Ford
€
Fulkerson algorithm, except that the search order when finding theaugmenting path is defined. The path found must be a shortest path that has available capacity. This can be found bya breadth-first search, as we let edges have unit length. The running time of
O
(
V
E
2
) is found by showing that eachaugmenting path can be found in
O
(
E
) time, that every time at least one of the
E
edges becomes saturated, that thedistance from the saturated edge to the source along the augmenting path must be longer than last time it wassaturated, and that the length is at most
V
. Another property of this algorithm is that the length of the shortestaugmenting path increases monotonically. There is an accessible proof in
Introduction to Algorithms
.
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, F)
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
EdmondsKarp algorithm2
input
: C, E, s, t, F
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 written on the edges, is the current flow, and is the capacity. The residual capacity from tois , the total capacity, minus the flow that is already used. If the net flow fromto is negative, it
contributes
to the residual capacity.
EdmondsKarp algorithm3
CapacityPathResulting network
Notice how the length of the augmenting path found by the algorithm (in red) never decreases. The paths found arethe shortest possible. The flow found is equal to the capacity across the minimum cut in the graph separating thesource and the sink. There is only one minimal cut in this graph, partitioning the nodes into the sets and , with the capacity
NotesReferences
1.Algorithms and Complexity (see pages 63
€
69). http:/
/
www.
cis.
upenn.
edu/
~wilf/
AlgComp3.
html
Article Sources and Contributors4
Article Sources and Contributors
Edmonds
€
Karp algorithm
Source
: http://en.wikipedia.org/w/index.php?oldid=568278410
Contributors
: Aaron Hurst, Amelio V•zquez, Balloonguy, Cburnett, Chopchopwhitey, Cosmi,Cquan, Darth Panda, Giftlite, Glrx, Hashproduct, Headbomb, Htmnssn, Jamelan, John of Reading, Katieh5584, Kristjan Wager, Kubek15, LiuZhaoliang, Magioladitis, Martani, Michael Hardy,Mihai Capot‚, Niloofar piroozi, Nils Grimsmo, Nixdorf, Nkojuharov, Ohad trabelsi, Parudox, Pkirlin, Poor Yorick, Pt, Pugget, RJFJR, Sesse, Simonfl, TPReal, WangPublic, Zero0000,
€•‚ƒ„… †‡ˆ
,66 anonymous edits
Image Sources, Licenses and Contributors
Image:Edmonds-Karp flow example 0.svg
Source
: http://en.wikipedia.org/w/index.php?title=File:Edmonds-Karp_flow_example_0.svg
License
: Creative Commons Attribution-ShareAlike3.0 Unported
Contributors
: en:User:Cburnett
Image:Edmonds-Karp flow example 1.svg
Source
: http://en.wikipedia.org/w/index.php?title=File:Edmonds-Karp_flow_example_1.svg
License
: Creative Commons Attribution-ShareAlike3.0 Unported
Contributors
: en:User:Cburnett
Image:Edmonds-Karp flow example 2.svg
Source
: http://en.wikipedia.org/w/index.php?title=File:Edmonds-Karp_flow_example_2.svg
License
: Creative Commons Attribution-ShareAlike3.0 Unported
Contributors
: en:User:Cburnett
Image:Edmonds-Karp flow example 3.svg
Source
: http://en.wikipedia.org/w/index.php?title=File:Edmonds-Karp_flow_example_3.svg
License
: Creative Commons Attribution-ShareAlike3.0 Unported
Contributors
: en:User:Cburnett
Image:Edmonds-Karp flow example 4.svg
Source
: http://en.wikipedia.org/w/index.php?title=File:Edmonds-Karp_flow_example_4.svg
License
: Creative Commons Attribution-ShareAlike3.0 Unported
Contributors
: en:User:Cburnett
License
Creative Commons Attribution-Share Alike 3.0 //creativecommons.org/licenses/by-sa/3.0/

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