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ps7sol - 6.889 Algorithms for Planar Graphs and Beyond...

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6.889: Algorithms for Planar Graphs and Beyond October 26, 2011 Problem Set 7 This problem set is due Wednesday, 11/2/2011 at noon. Problem: Give an O ( n log n )–time algorithm to compute an r –division ( O ( n/r ) pieces of size O ( r ) and boundary O ( r )) with the additional property that the boundary nodes of each piece lie on a constant number of faces (called “holes”). Note that a face of a piece is not necessarily a face of the graph. For simplicity, you may assume that the cycle separator theorem achieves perfect balance (meaning that, whenever we apply the separator theorem partitioning V into A, B, S each of the two components A S, B S has weight exactly w ( V ) / 2). Solution: The following algorithm was extracted from a preprint of Christian Wulff-Nilsen ( arXiv:1007.3609v2 ), wherein the algorithm is analyzed in great detail. Regard the entire graph G as a piece with no boundary nodes, split it recursively into two subpieces using a cycle separator, retriangulate each piece, and recurse. The recursion stops
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