lect13 - Lecture 13 Introduction to Maximum Flows Flow...

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Lecture 13 Introduction to Maximum Flows

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Flow Network A flow network ( , ) is a directed graph in which each edge ( , ) has a nonnegative capacity ( , ) 0. We distinguish two nodes: a source and a sink . A flow in is a function : such tha G V E x y E u x y s t G f V V R = × t (Capacity constraint) ( , ) ( , ) for all , , (Skew symmetry) ( , ) ( , ) for all , , (Flow conservation) ( , ) 0 for all { , }. | | ( , ) ( ( )) y V y V f x y u x y x y V f x y f y x x y V f x y x V s t f f s y v f = - = - = =
3

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The Ford Fulkerson Maximum Flow Algorithm Begin x := 0; create the residual network G(x); while there is some directed path from s to t in G(x) do begin let P be a path from s to t in G(x); := δ (P); send units of flow along P; update the r's; End end {the flow x is now maximum}.
The Ford-Fulkerson Augmenting Path Algorithm for the Maximum Flow Problem

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Ford-Fulkerson Max Flow 4 1 2 1 2 3 1 s 2 5 3 t This is the original network, plus reversals of the arcs.
Ford-Fulkerson Max Flow 4 1 1 2 1 2 3 1 s 2 5 3 t This is the original network, and the original residual network.

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4 1 1 2 1 2 3 1 Ford-Fulkerson Max Flow Find any s-t path in G(x) s 2 5 3 t
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lect13 - Lecture 13 Introduction to Maximum Flows Flow...

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