An Approximation Algorithm for Shortest Descending Paths

Abstract

A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. No efficient algorithm is known to find a shortest descending path (SDP) from s to t in a polyhedral terrain. We give a simple approximation algorithm that solves the SDP problem on general terrains. Our algorithm discretizes the terrain with O(n2 � 2 n X X/ǫ) Steiner points so that after an O ǫ log � � nX ǫ-time preprocessing phase for a given vertex s, we can determine a (1 + ǫ)approximate SDP from s to any point v in O(n) time if v is either a vertex of the terrain or a Steiner point, and in O(nX/ǫ) time otherwise. Here n is the size of the terrain, and X is a parameter of the geometry of the terrain.