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Unformatted text preview: Max Flow In optimal paths, every arc has a length, cost, or value. Now, every arc has a width. vi vj bi,j is the amount of flow that can go from vi to vj . 1 / 13 Max Flow 1 1 S 3 1 3 1 1 S 3 1 2 1 1 2 2 T 2 1 2 2 T 2 / 13 Max Matching S T 3 / 13 Ford Fulkerson i max v = ? +v xi,j  xj,k = 0 k v 0 xi,k bi,k j=s j = s, t j=t 4 / 13 Ford Fulkerson Max flow algorithm X X s t Capacity of a cut: C X, X =
iX,jX bi,j 5 / 13 Ford Fulkerson Labeling Process Step 0. vs X Step 1. If vi X and xi,j < bi,j then vj X Step 2. If vi X and xj,i > 0 then vj X 6 / 13 Max Flow a S 3 2 c 2 1 1 1 b 2 2 d 3 T S T 7 / 13 MultiTerminal Max Flow Undirected edges only. i.e. bi,j = bj,i Given edge capacities j 3 i Result of Max Flows j 7 i 8 7 k 5 4 k 8 / 13 Max Flow For any three nodes, Fi,j , Fj,k , Fi,k satisfy Fi,k min(Fi,j , Fj,k ) X X i k Fi,k = C[ X, X ] 9 / 13 Max Flow Fi,j min(Fi,k , Fk,j ) Fj,k min(Fj,i , Fi,k ) Among any three values, two values must be the same, and the third one is bigger or also the same value. 10 / 13 Max Flow Representation of j 7 i 8 7 k Given n = 100  node network
n 2 Max Flows = (n  1) Max Flows 11 / 13 Max Flow Edge Capacities e 2 j 3 k 1 1 2 s i Result of Max Flows e 3 j 3 k 2 s i 3 12 / 13 Max Flows For any undirected network, there is a treeshape network which has the same max flows and the same n  1 cut value. GomoryHu cut tree G  H tree Not only flow  equivalent, but also cut equivalent. 13 / 13 ...
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 Fall '06
 Hu
 Algorithms

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