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Unformatted text preview: CS 473: Algorithms Chandra Chekuri chekuri@cs.illinois.edu 3228 Siebel Center University of Illinois, UrbanaChampaign Fall 2009 Chekuri CS473ug Part I Network Flows: Introduction and Setup Chekuri CS473ug Transportation/Road Network Chekuri CS473ug Transportation/Road Network Chekuri CS473ug Internet Backbone Network Chekuri CS473ug Common Features of Flow Networks Network represented by a (directed) graph G = ( V , E ) Each edge e has a capacity c ( e ) 0 that limits amount of traffic on e Source(s) of traffic/data Sink(s) of traffic/data Traffic flows from sources to sinks Traffic is switched/interchanged at nodes Flow: abstract term to indicate stuff (traffic/data/etc) that flows from sources to sinks. Chekuri CS473ug Single Source Single Sink Flows Simple setting: single source s and single sink t every other node v is an internal node flow originates at s and terminates at t s 1 2 3 4 5 6 t 15 5 10 30 8 4 9 4 15 6 10 10 15 15 10 Each edge e has a capacity c ( e ) Source s V with no incoming edges Sink t V with no outgoing edges Assumptions: All capacities are integer, and every vertex has at least one edge incident to it. Chekuri CS473ug Definition of Flow Two ways to define flows: edge based path based They are essentially equivalent but have different uses. Edge based definition is more compact. Chekuri CS473ug Edge Based Definition of Flow Definition A flow in a network G = ( V , E ), is a function f : E R such that s 1 2 3 4 5 6 t 14 /15 4 /5 10 /10 14 /30 8 /8 /4 9 /9 /4 1 /15 4 /6 10 /10 9 /10 /15 /15 9 /10 Figure: Flow with value Chekuri CS473ug Edge Based Definition of Flow Definition A flow in a network G = ( V , E ), is a function f : E R such that s 1 2 3 4 5 6 t 14 /15 4 /5 10 /10 14 /30 8 /8 /4 9 /9 /4 1 /15 4 /6 10 /10 9 /10 /15 /15 9 /10 Figure: Flow with value Capacity Constraint: For each edge e , f ( e ) c ( e ) Chekuri CS473ug Edge Based Definition of Flow Definition A flow in a network G = ( V , E ), is a function f : E R such that s 1 2 3 4 5 6 t 14 /15 4 /5 10 /10 14 /30 8 /8 /4 9 /9 /4 1 /15 4 /6 10 /10 9 /10 /15 /15 9 /10 Figure: Flow with value Capacity Constraint: For each edge e , f ( e ) c ( e ) Conservation Constraint: For each vertex v 6 = s , t X e into v f ( e ) = X e out of v f ( e ) Chekuri CS473ug Edge Based Definition of Flow Definition A flow in a network G = ( V , E ), is a function f : E R such that s 1 2 3 4 5 6 t 14 /15 4 /5 10 /10 14 /30 8 /8 /4 9 /9 /4 1 /15 4 /6 10 /10 9 /10 /15 /15 9 /10 Figure: Flow with value Capacity Constraint: For each edge e , f ( e ) c ( e ) Conservation Constraint: For each vertex v 6 = s , t X e into v f ( e ) = X e out of v f ( e ) Value of flow: total flow out of source (or in to sink) Chekuri CS473ug More Definitions and Notation Notation The inflow into a vertex v is f in ( v ) = e into v f ( e ) and the outflow is f out ( v ) = e out of v f ( e ) For a set of vertices A , f in ( A ) = e into A f...
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