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lecture_20 - Introduction to Algorithms 6.046J/18.401J...

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Introduction to Algorithms 6.046J/18.401J L ECTURE 20 Network Flow I Flow networks Maximum-flow problem Flow notation Properties of flow Cuts Residual networks Augmenting paths Prof. Charles E. Leiserson
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Flow networks Definition. A flow network is a directed graph G = ( V , E ) with two distinguished vertices: a source s and a sink t . Each edge ( u , v ) E has a nonnegative capacity c ( u , v ) . If ( u , v ) E , then c ( u , v ) = 0 . © 2001–4 by Charles E. Leiserson Introduction to Algorithms November 24, 2004 L20.2
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Flow networks Definition. A flow network is a directed graph G = ( V , E ) with two distinguished vertices: a source s and a sink t . Each edge ( u , v ) E has a nonnegative capacity c ( u , v ) . If ( u , v ) E , then c ( u , v ) = 0 . Example: s s t t 3 2 3 3 2 2 3 3 1 2 1 © 2001–4 by Charles E. Leiserson Introduction to Algorithms November 24, 2004 L20.3
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Flow networks Definition. A positive flow on G is a function p : V × V R satisfying the following: Capacity constraint: For all u , v V , 0 p ( u , v ) c ( u , v ) . Flow conservation: For all u V – { s , t } , v u p ) u v p ) = 0 . ( , ( , v V v V © 2001–4 by Charles E. Leiserson Introduction to Algorithms November 24, 2004 L20.4
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Flow networks Definition. A positive flow on G is a function p : V × V R satisfying the following: Capacity constraint: For all u , v V , 0 p ( u , v ) c ( u , v ) . Flow conservation: For all u V – { s , t } , v u p ) u v p ) = 0 . ( , ( , v V v V The value of a flow is the net flow out of the source: v s p ) s v p ) . ( , ( , v V v V © 2001–4 by Charles E. Leiserson Introduction to Algorithms November 24, 2004 L20.5
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A flow on a network positive capacity flow 1:1 2:2 1:3 2:3 s s 0:1 1:3 2:3 1:2 t t 2:2 1:2 2:3 © 2001–4 by Charles E. Leiserson Introduction to Algorithms November 24, 2004 L20.6
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A flow on a network positive capacity flow 1:1 2:2 1:3 2:3 s s 0:1 1:3 2:3 1:2 t t 2:2 u 2:3 1:2 Flow conservation (like Kirchoff’s current law): Flow into u is 2 + 1 = 3 . Flow out of u is 0 + 1 + 2 = 3 . The value of this flow is 1 – 0 + 2 = 3 . © 2001–4 by Charles E. Leiserson Introduction to Algorithms November 24, 2004 L20.7
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The maximum-flow problem Maximum-flow problem: Given a flow network G , find a flow of maximum value on G .
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