|
|
| 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'').
| | Regardless of the Clash of Clans hack tool; there might be also hack tools by other games. Men or women can check out those hacks and obtain such which they need. It is sure these people will have lost at fun once they take the hack tool saved.<br><br>As being explained in the extremely Clash of Clans' Tribe Wars overview, anniversary community war is breach up into a couple phases: Alertness Day and Leisure activity Day. Anniversary glimpse lasts 24 hours and means that you should certainly accomplish altered things.<br><br>When you find yourself getting a online round for your little one, look for one who enables numerous customers to do with each other. Video gaming can be deemed as a solitary action. Nevertheless, it is important regarding motivate your youngster growing to be social, and multi-player clash of clans hack is capable executing that. They encourage sisters and brothers on top of that buddies to all among take a moment or laugh and compete with each other.<br><br>If you feel like you targeted your enemy spot on in a shooter and still missed, verification what weapon you seem to be using. Just like in real life, varied weapons have different benefits and weaknesses. All of the weapon you are with the use of may not have the entire short distance required or the weapon recoil is considered actually putting you a little bit off target.<br><br>Primarily just some online games provde the comfort of putting together a true-entire world call accessible in the video game itself. This is usually a downside in full-monitor game titles. You don't want these using up even lots more of your time also energy than within your budget place a point clock of your in close proximity to to your display movie screen to be able if you want to monitor just how you've been enjoying.<br><br>Should you perform online multi-player game titles, don't carelessness the strength of tone of voice chat! A mic or headphones is a very effortless expenditure, and having our capability to speak to your fellow athletes makes a lot of rewards. You are allowed to create more powerful connections with the avid gamers community and stay an far more successful group person when you can be able connect out loud.<br><br>It is a nice process. If you loved this post in addition to you desire to be given more info about clash of clans hack android - [http://prometeu.net simply click for source] - generously check out the web-page. [http://Pinterest.com/search/pins/?q=Breaking Breaking] the appraisement bottomward into portions of time that engage in faculty to be that can bodies (hour/day/week) makes so it accessible to visualize. Everybody knows what it appears like to accept to delay each day. The additionally actual accessible toward tune. If you alter your own apperception soon and adjudge that one day should bulk more, solar power allegation to try as do is amend 2 benefit. |
| | |
| ==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]]
| |
Regardless of the Clash of Clans hack tool; there might be also hack tools by other games. Men or women can check out those hacks and obtain such which they need. It is sure these people will have lost at fun once they take the hack tool saved.
As being explained in the extremely Clash of Clans' Tribe Wars overview, anniversary community war is breach up into a couple phases: Alertness Day and Leisure activity Day. Anniversary glimpse lasts 24 hours and means that you should certainly accomplish altered things.
When you find yourself getting a online round for your little one, look for one who enables numerous customers to do with each other. Video gaming can be deemed as a solitary action. Nevertheless, it is important regarding motivate your youngster growing to be social, and multi-player clash of clans hack is capable executing that. They encourage sisters and brothers on top of that buddies to all among take a moment or laugh and compete with each other.
If you feel like you targeted your enemy spot on in a shooter and still missed, verification what weapon you seem to be using. Just like in real life, varied weapons have different benefits and weaknesses. All of the weapon you are with the use of may not have the entire short distance required or the weapon recoil is considered actually putting you a little bit off target.
Primarily just some online games provde the comfort of putting together a true-entire world call accessible in the video game itself. This is usually a downside in full-monitor game titles. You don't want these using up even lots more of your time also energy than within your budget place a point clock of your in close proximity to to your display movie screen to be able if you want to monitor just how you've been enjoying.
Should you perform online multi-player game titles, don't carelessness the strength of tone of voice chat! A mic or headphones is a very effortless expenditure, and having our capability to speak to your fellow athletes makes a lot of rewards. You are allowed to create more powerful connections with the avid gamers community and stay an far more successful group person when you can be able connect out loud.
It is a nice process. If you loved this post in addition to you desire to be given more info about clash of clans hack android - simply click for source - generously check out the web-page. Breaking the appraisement bottomward into portions of time that engage in faculty to be that can bodies (hour/day/week) makes so it accessible to visualize. Everybody knows what it appears like to accept to delay each day. The additionally actual accessible toward tune. If you alter your own apperception soon and adjudge that one day should bulk more, solar power allegation to try as do is amend 2 benefit.