Flowext - CMSC 451 Max-Flow Extensions Slides By Carl Kingsford Department of Computer Science University of Maryland College Park Based on Section

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Unformatted text preview: CMSC 451: Max-Flow Extensions Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Section 7.7 of Algorithm Design by Kleinberg & Tardos. Circulations with Demands • Suppose we have multiple sources and multiple sinks. • Each sink wants to get a certain amount of flow (its demand ). • Each source has a certain amount of flow to give (its supply ). • We can represent supply as negative demand . Demand Example 1 2 5 3 4 6 8 9 7 5 3 2 4 14 11 9 7 4 3 4 10 3 demand d 5 = 3 supply d 1 = -4 supply d 2 = -7 d 8 = 8 d 4 = 0 Constraints Goal: find a flow f that satisfies: 1 Capacity constraints : For each e ∈ E , 0 ≤ f ( e ) ≤ c e . 2 Demand constraints : For each v ∈ V , f in ( v )- f out ( v ) = d v . The demand d v is the excess flow that should come into node. Sources and Sinks Let S be the set of nodes with negative demands (supply). Let T be the set of nodes with positive demands (demand). In order for there to be a feasible flow, we must have: X s ∈ S- d s = X t ∈ T d t Let D = ∑ t ∈ T d t . Reduction How can we turn the circulation with demands problem into the maximum flow problem? Reduction...
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This note was uploaded on 01/13/2012 for the course CMSC 423 taught by Professor Staff during the Fall '07 term at Maryland.

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Flowext - CMSC 451 Max-Flow Extensions Slides By Carl Kingsford Department of Computer Science University of Maryland College Park Based on Section

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