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Lecture7 - CME 305 Discrete Mathematics and Algorithms...

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CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi ([email protected]) January 26, 2010 Lecture 7: Network Flow and Minimum Cut In the next two lectures, we will study the network flow problem which has important applications in commu- nication networks. Moreover, many seemingly unrelated problems can be viewed as network flow problems. Definition: A network N is a set containing: a directed graph G ( V, E ); a vertex s V which has only outgoing edges, we call s the source node; a vertex t V which has only incoming edges, we call t the sink node; a capacity function c : E 7→ IR + , where IR + is the set of non-negative real numbers. Definition: A flow f on a network N is a function f : E 7→ IR + . Flow f is a feasible flow if it satisfies the following two conditions: 1. Edge capacity limit: e E, 0 f ( e ) c ( e ) 2. Conservation of flow: v V \ { s, t } , X e leaving v f ( e ) = X e entering v f ( e ) A network flow can be used as a model of packet routing in computer networks, finding a route from point a to point b in traffic/congestion grids, a supply chain problem, flow of water through pipes, or electricity flow in a circuit.
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