Lecture7 - CME 305: Discrete Mathematics and Algorithms...

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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 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, ≤ 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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Lecture7 - CME 305: Discrete Mathematics and Algorithms...

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