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 $\mathit{dist}(s,p)<\mathit{dist}(s,q)$ then $z(p)⩾z(q)$, where $\mathit{dist}(s,p)$ 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.