A fully dynamic approximation scheme for all-pairs shortest paths in planar graphs
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≤1, our algorithm maintains approximate allpairs 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 wis O(ε−1n2/3 log2n log D), where n is the number of nodes in G and D is the sum of its edge lengths.
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.
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