Lecture15

# Lecture15 - Graphs and Network Flows IE411 Lecture 15 Dr...

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Graphs and Network Flows IE411 Lecture 15 Dr. Ted Ralphs

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IE411 Lecture 15 1 Preflow-Push Algorithms First developed by A. V. Goldberg in 1985. Best preflow-push algorithms outperform best augmenting path algorithms in theory and in practice. Push flows on individual arcs instead of augmenting paths. Do not satisfy mass balance constraints at intermediate stages. Work towards feasibility . Can be seen as a generalization of SAP (SAP is a special case).
IE411 Lecture 15 2 Definitions Distance Labels Valid distance labels d with respect to a flow x satisfy d ( t ) = 0 and d ( i ) d ( j ) + 1 ( i, j ) G ( x ) Admissible Arc An arc ( i, j ) G ( x ) is admissible if d ( i ) = d ( j ) + 1 Preflow Preflow x in a network is defined as one which 1. Satisfies the capacity constraints 0 x ij u ij ( i, j ) 2. Has incoming flow at any node other than the source at least as large as outgoing flow X j :( j,i ) A x ji - X j :( i,j ) A x ij 0 i N - { s }

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IE411 Lecture 15 3 Definitions (con’t) Excess Excess of a node i , denoted e ( i ) , is defined with respect to a flow x e ( i ) = X j :( j,i ) A x ji - X j :( i,j ) A x ij Active Node A node i = t is called active if e ( i ) > 0
IE411 Lecture 15 4 Basic Idea of Preflow-Push Algorithm

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• Spring '14
• TedRalphs
• Graph Theory, Record label, preflow-push algorithm, generic preﬂow-push algorithm

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