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NCApproximation Schemes for NP and PSPACEHard Problems for Geometric Graphs
, 1997
"... We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance tradeoff as the best known approximation schemes for planar gr ..."
Abstract

Cited by 93 (1 self)
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We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance tradeoff as the best known approximation schemes for planar graphs. We also define the concept of precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance tradeoff than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for precision unit disk graphs, many more graph problems have efficient approximation schemes. Our NC approximation schemes can also be extended to obtain efficient NC approximation schemes for several PSPACEhard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann and Widmayer. The approximation schemes for hierarchically specified un...
A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs
, 1998
"... In this paper we give a fully dynamic approximation scheme for maintaining allpairs shortest paths in planar networks. Given an error parameter ε such that 0 < ε, our algorithm maintains approximate allpairs shortest paths in an undirected planar graph G with nonnegative edge lengths. The approxi ..."
Abstract

Cited by 17 (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 ε such that 0 < ε, 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 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 allpairs shortest paths among a selected subset of the nodes by a sparse substitute graph.