Deterministic minimal time vessel routing
Operations Research
International Journal of Robotics Research
International Journal of Robotics Research
The weighted region problem: finding shortest paths through a weighted planar subdivision
Journal of the ACM (JACM)
A new algorithm for computing shortest paths in weighted planar subdivisions (extended abstract)
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximating weighted shortest paths on polyhedral surfaces
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximation algorithms for geometric shortest path problems
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
An epsilon-Approximation for Weighted Shortest Paths on Polyhedral Surfaces
SWAT '98 Proceedings of the 6th Scandinavian Workshop on Algorithm Theory
Movement Planning in the Presence of Flows
Algorithmica
Querying approximate shortest paths in anisotropic regions
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Approximate shortest paths in anisotropic regions
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for shortest descending paths in terrains
Journal of Discrete Algorithms
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The main result of this paper is an approximation algorithm for the weighted region optimal path problem. In this problem, a point robot moves in a planar space composed of n triangular regions, each of which is associated with a positive unit weight. The objective is to find, for given source and destination points s and t, a path from s to t with the minimum weighted length. Our algorithm, BUSHWHACK, adopts a traditional approach (see [M. Lanthier, A. Maheshwari, J.-R. Sack, Approximating weighted shortest paths on polyhedral surfaces, in: Proceedings of the 13th Annual ACM Symposium on Coputational Geometry, 1997, pp. 274-283; L. Aleksandrov, M. Lanthier, A. Maheshwari, J.-R. Sack, An ε-approximation algorithm for weighted shortest paths on polyhedral surfaces, in: Proceedings of the 6th Scandinavian Workshop on Algorithm Theory, in: Lecture Notes in Comput. Sci., vol. 1432, 1998, pp. 11-22; L. Aleksandrov, A. Maheshwari, J.-R. Sack, Approximation algorithms for geometric shortest path problems, in: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000, pp. 286-295]) that converts the original continuous geometric search space into a discrete graph G by placing representative points on boundary edges. However, by exploiting geometric structures that we call intervals, BUSHWHACK computes an approximate optimal path more efficiently as it accesses only a sparse subgraph of g. Combined with the logarithmic discretization scheme introduced by Aleksandrov et al. [Approximation algorithms for geometric shortest path problems, in: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000, pp. 286-295], BUSHWHACK can compute an ε-approximation in O(n/ε(log 1/ε + log n) log 1/ε) time. By reducing complexity dependency on ε, this result improves on all previous results with the same discretization approach. We also provide an improvement over the discretization scheme of [L. Aleksandrov, A. Maheshwari, J.-R. Sack, Approximation algorithms for geometric shortest path problems, in: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000, pp. 286-295] so that the size of g is no longer dependent on unit weight ratio, the ratio between the maximum and minimum unit weights. This leads to the first ε-approximation algorithm whose time complexity does not depend on unit weight ratio.