# 10 - Algorithms – Flows and Cuts Flows and Cuts Design...

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Unformatted text preview: Algorithms – Flows and Cuts Flows and Cuts Design and Analysis of Algorithms Andrei Bulatov Algorithms – Flows and Cuts 10-2 Flows and Flow Networks A flow network is a digraph with a unique source and sink nodes Arcs have capacities A flow is a function f: E → such that (1) ( Capacity condition ) For each e ∈ E, we have ) ( onservation condition irchhoff principle e c e f ≤ ≤ ) ( + R (2) ( Conservation condition , Kirchhoff principle for each node except s and t The value of the flow is ∑ ∑ = v e v e e f e f of out into ) ( ) ( ∑ s e e f of out ) ( u v s t 20 10 20 10 30 Algorithms – Flows and Cuts 10-3 Residual Graph Starting with the zero flow- push a flow along (s,u), (u,v), (v,t) such that f(s,u) = f(u,v) = f(v,t) = 20- construct the residual graph w.r.t. f- push a flow along (s,v), (v,u), (u,t) s. t. g(s,v) = g(v,u) = g(u,t) = 10 nstruct the residual graph .r.t g- construct the residual graph w.r.t. g- we cannot push any flow anymore. - is f + g maximal? u v s t 20 10 20 10 30 u v s t 20 10 20 10 20 10 u v s t 20 10 20 10 10 20 Algorithms – Flows and Cuts 10-4 Ford-Falkerson: Analysis Let Theorem If all the capacities are integers then the Ford-Falkerson algorithm ∑ = s e e c C of out L can be implemented to run in O(mC) time Theorem If all the capacities are integers then the Ford-Falkerson algorithm finds a maximal flow. Algorithms – Flows and Cuts...
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10 - Algorithms – Flows and Cuts Flows and Cuts Design...

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