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Ford-Fulkerson algorithm for the Maximum F low problem

Ford-Fulkerson algorithm for the Maximum F low problem. Maximum Flow Problem Given a connected graph G=(V,E), a capacity c:E->R+, and two nodes s and t, find a maximum s-t flow. Minimum Cut Problem

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Ford-Fulkerson algorithm for the Maximum F low problem

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  1. Ford-Fulkerson algorithm for the Maximum Flow problem • Maximum Flow Problem • Given a connected graph G=(V,E), a capacity c:E->R+, and two nodes s and t, find a maximum s-t flow. • Minimum Cut Problem • Given a connected graph G=(V,E), a capacity c:E->R+, and two nodes s and t, find a minimum s-t cut. • Ford-Fulkerson Labeling Algorithm • (Initialization) Let x be an initial feasible flow (e.g. x(e) = 0 for all e in E). • (Flow augmentation) If there are no augmenting path from s to t on the residual network, then stop. The present x is a max flow. If there is a flow augmenting path p, replace the flow x as • x(e)=x(e)+delta if e is a forward arc on p. • x(e)=x(e)-delta if e is a backward arc on p. • where delta is a minimum value of residual capacity on p. Repeat this step.

  2. Ford-Fulkerson Algorithm 0 flow 2 4 4 capacity G: 0 0 0 6 0 8 10 10 2 0 0 0 0 10 s 3 5 t 10 9 Flow value = 0

  3. 8 X 8 X 8 X Ford-Fulkerson Algorithm 0 flow 2 4 4 capacity G: 0 0 0 6 0 8 10 10 2 0 0 0 0 10 s 3 5 t 10 9 Flow value = 0 2 4 4 residual capacity Gf: 6 8 10 10 2 10 s 3 5 t 10 9 Gf is the “differential” graph of G vis-s-vis the flow f

  4. 10 X X 2 10 X 2 X Ford-Fulkerson Algorithm 0 2 4 4 G: 0 8 8 6 0 8 10 10 2 0 0 8 0 10 s 3 5 t 10 9 Flow value = 8 2 4 4 Gf: 8 6 8 10 2 2 10 s 3 5 t 2 9 8 Gf is the “differential” graph of G vis-s-vis the flow f

  5. 6 X X 6 6 X 8 X Ford-Fulkerson Algorithm 0 2 4 4 G: 0 10 8 6 0 8 10 10 2 2 0 10 2 10 s 3 5 t 10 9 Flow value = 10 2 4 4 Gf: 6 8 10 10 2 10 s 3 5 t 10 7 2

  6. 2 X 8 X X 0 8 X Ford-Fulkerson Algorithm 0 2 4 4 G: 6 10 8 6 6 8 10 10 2 2 6 10 8 10 s 3 5 t 10 9 Flow value = 16 2 4 4 Gf: 6 6 8 4 10 2 4 s 3 5 t 10 1 6 8

  7. 3 X 9 X 7 X 9 X 9 X Ford-Fulkerson Algorithm 2 2 4 4 G: 8 10 8 6 6 8 10 10 2 0 8 10 8 10 s 3 5 t 10 9 Flow value = 18 2 2 4 2 Gf: 8 6 8 2 10 2 2 s 3 5 t 10 1 8 8

  8. Ford-Fulkerson Algorithm 3 2 4 4 G: 9 10 7 6 6 8 10 10 2 0 9 10 9 10 s 3 5 t 10 9 Flow value = 19 3 2 4 1 Gf: 9 1 6 7 1 10 2 1 s 3 5 t 10 9 9

  9. Ford-Fulkerson Algorithm 3 2 4 4 G: 9 10 7 6 6 8 10 10 2 0 9 10 9 10 s 3 5 t 10 9 Cut capacity = 19 Flow value = 19 3 2 4 1 Gf: 9 1 6 7 1 10 2 1 s 3 5 t 10 9 9

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