Shortest monotone descent path problem in polyhedral terrain

Abstract

Given a polyhedral terrain with n vertices, the shortest monotone descent path problem deals with finding the shortest path between a pair of points, called source ( s) and destination ( t) such that the path is constrained to lie on the surface of the terrain, and for every pair of points p = ( x ( p ) , y ( p ) , z ( p ) ) and q = ( x ( q ) , y ( q ) , z ( q ) ) on the path, if dist ( s , p ) < dist ( s , q ) then z ( p ) ⩾ z ( q ) , where dist ( s , p ) denotes the distance of p from s along the aforesaid path. This is posed as an open problem by Berg and Kreveld [M. de Berg, M. van Kreveld, Trekking in the Alps without freezing or getting tired, Algorithmica 18 (1997) 306–323]. We show that for some restricted classes of polyhedral terrain, the optimal path can be identified in polynomial time.