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(n log n) time and requires O(n log n) space, where n is the number of edges of P. The algorithm is based on the O(n log n) algorithm of Hershberger and Suri for shortest paths in the plane [22], 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 used in [22], 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 [32] that implicitly encodes the shortest paths from s to all other points of the surface. This structure allows us to a...

• Type: Conference
• Year: 2006
• Where: COMPGEOM
• Authors: Yevgeny Schreiber, Micha Sharir
• Keywords: Algorithm, COMPGEOM 2006, Convex Polytope, Shortest Paths