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# Module6_08 - Flow Networks Topics Flow Networks Residual...

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11/19/09 CSE 5311 Kumar 1 Flow Networks Topics Flow Networks Residual networks Ford-Fulkerson’s algorithm Ford-Fulkerson's Max-flow Min-cut Algorithm Chapter 7 Algorithm Design Kleinberg and Tardos

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11/19/09 CSE 5311 Kumar 2 Flow Networks A directed graph can be interpreted as a flow network to analyse material flows through networks. Material courses through a system from a source (where it is produced) to a sink (where it is consumed). Examples : Water through pipelines Newspapers through distribution system Electricity through cables Cars on a production line on roads The source produces the material at a steady rate . The sink consumes the material at a steady rate
11/19/09 CSE 5311 Kumar 3 Flow: the rate at which the material moves from one point to another 100 litres of water per hour in a pipe 30 Amperes of electric current in a circuit 5 litres/hour 30 liters/hour 25 litres/hour The rate at which a material enters a vertex = the rate at which the material leaves the vertex

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11/19/09 CSE 5311 Kumar 4 The flow network G =(V,E) is a directed graph in which each edge (u,v) E has a nonnegative capacity c(u,v) 0. If (u,v) E then c(u,v) = 0. A flow network has a source vertex s , and a sink vertex t . For every vertex v V there is a path from s to v and v to t in a connected graph. source sink s t
11/19/09 CSE 5311 Kumar 5 A flow in G is a real-valued function f : V × V R that satisfies the following three properties: 1. Capacity constraint : For all u,v V , we require f(u,v) c(u,v). The net flow from one vertex to another must not exceed the given capacity. 2. Skew symmetry : For all u,v V , we require f(u,v) = -f( v,u ). The net flow from a vertex u to a vertex v is the negative of the net flow in the reverse direction. The net flow from a vertex to itself is zero for all u V , that is f(u,u) = 0. 3. Flow conservation : For all u V - {s,t}, we require The total net flow out of a vertex other than the source or sink is zero. = V v v u f 0 ) , (

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11/19/09 CSE 5311 Kumar 6 The quantity f(u,v) can be negative or positive, it is called the net flow from vertex u to v . The value of a flow is defined as = V v v s f f ) , ( In the maximum-flow problem, we are given a flow network G with source s and sink t , and we wish to find a flow of maximum value from s to t . There is no net flow between u and v if there is no edge between them. If (u,v) E and (v,u) E , then c(u,v) = c(v,u) = 0.
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