Approximate Euclidean shortest paths amid convex obstacles
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Shortest Path Problems on a Polyhedral Surface
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
A survey of geodesic paths on 3D surfaces
Computational Geometry: Theory and Applications
Near optimal algorithm for the shortest descending path on the surface of a convex terrain
Journal of Discrete Algorithms
On the expected complexity of voronoi diagrams on terrains
Proceedings of the twenty-eighth annual symposium on Computational geometry
An improved algorithm for the shortest descending path on a convex terrain
Journal of Discrete Algorithms
Hi-index | 0.00 |
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)).