Navigating Weighted Regions with Scattered Skinny Tetrahedra

We propose an algorithm for finding a \((1+\varepsilon )\)-approximate shortest path through a weighted 3D simplicial complex \(\mathcal T\). 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 \(\mathcal T\). Let \(\rho \) be some arbitrary constant. Let \(\kappa \) be the size of the largest connected component of tetrahedra whose aspect ratios exceed \(\rho \). There exists a constant C dependent on \(\rho \) but independent of \(\mathcal T\) such that if \(\kappa \le \frac{1}{C}\log \log n + O(1)\), the running time of our algorithm is polynomial in n, \(1/\varepsilon \) and \(\log (NW)\). If \(\kappa = O(1)\), the running time reduces to \(O(n \varepsilon ^{-O(1)}(\log (NW))^{O(1)})\).