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

Abstract

We present an approximation algorithm for computing shortest paths in weighted three-dimensional domains. Given a polyhedral domain D, consisting of n tetrahedra with positive weights, and a real number ε ∈ (0,1), our algorithm constructs paths in D from a fixed source vertex to all vertices of D, the costs of which are at most 1 + ε times the costs of (weighted) shortest paths, in O(C(D)n/ε2.5logn/εlog31/ε) time, where 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. © 2013 Springer Science+Business Media New York.