Approximating Shortest Paths on a Nonconvex Polyhedron

Abstract

We present an approximation algorithm that, given the boundary P of a simple, nonconvex polyhedron in ${\mathbb R}^3$ and two points s and t on P, constructs a path on P between s and t whose length is at most ${7(1+{\varepsilon})} dP(s,t), where dP(s,t) is the length of the shortest path between s and t on P, and ${\varepsilon} 0$ is an arbitrarily small positive constant. The algorithm runs in O(n5/3 log5/3 n) time, where n is the number of vertices in P. We also present a slightly faster algorithm that runs in O(n8/5 log8/5 n) time and returns a path whose length is at most ${15(1+{\varepsilon})} d_{P}(s,t)$.