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ps10 - that there are no counterclockwise residual cycles...

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6.889: Algorithms for Planar Graphs and Beyond November 16, 2011 Problem Set 10 This problem set is due Wednesday, November 23 at noon. In this problem set you will develop an algorithm for canceling flow cycles in a given flow assignment. In general graphs this can be done in O ( m log n ) time using Sleator’s and Tarjan’s dynamic trees. You will use the relation between dual shortest paths and circulations to give a linear time algorithm in planar graphs. 1. Let G be a planar graph with non-negative capacities on its arcs. Let φ be shortest path distances from f in G * . Let θ be the circulation induced by φ . That is, θ ( d ) = φ (head( d )) - φ (tail( d )) . Recall that a residual path is a path whose darts all have strictly positive capacities. Show
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Unformatted text preview: that there are no counterclockwise residual cycles in the residual graph G θ . 2. What price function φ would you use to get the same property as in (1), but with no clockwise residual cycles? 3. Use parts (1) and (2) to give a linear time algorithm that, given a flow assignment γ in G makes γ acyclic by removing all flow cycles in γ . That is, it produces another flow assignment γ s.t. (a) γ-γ is a circulation (b) for every arc a , γ (( a, 1)) ≤ γ (( a, 1)) (c) for any cycle C there is a dart d ∈ C s.t. γ ( d ) ≤...
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