An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions

We present an optimal-time algorithm for computing (an implicit representation of) the shortest-path map from a fixed source s on the surface of a convex polytope P in three dimensions. Our algorithm runs in O(nlog n) time and requires O(nlog n) space, where n is the number of edges of P. The algorithm is based on the O(nlog n) algorithm of Hershberger and Suri for shortest paths in the plane (Hershberger, J., Suri, S. in SIAM J. Comput. 28(6):2215–2256, 1999), and similarly follows the continuous Dijkstra paradigm, which propagates a “wavefront” from s along ∂ P. This is effected by generalizing the concept of conforming subdivision of the free space introduced by Hershberger and Suri and by adapting it for the case of a convex polytope in ℝ³, allowing the algorithm to accomplish the propagation in discrete steps, between the “transparent” edges of the subdivision. The algorithm constructs a dynamic version of Mount’s data structure (Mount, D.M. in Discrete Comput. Geom. 2:153–174, 1987) that implicitly encodes the shortest paths from s to all other points of the surface. This structure allows us to answer single-source shortest-path queries, where the length of the path, as well as its combinatorial type, can be reported in O(log n) time; the actual path can be reported in additional O(k) time, where k is the number of polytope edges crossed by the path. The algorithm generalizes to the case of m source points to yield an implicit representation of the geodesic Voronoi diagram of m sites on the surface of P, in time O((n+m)log (n+m)), so that the site closest to a query point can be reported in time O(log (n+m)). Work on this paper was supported by NSF Grants CCR-00-98246 and CCF-05-14079, by a grant from the U.S.-Israeli Binational Science Foundation, by grant 155/05 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. The paper is based on the Ph.D. Thesis of the first author, supervised by the second author. A preliminary version has been presented in Proc. 22nd Annu. ACM Sympos. Comput. Geom., pp. 30–39, 2006.