6.889: Algorithms for Planar Graphs and BeyondOctober 26, 2011Problem Set 7This problem set is due Wednesday, 11/2/2011 at noon.Problem:Give anO(nlogn)–time algorithm to compute anr–division (O(n/r) pieces ofsizeO(r) and boundaryO(√r)) with the additional property that the boundary nodes ofeach piece lie on a constant number of faces (called “holes”). Note that a face of a piece isnot 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 partitioningVintoA, B, Seachof the two componentsA∪S, B∪Shas weight exactlyw(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 graphGas a piece with no boundary nodes, split it recursively into twosubpieces using a cycle separator, retriangulate each piece, and recurse. The recursion stops
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