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 τ. 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 τ. Let ρ be some arbitrary constant. Let Κ be the size of the largest connected component of tetrahedra whose aspect ratios exceed Κ. There exists a constant C dependent on Κ but independent of such that if Κ ≤ 1 Cloglog n + O(1), the running time of our algorithm is polynomial in n, 1/ and log(NW). If Κ = O(1), the running time reduces to O(nϵ-O(1)(log(NW))O(1)).