Navigating Weighted Regions with Scattered Skinny Tetrahedra

Abstract

We propose an algorithm for finding a [Formula: see text]-approximate shortest path through a weighted 3D simplicial complex [Formula: see text]. The weights are integers from the range [Formula: see text] and the vertices have integral coordinates. Let [Formula: see text] be the largest vertex coordinate magnitude, and let [Formula: see text] be the number of tetrahedra in [Formula: see text]. Let [Formula: see text] be some arbitrary constant. Let [Formula: see text] be the size of the largest connected component of tetrahedra whose aspect ratios exceed [Formula: see text]. There exists a constant [Formula: see text] dependent on [Formula: see text] but independent of [Formula: see text] such that if [Formula: see text], the running time of our algorithm is polynomial in [Formula: see text], [Formula: see text] and [Formula: see text]. If [Formula: see text], the running time reduces to [Formula: see text].