On the number of shortest descending paths on the surface of a convex terrain

• Authors: Mustaq Ahmed;Anil Maheshwari;Subhas C. Nandy;Sasanka Roy
• Venue: Journal of Discrete Algorithms
• Year: 2011

The shortest paths on the surface of a convex polyhedron can be grouped into equivalence classes according to the sequences of edges, consisting of n-triangular faces, that they cross. Mount (1990) [7] proved that the total number of such equivalence classes is @Q(n^4). In this paper, we consider descending paths on the surface of a 3D terrain. A path in a terrain is called a descending path if the z-coordinate of a point p never increases, if we move p along the path from the source to the target. More precisely, a descending path from a point s to another point t is a path @P such that for every pair of points p=(x(p),y(p),z(p)) and q=(x(q),y(q),z(q)) on @P, if dist(s,p)=z(q). Here dist(s,p) denotes the distance of p from s along @P. We show that the number of equivalence classes of the shortest descending paths on the surface of a convex terrain is @Q(n^4). We also discuss the difficulty of finding the number of equivalence classes on a convex polyhedron.