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# hw3(12) - CS CS 473g Algorithms Fall 2007 Homework 3(due...

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CS CS 473g: Algorithms, Fall 2007 Homework 3 (due Tuesday, October 23, 2007 at 11:59.99 p.m.) Version 1.02 Required Problems 1. The good, the bad, and the middle. [20 Points] Suppose you’re looking at a flow network G with source s and sink t , and you want to be able to express something like the following intuitive notion: Some nodes are clearly on the “source side” of the main bottlenecks; some nodes are clearly on the “sink side” of the main bottlenecks; and some nodes are in the middle. However, G can have many minimum cuts, so we have to be careful in how we try making this idea precise. Here’s one way to divide the nodes of G into three categories of this sort. We say a node v is upstream if, for all minimum s - t cuts ( A, B ) , we have v A – that is, v lies on the source side of every minimum cut. We say a node v is downstream if, for all minimum s - t cuts ( A, B ) , we have v B – that is, v lies on the sink side of every minimum cut. We say a node v is central if it is neither upstream nor downstream; there is at least one minimum s - t cut ( A, B ) for which v A , and at least one minimum s - t cut ( A 0 , B 0 ) for which v B 0 . Give an algorithm that takes a flow network G and classifies each of its nodes as being upstream, downstream, or central. The running time of your algorithm should be within a constant factor of the time required to compute a single maximum flow.

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hw3(12) - CS CS 473g Algorithms Fall 2007 Homework 3(due...

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