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A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs
, 1993
"... In this paper we give a fully dynamic approximation scheme for maintaining allpairs shortest paths in planar networks. Given an error parameter ffl such that 0 ! ffl, our algorithm maintains approximate allpairs shortestpaths in an undirected planar graph G with nonnegative edge lengths. The a ..."
Abstract

Cited by 19 (1 self)
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In this paper we give a fully dynamic approximation scheme for maintaining allpairs shortest paths in planar networks. Given an error parameter ffl such that 0 ! ffl, our algorithm maintains approximate allpairs shortestpaths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a 1 + ffl factor. The time bounds for both query and update for our algorithm is O(ffl \Gamma1 n 2=3 log 2 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 adds is amortized. Our approximation algorithm is based upon a novel technique for approximately representing allpairs shortest paths among a selected subset of the nodes by a sparse substitute graph. Research supported by NSF grant CCR9012357 and NSF PYI award CCR9157620, together with PYI matching funds from Thinking Machines Corporation and Xerox Corporation. Addit...