A separator theorem for graphs of bounded genus
Journal of Algorithms
Fast algorithms for shortest paths in planar graphs, with applications
SIAM Journal on Computing
Linear algorithms for graph separation problems
No. 318 on SWAT 88: 1st Scandinavian workshop on algorithm theory
Shortest paths on a polyhedron
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Linear Algorithms for Partitioning Embedded Graphs of BoundedGenus
SIAM Journal on Discrete Mathematics
Approximating shortest paths on a convex polytope in three dimensions
Journal of the ACM (JACM)
Star Unfolding of a Polytope with Applications
SIAM Journal on Computing
Constructing Approximate Shortest Path Maps in Three Dimensions
SIAM Journal on Computing
Two-point Euclidean shortest path queries in the plane
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Partitioning Planar Graphs with Costs and Weights
ALENEX '02 Revised Papers from the 4th International Workshop on Algorithm Engineering and Experiments
Determining approximate shortest paths on weighted polyhedral surfaces
Journal of the ACM (JACM)
Sublinear Geometric Algorithms
SIAM Journal on Computing
Querying approximate shortest paths in anisotropic regions
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Line Segment Facility Location in Weighted Subdivisions
AAIM '09 Proceedings of the 5th International Conference on Algorithmic Aspects in Information and Management
Shortest Path Problems on a Polyhedral Surface
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Fréchet Distance Problems in Weighted Regions
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Shortest path queries between geometric objects on surfaces
ICCSA'07 Proceedings of the 2007 international conference on Computational science and its applications - Volume Part I
Querying Approximate Shortest Paths in Anisotropic Regions
SIAM Journal on Computing
Field D* path-finding on weighted triangulated and tetrahedral meshes
Autonomous Agents and Multi-Agent Systems
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We consider the classical geometric problem of determining shortest paths between pairs of points lying on a weighted polyhedral surface P consisting of n triangular faces. We present query algorithms that compute approximate distances and/or approximate (weighted) shortest paths. Our algorithm takes as input an approximation parameter ε∈(0,1) and a query time parameter $\mathfrak{q}$ and builds a data structure which is then used for answering ε-approximate distance queries in $O(\mathfrak{q})$ time. This algorithm is source point independent and improves significantly on the best previous solution. For the case where one of the query points is fixed we build a data structure that can answer ε-approximate distance queries to any query point in P in $O(\log\frac{1}{\varepsilon})$ time. This is an improvement upon the previously known solution for the Euclidean fixed source query problem. Our algorithm also generalizes the setting from previously studied unweighted polyhedral to weighted polyhedral surfaces of arbitrary genus. Our solutions are based on a novel graph separator algorithm introduced here which extends and generalizes previously known separator algorithms.