|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| 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 ''[[big O notation|O]]''(''V'' ''E''<sup>2</sup>) time. It is asymptotically slower than the [[Push-relabel maximum flow algorithm#Relabel-to-front algorithm, ie. using FIFO heuristic|relabel-to-front algorithm]], which runs in ''O''(''V''<sup>3</sup>) time, but it is often faster in practice for [[sparse graph]]s. 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'').
| | Hi there! :) My name is Virgilio, I'm a student studying Agriculture and Life Sciences from Brooklyn, United States.<br><br>Feel free to surf to my web site: [http://Natorebd.net/wp/?p=70875 fifa 15 Coin Hack] |
| | |
| ==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''<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, 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==
| |
| {{Wikibooks|Algorithm implementation|Graphs/Maximum flow/Edmonds-Karp|Edmonds-Karp}}
| |
| | |
| :''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
| |
| '''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:
| |
| | |
| [[Image:Edmonds-Karp flow example 0.svg|300px]]
| |
| | |
| 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.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| !rowspan="2"| Capacity
| |
| ! Path
| |
| |-
| |
| ! Resulting network
| |
| |-
| |
| |rowspan="2"| <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>
| |
| |align="center"| <math>A,D,E,G</math>
| |
| |-
| |
| | [[Image:Edmonds-Karp flow example 1.svg|300px]]</td>
| |
| |-
| |
| |rowspan="2"| <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>
| |
| |align="center"| <math>A,D,F,G</math>
| |
| |-
| |
| | [[Image:Edmonds-Karp flow example 2.svg|300px]]</td>
| |
| |-
| |
| |rowspan="2"| <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>
| |
| |align="center"| <math>A,B,C,D,F,G</math>
| |
| |-
| |
| | [[Image:Edmonds-Karp flow example 3.svg|300px]]</td>
| |
| |-
| |
| |rowspan="2"| <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>
| |
| |align="center"| <math>A,B,C,E,D,F,G</math>
| |
| |-
| |
| | [[Image:Edmonds-Karp flow example 4.svg|300px]]</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
| |
| :<math>c(A,D)+c(C,D)+c(E,G)=3+1+1=5.\ </math>
| |
| | |
| ==Notes==
| |
| {{reflist|30em}}
| |
| | |
| ==References==
| |
| # Algorithms and Complexity (see pages 63–69). http://www.cis.upenn.edu/~wilf/AlgComp3.html
| |
| | |
| {{DEFAULTSORT:Edmonds-Karp Algorithm}}
| |
| [[Category:Network flow]]
| |
| [[Category:Graph algorithms]]
| |
Hi there! :) My name is Virgilio, I'm a student studying Agriculture and Life Sciences from Brooklyn, United States.
Feel free to surf to my web site: fifa 15 Coin Hack