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- W1798872925 abstract "We present an approximation algorithm for computing shortest paths in weighted three-dimensional domains. Given a polyhedral domain $$mathcal D $$D, consisting of $$n$$n tetrahedra with positive weights, and a real number $$varepsilon in (0,1)$$źź(0,1), our algorithm constructs paths in $$mathcal D $$D from a fixed source vertex to all vertices of $$mathcal D $$D, the costs of which are at most $$1+varepsilon $$1+ź times the costs of (weighted) shortest paths, in $$O(mathcal{C }(mathcal D )frac{n}{varepsilon ^{2.5}}log frac{n}{varepsilon }log ^3frac{1}{varepsilon })$$O(C(D)nź2.5lognźlog31ź) time, where $$mathcal{C }(mathcal D )$$C(D) is a geometric parameter related to the aspect ratios of tetrahedra. The efficiency of the proposed algorithm is based on an in-depth study of the local behavior of geodesic paths and additive Voronoi diagrams in weighted three-dimensional domains, which are of independent interest. The paper extends the results of Aleksandrov et al. (J ACM 52(1):25---53, 2005), to three dimensions." @default.
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- W1798872925 date "2013-06-01" @default.
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- W1798872925 title "An Approximation Algorithm for Computing Shortest Paths in Weighted 3-d Domains" @default.
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