Data Structures and Algorithms Network Flows - dgraf.chdgraf.ch/da16/files/slides12.pdfData...

Data Structures and AlgorithmsNetwork Flows

Daniel Graf

ETH Zürich

May 25, 2016

Network Flow: Example

Network Flow: Example

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Network Flow: Problem Statement

Input: A flow network consisting ofdirected graph G = (V ,E )source and sink s, t ∈ Vedge capacity c : E → N.

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Output: A flow function f : E → N suchthat:

all capacity constraints are satisfied:∀u, v ∈ V : 0 ≤ f (u, v) ≤ c(u, v)(no pipe is overflowed)flow is conserved at every vertex:∀u ∈ V \ {s, t} :∑

(v ,u)∈E f (v , u) =∑

(u,v)∈E f (u, v)(no vertex is leaking)the total flow is maximal:|f | =

∑v∈V f (s, v)−

∑v∈V f (v , s) =∑

u∈V f (u, t)−∑

u∈V f (t, u)

Network Flow Algorithms: Ford-Fulkerson and Edmonds-Karp

Take any s-t-path and increase the flow along it.Update capacities and repeat as long as we can.Problem: We can get stuck at a local optimum.

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Solution: Keep track of the flow and allow paths that reroute units of flow.These are called augmenting paths in the residual network.Ford-Fulkerson: Repeatedly take any augmenting path: running time O(m|f |).Edmonds-Karp: Repeatedly take the shortest augmenting path:running time: best of O(m|f |), O(nmmax c) and O(nm2). [BGL-Doc].

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http://www.boost.org/doc/libs/1_59_0/libs/graph/doc/edmonds_karp_max_flow.html



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Edmonds-Karp Maximum Flow Algorithm

The headers and edge struct:1 // Sample implementation of the Edmonds Karp Algorithm2 // Daniel Graf , [email protected], 7.11.20153 #i n c l u d e <ios t r eam >4 #i n c l u d e <vec to r >5 #i n c l u d e <c a s s e r t >6 #i n c l u d e <queue>7

8 #d e f i n e INF 10000000009 typedef long long int i n ;

10 using namespace s t d ;11

12 struct Edge {13 i n from , to , f low , cap , r e v ;14 i n r e s i d u a l _ c a p a c i t y ( ) {15 return cap−f l ow ;16 }17 } ;18 . . .


Graph struct, add_edge function:1 . . .2 struct Graph {3 i n s , t ;4 vec to r <vec to r <Edge> > E ; // adjacency -list of edges5 vec to r <Edge∗> P ; // predecessor map for the BFS6

7 Graph ( i n N) {8 E = vec to r <vec to r <Edge> >(N) ;9 }

10

11 void add_edge ( i n from , i n to , i n cap ) {12 if ( from==to ) return ;13 E [ from ] . push_back ({ from , to , 0 , cap , ( i n )E [ to ] . s i z e ( ) } ) ;14 E [ to ] . push_back ({ to , from , 0 , 0 , ( i n )E [ from ] . s i z e () −1}) ;15 }16 . . .


Reset all flow values:1 . . .2 void r e s e t _ f l o w ( ) {3 for ( i n v=0; v<E . s i z e ( ) ; v++) {4 for ( i n e =0; e<E [ v ] . s i z e ( ) ; e++) {5 E [ v ] [ e ] . f l ow = 0 ;6 }7 }8 }9 . . .


Can we find a path from s to t?1 bool f i n d _ f l o w ( ) {2 P = vec to r <Edge∗>(E . s i z e ( ) ,NULL ) ;3 // Breadth First Search through the edges with remaining capacity4 queue<in> Q; Q. push ( s ) ;5 while ( !Q. empty ( ) && P[ t ]==NULL) {6 i n v = Q. f r o n t ( ) ; Q. pop ( ) ;7 for ( i n e =0; e<E [ v ] . s i z e ( ) ; e++) {8 if (E [ v ] [ e ] . r e s i d u a l _ c a p a c i t y ()==0) {9 continue ;

10 }11 i n w = E [ v ] [ e ] . to ;12 if (P [w]==NULL) {13 P[w] = &(E [ v ] [ e ] ) ;14 Q. push (w) ;15 }16 }17 } . . .


How much can we fit through that path from s to t?1 . . .2 // Check if there is a path to t3 if (P [ t ] == NULL) {4 return 0 ;5 }6 // Check the minimum capacity7 i n f l ow = INF ;8 i n pos = t ;9 while ( pos != s ) {

10 f l ow = min ( f low , P [ pos]−> r e s i d u a l _ c a p a c i t y ( ) ) ;11 pos = P[ pos]−>from ;12 }13 return f l ow ;14 }


Increase the flow along the path and reduce the reverse edges.1 void update_f low ( i n f l ow ) {2 i n pos = t ;3 while ( pos != s ) {4 // cout << "update at vertex " << pos << endl;5 P[ pos]−>f l ow += f l ow ;6 E [P [ pos]−>to ] [ P [ pos]−>r e v ] . f l ow −= f l ow ;7 pos = P[ pos]−>from ;8 }9 }


Repeatedly search for the shortest augmenting path while one exists.1 i n edmonds_karp_max_flow ( i n _s , i n _t ) {2 s = _s ;3 t = _t ;4 r e s e t _ f l o w ( ) ;5 i n f l ow = 0 ;6 i n new_flow ;7 do {8 new_flow = f i n d _ f l o w ( ) ;9 f l ow += new_flow ;

10 if ( new_flow > 0) {11 update_f low ( new_flow ) ;12 }13 } while ( new_flow > 0 ) ;14 return f l ow ;15 }16 } ; // end of the Graph struct


Read graph from input and call the algorithm.1 void read_graph_from_std in ( Graph &G) {2 i n N,M;3 c i n >> N >> M;4 G = Graph (N) ;5 for ( i n m=0; m<M; m++) {6 i n a , b , c ;7 c i n >> a >> b >> c ;8 G. add_edge ( a , b , c ) ;9 }

10 }11 int main ( ) {12 Graph G( 0 ) ;13 read_graph_from_std in (G ) ;14 i n s , t ; c i n >> s >> t ;15 i n r e s = G. edmonds_karp_max_flow ( s , t ) ;16 cout << "max␣flow:␣" << r e s << en d l ;17 }

Tutorial Problem: Soccer Prediction

Swiss Soccer Championship, two rounds before the end.2 points awarded per game, split 1-1 if game ends in a tie.Goal difference used for tie breaking in the final standings.

Team Points RemainingGames

FC St. Gallen (FCSG) 37 FCB, FCWBSC Young Boys (YB) 36 FCW, FCBFC Basel (FCB) 35 FCSG, YBFC Luzern (FCL) 33 FCZ, GCZFC Winterthur (FCW) 31 YB, FCSG

Can FC Luzern still win the Championship? 37 points still possible, so yes?Is this a flow problem?

Tutorial Problem: Modelling

Can we let the points flow from the games to the teams so that all the teams end upwith at most 37 points?

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Tutorial Problem: Analysis

Is this a flow problem? Yes.What does a unit of flow stand for? A point in the soccer ranking.How large is this flow graph?For N teams, M games, we have n = 2 + N + M nodes and m = N + 3M edges.Flow is at most 2M, edge capacity is at most 2M.What algorithm should we use?Edmonds-Karp Max-Flow runs in O(m|f |) = O(n2).

Common tricks

Multiple sources/sinks:

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Undirected Graphs

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Vertex Capacities

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Minimum Flow per Edge

[Exercise]

Common tricks


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[Exercise]

Common tricks


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[Exercise]

Common tricks


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[Exercise]

Common tricks


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[Exercise]

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[Exercise]

Flow Application: Edge Disjoint Paths

How many ways are there to get from HB to CAB without using the same street twice?

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Map:search.ch,TomTom,swisstopo,OSM


How many ways are there to get from HB to CAB without using the same streettwice?

Is this a flow problem? No.Can it be turned into a flow problem? Maybe.Build directed street graph by adding edges in both directions.Set all capacities to 1.

LemmaIn a directed graph with unit capacities, the maximum number of edge-disjoints-t-paths is equal to the maximum flow from s to t.


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Data Structures and Algorithms Network Flows - dgraf.chdgraf.ch/da16/files/slides12.pdfData...

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