TR-11-02: An Approximation Algorithm for Computing Shortest Paths in Weighted 3-d Domains

Abstract

We present the first polynomial time 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 $epsin(0,1)$, our algorithm constructs paths in D from a fixed source vertex to all vertices of D, whose costs are at most $1+eps$ times the costs of (weighted) shortest paths, in $O(C(D)frac{n}{eps^{2.5}}logfrac{n}{eps}log^3frac{1}{eps})$ 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, Maheshwari and Sack [JACM 2005] to three dimensions.

TR-11-02.pdf