Approximate Shortest Paths and Geodesic Diameter on a Convex Polytope in Three Dimensions

Abstract

Given a convex polytope P with n edges in $\Bbb R$ 3 , we present a relatively simple algorithm that preprocesses P in O(n) time, such that, given any two points $s,t \in \partial P$ , and a parameter 0 < $\varepsilon \le$ 1, it computes, in O(log n) /ɛ 1.5 + 1/ ɛ 3 ) time, a distance Δ P (s,t) , such that d P (s,t) $\leq$ Δ P (s,t) $\leq$ (1+ɛ )d P (s,t) , where d P (s,t) is the length of the shortest path between s and t on $\partial{P}$ . The algorithm also produces a polygonal path with O (1/ɛ 1.5 ) segments that avoids the interior of P and has length Δ P (s,t) . Our second related result is: Given a convex polytope P with n edges in $\Bbb R$ 3 , and a parameter 0 < $\varepsilon \leq$ 1, we present an O (n + 1/ ɛ 5 )-time algorithm that computes two points ${\frak{s}}, {\frak{t}} \in {\partial}{P}$ such that $d_P({\frak{s}}, {\frak{t}}) \geq (1-{\varepsilon}){\cal D}_P$ , where ${\cal D}_P = \max_{s, t \in {\partial}{P}} d_P(s, t)$ is the geodesic diameter of P .