Navigating Weighted Regions with Scattered Skinny Tetrahedra

Abstract

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 $O(n \varepsilon ^{-O(1)}(\log (NW))^{O(1)})$.

S.-W. Cheng—Supported by Research Grants Council, Hong Kong, China (project no. 611812).