3-dimensional shortest paths in the presence of polyhedral obstacles

Abstract

We consider the problem of finding a minimum length path between two points in 3-dimensional Euclidean space which avoids a set of (not necessarily convex) polyhedral obstacles; we let n denote the number of the obstacle edges and k denote the number of "islands" in the obstacle space. An island is defined to be a maximal convex obstacle surface such that for any two points contained in the interior of the island, a minimal length path between these two points is strictly contained in the interior of the island; for example, a set of i disconnected convex polyhedra forms a set of i islands, however, a single non-convex polyhedron will constitute more that one island. Prior to this work, the best known algorithm required double-exponential time. We present an algorithm that runs in $n^{k^{0(1)} }$ time and also one that runs in O(n ^log(k)) space.

This work was partially supported by the Office of Naval Research grant number N000-14-80-C-0647 and was completed while this author was visiting the Laboratory for Computer Science at MIT.

This work was partially supported by NSF grant number DCR-8403244.