Paper Info
Title
An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions
Abstract
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)).
Year
DOI
Venue
2008
10.1007/s00454-007-9031-0
Proceedings of the twenty-second annual symposium on Computational geometry
Keywords
Field
DocType
convex polytope,algorithm generalizes,additional o,optimal-time algorithm,convex polytope p,shortest path,discrete comput,actual path,implicit representation,polytope edge,shortest paths,wavefront,point source,data structure,geodesic,geodesics,three dimensions,subdivision,free field,voronoi diagram,computational geometry
Discrete mathematics,Binary logarithm,Combinatorics,Shortest path problem,Computational geometry,Algorithm,Polytope,Convex polytope,Voronoi diagram,Geodesic,Mathematics,Dijkstra's algorithm
Journal
Volume
Issue
ISSN
39
1
1432-0444
ISBN
Citations 
PageRank 
1-59593-340-9
27
1.16
References 
Authors
33
2
Name
Order
Citations
PageRank
Yevgeny Schreiber1382.86
Micha Sharir284051183.84