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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|O]]''(''V'' ''E''<sup>2</sup>) time. 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'').
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==Algorithm==
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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 (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 ''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>
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==Pseudocode==
'''MathML'''
{{Wikibooks|Algorithm implementation|Graphs/Maximum flow/Edmonds-Karp|Edmonds-Karp}}
:<math forcemathmode="mathml">E=mc^2</math>


:''For a more high level description, see [[Ford–Fulkerson algorithm]].
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


'''algorithm''' EdmondsKarp
'''source'''
    '''input''':
:<math forcemathmode="source">E=mc^2</math> -->
        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
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
    '''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.offer(s)
    '''while''' Q.size() > 0
        u := Q.poll()
        '''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.offer(v)
                '''else'''
                    '''return''' M[t], P
    '''return''' 0, P


==Demos==


:''EdmondsKarp pseudo code using Adjacency nodes.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


'''algorithm''' EdmondsKarp
    '''input''':
        graph ''(Graph with list of Adjacency nodes with capacities,flow,reverse and destinations)''
        s            ''(Source)''
        t            ''(Sink)''
    '''output''':
        flow            ''(Value of maximum flow)''
    flow := 0 ''(Initial flow to zero)''
    q := '''array'''(1..n) ''(Initialize q to graph length)''
    '''while''' true
        qt := 0 ''(Variable to iterate over all the corresponding edges for a source)''
        q[qt+1] := s    ''(initialize source array)''
        pred := '''array'''(q.length)    ''(Initialize predecessor List with the graph length)''
        '''for''' qh=0;qh < qt && pred[t] := null
            cur := q[qh]
            for (graph[cur]) ''(Iterate over list of Edges)''
                  Edge[] e :=  graph[cur]  ''(Each edge should be associated with Capacity)''
                  if pred[e.t] == null && e.cap > e.f
                    pred[e.t] := e
                    q[qt++] : = e.t
        if pred[t] = null
            break
        int df := MAX VALUE ''(Initialize to max integer value)''
        for u = t; u != s; u = pred[u].s
            df := min(df, pred[u].cap - pred[u].f)
        for u = t; u != s; u = pred[u].s
            pred[u].f  := pred[u].f + df
            pEdge := '''array'''(PredEdge)
            pEdge := graph[pred[u].t]
            pEdge[pred[u].rev].f := pEdge[pred[u].rev].f - df;
        flow := flow + df
    '''return''' flow


==Example==
* accessibility:
Given a network of seven nodes, source A, sink G, and capacities as shown below:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


[[Image:Edmonds-Karp flow example 0.svg|300px]]
==Test pages ==


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.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


{| class="wikitable"
*[[Inputtypes|Inputtypes (private Wikis only)]]
|-
*[[Url2Image|Url2Image (private Wikis only)]]
!rowspan="2"| Capacity
==Bug reporting==
! Path
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
|-
! 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]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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