ps10sol - 6.889: Algorithms for Planar Graphs and Beyond...

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6.889: Algorithms for Planar Graphs and Beyond Problem Set 10 - Solutions 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 that there are no counterclockwise residual cycles in the residual graph G θ . Solution: Let T * be a shortest path tree in G * rooted at f , and let C be a counterclockwise cycle. Since C encloses at least one face (and by definition does not enclose f ), and since T *
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This note was uploaded on 01/20/2012 for the course CS 6.889 taught by Professor Erikdemaine during the Fall '11 term at MIT.

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