On Point Location and Motion Planning among Simplices

Abstract

Let $\SS$ be a set of $n$ possibly intersecting $(d-1)$-simplices in $d$-space for $d \geq 2$, and let ${\cal A}(\SS)$ be the arrangement of $\SS$. Let $K = |{\cal A}(\SS)|$ be the number of faces of any dimension in the arrangement of $\SS$. A data structure is described that uses storage $O(n^{d-1+\eps} +K)$ and is built {\em deterministically} in time $O(n^{d-1+\eps} +K\log n)$, where $\eps >0$ is an arbitrarily small constant, such that the face of ${\cal A}(\SS)$ containing a query point is located in time $O(\log^3 n)$. If two query points are in the same cell of ${\cal A}(\SS)$, a collision-free path connecting them is produced. This result is obtained by exploiting powerful and so far overlooked properties of sparse nets introduced by Chazelle [{\em Discrete Comput. Geom.}, 9 (1993), pp. 145--158]. If the $(d-1)$-simplices in $\SS$ have pairwise-disjoint interiors and $d \geq 3$, improved bounds are obtained. A data structure is described that uses $O(n^{d-1})$ storage and is built deterministically in time $O(n^{d-1})$ such that point-location queries are solved in time $O(\log n)$. Also, as a by-product, this method gives the first optimal worst-case algorithm for triangulating a nonsimple polyhedron in 3-space.