Shortest Paths on Convex Polyhedral Surfaces

Paper

Abstract

Let

be the surface of a convex polyhedron with

vertices. We consider the two-point shortest path query problem for

: Constructing a data structure so that given any two query points

and

, a shortest path from

can be computed efficiently. To achieve

query time (for computing the shortest path length), the previously best result uses

preprocessing time and space [Aggarwal, Aronov, O'Rourke, and Schevon, SICOMP 1997], where

is an arbitrarily small positive constant. In this paper, we present a new data structure of

preprocessing time and space, with

query time. For a special case where one query point is required to lie on one of the edges of

, the previously best work uses

preprocessing time and space to achieve

query time. We improve the preprocessing time and space to

, with

query time. Furthermore, we present a new algorithm to compute the exact set of shortest path edge sequences of

, which are known to be

in number and have a total complexity of

in the worst case. The previously best algorithm for the problem takes roughly

time, while our new algorithm runs in

time.