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Structured Recursive Separator Decompositions for Planar Graphs in Linear Time
, 2012
"... Given a planar graph G on n vertices and an integer parameter r < n, an r–division of G with few holes is a decomposition of G into O(n/r) regions of size at most r such that each region contains at most a constant number of faces that are not faces of G (also called holes), and such that, for ea ..."
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Given a planar graph G on n vertices and an integer parameter r < n, an r–division of G with few holes is a decomposition of G into O(n/r) regions of size at most r such that each region contains at most a constant number of faces that are not faces of G (also called holes), and such that, for each region, the total number of vertices on these faces is O( r). We provide a lineartime algorithm for computing r–divisions with few holes. In fact, our algorithm computes a structure, called decomposition tree, which represents a recursive decomposition of G that includes r–divisions for essentially all values of r. In particular, given an increasing sequence r = (r1, r2,...), our algorithm can produce a recursive r–division with few holes in linear time. r–divisions with few holes have been used in efficient algorithms to compute shortest paths, minimum cuts, and maximum flows. Our lineartime algorithm improves upon the decomposition algorithm used in the stateoftheart algorithm for minimum st–cut (Italiano, Nussbaum, Sankowski, and WulffNilsen, STOC 2011), removing one of the bottlenecks in the overall running time of their algorithm (analogously for minimum cut in planar and boundedgenus graphs).
Faster Shortest Paths in Dense Distance Graphs, with Applications
, 2014
"... We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the lineartime shortestpath algorithm of Henzinger, Klein, Subramanian, and Rao [STOC’94]. The second is Fakcharoenphol and Rao’s algorithm [FOCS’01] for emulating Dijkstra’s alg ..."
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We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the lineartime shortestpath algorithm of Henzinger, Klein, Subramanian, and Rao [STOC’94]. The second is Fakcharoenphol and Rao’s algorithm [FOCS’01] for emulating Dijkstra’s algorithm on the dense distance graph (DDG). A DDG is defined for a decomposition of a planar graph G into regions of at most r vertices each, for some parameter r < n. The vertex set of the DDG is the set of Θ(n/ r) vertices of G that belong to more than one region (boundary vertices). The DDG has Θ(n) arcs, such that distances in the DDG are equal to the distances in G. Fakcharoenphol and Rao’s implementation of Dijkstra’s algorithm on the DDG (nicknamed FRDijkstra) runs in O(n log(n)r−1/2 log r) time, and is a key component in many stateoftheart planar graph algorithms for shortest paths, minimum cuts, and maximum flows. By combining these two techniques we remove the logn dependency in the running time of the shortestpath algorithm, making it O(nr−1/2 log2 r). This work is part of a research agenda that aims to develop new techniques that would lead to faster, possibly lineartime, algorithms for problems such as minimumcut, maximumflow, and shortest paths with negative arc lengths. As immediate applications, we show how to compute maximum flow in directed weighted planar graphs in O(n log p) time, where p is the minimum number of edges on any path from the source to the sink. We also show how to compute any part of the DDG that corresponds to a region with r vertices and k boundary vertices in O(r log k) time, which is faster than has been previously known for small values of k.