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

We present the first polynomial time approximation algorithm for computing short-est paths in weighted three-dimensional domains. Given a polyhedral domain D, con-sisting of n tetrahedra with positive weights, and a real number ε ∈ (0, 1), our algorithmconstructs paths in D from a fixed source vertex to all vertices of D, whose costs areat most 1 + ε times the costs of (weighted) shortest paths, in O(C(D) nε2.5 log nε log 3 1ε )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 thelocal behavior of geodesic paths and additive Voronoi diagrams in weighted three-dimensional domains, which are of independent interest. The paper extends the resultsof Aleksandrov et al. [4] to three dimensions.