08 - The preflow-push algorithm for maximum flow

08 - The preflow-push algorithm for maximum flow - 20...

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2/29/08 - The prefow-push algorithm. .. The Prefow-Push Algorithm DeF. A prefow in a fow network is a Function F: E -> + s.t. (i) 0 F(e) c e (ii) v s e In(v) F(e) - e Out(v) F(e) =: e F (v) 0 “excess at v” DeF. A labeling is a Function h: V -> 0 . It is compatible with a prefow F iF (i) h(s) = n, h(t) = 0. n = |V(G)| (ii) “steepness” IF (v,w) E(G F ) then h(v) 1 + h(w) Lemma. IF F is a fow and (F, h) are compatible then F is a maximum fow. ProoF. IF F is not a max fow then path s = v 0 -> v 1 -> v 2 -> . .. -> v k = t in the residual graph. h(v k ) = 0, h(v k-1 ) 1, . .., h(s) = h(v 0 ) k. < n =><= Init h(s) = n h(v) = 0 v s. F(e) = {c e iF e Out(s) {0 otherwise Push(F,h,v,w) Require e F (v) > 0 h(v) > h(w) (v,w) E(G F ) iF e=(v,w) is Forward δ = min{e F (v), c e - F(e)} F(e) <- F(e) + δ iF e = (v,w) is backward 20-1 20
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δ = min{e f (v), f(e)} f(e) + f(e) - δ return (f,h) Relabel(f,h,v) Require e f (v) > 0 e = (v,w)
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This note was uploaded on 10/02/2008 for the course CS 482 taught by Professor Kleinberg during the Spring '08 term at Cornell.

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08 - The preflow-push algorithm for maximum flow - 20...

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