Shortest paths on polyhedral surfaces and terrains

Abstract

We present an algorithm for computing shortest paths on polyhedral surfaces under convex distance functions. Let n be the total number of vertices, edges and faces of the surface. Our algorithm can be used to compute L1 and L∞ shortest paths on a polyhedral surface in O(n2 log4 n) time. Given an ε ∈ (0, 1), our algorithm can find (1 + ε)-approximate shortest paths on a terrain with gradient constraints and under cost functions that are linear combinations of path length and total ascent. The running time is O[EQUATION]. This is the first efficient PTAS for such a general setting of terrain navigation.