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Unformatted text preview: CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi (saberi@stanford.edu) 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 nonnegative 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, ≤ 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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 Spring '10
 JoeWhite
 Algorithms

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