Lecture 7

# Lecture 7 - Maximum Flow Lecture 7 The Maximum Flow problem...

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1 Maximum Flow: Lecture 7 The Maximum Flow problem The Ford Fulkerson Algorithm Edmonds-Karp Algorithm Applications

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2 Maximum Flow Problems Oil/Gas Pipeline Transportation Wells, intermediate boosters and pumping stations, refineries, and pipelines form a crude oil transportation network Each pipeline has a finite capacity What is the maximum capacity of the network between the wells and the refineries?
3 The Capacitated Network s 1 3 4 5 6 7 t 9 8 Source Wells Boosters Refineries Sink Each arc has finite capacity 2 7 6 6 5 9 4 5 6 8 8 9 6 3 3

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4 Power Transmission Electric Power Supply for Show @ WanChai Lamma Power Station Wan Chai North Point Repulse Bay Aberdeen PokFuLam Western Central HappyValley 30 50 40 20 20 20 20 10 20 5 40 15 15 40 25 15 20 The great blackout of NA
5 Problem Statement What is the maximum amount of flow from a source node s to a sink node t without exceeding the capacity of any arc in a capacitated network? s 1 2 t 10 8 1 10 6

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6 s 1 2 t 10 , 8 8, 7 1, 1 10, 6 6, 5 s 1 2 t 2 1 1 4 1 8 5 6 7 i j u , ij x ij i j u - x x ij ij ij We let r ij denote the residual capacity of arc ( i, j ) The Residual Network G ( x ) The Residual Network The Ford Fulkerson Algorithm
7 A Useful Idea: Augmenting Paths An augmenting path is a path from s to t with positive capacities in the residual network The residual capacity of the augmenting path P is d ( P ) = min{r ij : (i,j) P }. To augment along P is to send d(P) units of flow along each arc of the path. We modify x and the residual capacities appropriately. r ij := r ij - d(P) and r ji := r ji + d(P) for (i,j) P. s 1 2 t 2 1 4 8 5 6 7 s 1 2 t 2 1 4 8 6 6 8

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8 The Ford Fulkerson Algorithm Begin x := 0; create the residual network G ( x ); while there is some augmenting path from s to t in G ( x ) do begin let P be an augmenting path from s to t in G ( x ); 2200 := d ( P ); send units of flow along P;
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• Fall '10
• D
• Shortest path problem, Flow network, Maximum flow problem, Graph algorithms, Network flow, ford Fulkerson algorithm

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Lecture 7 - Maximum Flow Lecture 7 The Maximum Flow problem...

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