Navigating Weighted Regions with Scattered Skinny Tetrahedra

Abstract

We propose an algorithm for finding a (1 + ε)-approximate shortest path through a weighted 3D simplicial complex T. The weights are integers from the range [1,W] and the vertices have integral coordinates. Let N be the largest vertex coordinate magnitude, and let n be the number of tetrahedra in T. Let ρ be some arbitrary constant. Let κ be the size of the largest connected component of tetrahedra whose aspect ratios exceed ρ. There exists a constant C dependent on ρ but independent of T such that if κ ≤ 1 C log log n + O(1), the running time of our algorithm is polynomial in n, 1/ε and log(NW). If κ = O(1), the running time reduces to O(nε−O(1)(log(NW))O(1)).