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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 15

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online