Optimistic and Pessimistic Shortest Paths on Uncertain Terrains

We consider the problem of finding shortest paths on the surface of uncertain terrains. In this paper, a terrain is a triangulated 2D surface in 3D such that every vertical line intersects the surface at most once. Terrains of this type are used to represent, for example, a piece of the earth’s surface, and are typically inexact. We model their uncertainty by allowing the terrain vertices to have a range of possible heights (Z-coordinates), while fixing the triangulation (i.e. the adjacency of vertices). This defines a set of feasible (certain) terrains. We are looking for a “shortest” path between two vertices s and t (defined by its projection to the XY -plane) but the length of any particular path may depend on the actual feasible terrain. We consider both pessimistic (a path’s length is its maximum length over all feasible terrains) and optimistic (a path’s length is its minimum feasible length) scenarios. If we are allowed to walk on the faces of the terrain, the problem is NP-hard in both pessimistic and optimistic scenarios [5, 4]. In this paper, we prove that if we can walk only on terrain edges, the pessimistic problem is still NP-hard (and we give a fully-polynomial time approximation scheme for it) while the optimistic problem is solvable in polynomial time.