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

Given a convex polytope P with n edges in R3, we present a relatively simple algorithm that preprocesses P in O(n) time, such that, given any two points s, t ∈ ∂P , and a parameter 0 < ε ≤ 1, it computes, in O((log n)/ε1.5 + 1/ε3) time, a distance 1P(s, t), such that dP(s, t) ≤ 1P(s, t) ≤ (1+ ε)dP(s, t), where dP(s, t) is the length of the shortest path between s and t on ∂P . The algorithm also produces a polygonal path with O(1/ε1.5) segments that avoids the interior of P and has length 1P(s, t). Our second related result is: Given a convex polytope P with n edges in R3, and a parameter 0 < ε ≤ 1, we present an O(n+ 1/ε5)-time algorithm that computes two points s, t ∈ ∂P such that dP(s, t) ≥ (1− ε)DP , where DP = maxs,t∈∂P dP(s, t) is the geodesic diameter of P .