Computational geometry: an introduction
Computational geometry: an introduction
The weighted region problem: finding shortest paths through a weighted planar subdivision
Journal of the ACM (JACM)
Approximating weighted shortest paths on polyhedral surfaces
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Voronoi diagrams based on convex distance functions
SCG '85 Proceedings of the first 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
Movement Planning in the Presence of Flows
Algorithmica
Determining approximate shortest paths on weighted polyhedral surfaces
Journal of the ACM (JACM)
On finding approximate optimal paths in weighted regions
Journal of Algorithms
On finding energy-minimizing paths on terrains
IEEE Transactions on Robotics
Querying approximate shortest paths in anisotropic regions
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
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
Fréchet Distance Problems in Weighted Regions
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Approximation algorithms for shortest descending paths in terrains
Journal of Discrete Algorithms
Line facility location in weighted regions
Journal of Combinatorial Optimization
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Our goal is to find an approximate shortest path for a point robot moving in a planar subdivision with n vertices. Let ρ ≥ 1 be a real number. Distances in each face of this subdivision are measured by a convex distance function whose unit disk is contained in a concentric unit Euclidean disk, and contains a concentric Euclidean disk with radius 1/ρ. Different convex distance functions may be used for different faces, and obstacles are allowed. These convex distance functions may be asymmetric. For all ε ∈ (0, 1), and for any two points vs and vd, we give an algorithm that finds a path from vs to vd whose cost is at most (1 + ε) times the minimum cost. Our algorithm runs in O (ρ2logρ/ε2n3 log (ρn/ε)) time. This bound does not depend on any other parameters; in particular, it does not depend on the minimum angle in the subdivision. We give applications to two special cases that have been considered before: the weighted region problem and motion planning in the presence of uniform flows. For the weighted region problem with weights in [1, ρ] ∪ {∞}, the time bound of our algorithm improves to O (ρ2logρ/εn3 log (ρn/ε)).