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
BUSHWHACK: An Approximation Algorithm for Minimal Paths through Pseudo-Euclidean Spaces
ISAAC '01 Proceedings of the 12th International Symposium on Algorithms and Computation
On discretization methods for approximating optimal paths in regions with direction-dependent costs
Information Processing Letters
Shortest Gently Descending Paths
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
Parallel Optimal Weighted Links
Transactions on Computational Science III
Line Segment Facility Location in Weighted Subdivisions
AAIM '09 Proceedings of the 5th International Conference on Algorithmic Aspects in Information and Management
Fréchet Distance Problems in Weighted Regions
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
On discretization methods for approximating optimal paths in regions with direction-dependent costs
Information Processing Letters
Querying Approximate Shortest Paths in Anisotropic Regions
SIAM Journal on Computing
Line facility location in weighted regions
Journal of Combinatorial Optimization
A survey of geodesic paths on 3D surfaces
Computational Geometry: Theory and Applications
Approximate distance queries for weighted polyhedral surfaces
ESA'11 Proceedings of the 19th European conference on Algorithms
Approximating generalized distance functions on weighted triangulated surfaces with applications
Journal of Computational and Applied Mathematics
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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 @?-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 @?-approximation in O(n@?(log1@?+logn)log1@?) time. By reducing complexity dependency on @?, 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 @?-approximation algorithm whose time complexity does not depend on unit weight ratio.