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flowext - CMSC 451 Max-Flow Extensions Slides By Carl...

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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.
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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 .
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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
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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.
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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 .
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Reduction How can we turn the circulation with demands problem into the maximum flow problem?
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