An Approximation Algorithm for Computing Shortest Paths in Weighted 3-d Domains

02/15/2011

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by   Lyudmil Aleksandrov, et al.

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We present the first polynomial time approximation algorithm for computing shortest paths in weighted three-dimensional domains. Given a polyhedral domain , consisting of n tetrahedra with positive weights, and a real number ∈(0,1), our algorithm constructs paths in from a fixed source vertex to all vertices of , whose costs are at most 1+ times the costs of (weighted) shortest paths, in O(()n/^2.5n/^31/) time, where () 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, Maheshwari and Sack [JACM 2005] to three dimensions.