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COMPGEOM
2006
ACM

An optimal-time algorithm for shortest paths on a convex polytope in three dimensions

12 years 7 months ago
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...
Yevgeny Schreiber, Micha Sharir
Added 13 Jun 2010
Updated 13 Jun 2010
Type Conference
Year 2006
Where COMPGEOM
Authors Yevgeny Schreiber, Micha Sharir
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