Shortest monotone descent path problem in polyhedral terrain

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. (c) 2006 Elsevier B.V. All rights reserved.