A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs

13 years 4 months ago

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In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks. Given an error parameter ε such that 0 < ε, our algorithm maintains approximate all-pairs shortest paths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a 1 + ε factor. The time bounds for both query and update for our algorithm is O(ε−1n2/3 log2 n log D), where n is the number of nodes in G and D is the sum of its edge lengths. The time bound for the queries is worst case, while that for the additions is amortized. Our approximation algorithm is based upon a novel technique for approximately representing all-pairs shortest paths among a selected subset of the nodes by a sparse substitute graph. Key Words. Dynamic algorithm, Graph algorithm, Shortest path, Minimum-cost path, Planar graph.

Philip N. Klein, Sairam Subramanian

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