The weighted region problem: finding shortest paths through a weighted planar subdivision
Journal of the ACM (JACM)
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
Voronoi diagrams based on convex distance functions
SCG '85 Proceedings of the first annual symposium on Computational geometry
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
SIAM Journal on Computing
Movement Planning in the Presence of Flows
Algorithmica
Determining approximate shortest paths on weighted polyhedral surfaces
Journal of the ACM (JACM)
An optimal-time algorithm for shortest paths on a convex polytope in three dimensions
Proceedings of the twenty-second annual symposium on Computational geometry
Querying approximate shortest paths in anisotropic regions
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Approximate Shortest Paths in Anisotropic Regions
SIAM Journal on Computing
Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets
Computational Geometry: Theory and Applications
On finding approximate optimal paths in weighted regions
Journal of Algorithms
Approximate shortest path queries on weighted polyhedral surfaces
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
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We present a data structure for answering approximate shortest path queries in a planar subdivision from a fixed source. Let $\rho\geqslant1$ be a real number. Distances in each face of this subdivision are measured by a possibly asymmetric convex distance function whose unit disk is contained in a concentric unit Euclidean disk and contains a concentric Euclidean disk with radius $1/\rho$. Different convex distance functions may be used for different faces, and obstacles are allowed. Let $\varepsilon$ be any number strictly between 0 and 1. Our data structure returns a $(1+\varepsilon)$ approximation of the shortest path cost from the fixed source to a query destination in $O(\log\frac{\rho n}{\varepsilon})$ time. Afterwards, a $(1+\varepsilon)$-approximate shortest path can be reported in $O(\log n)$ time plus the complexity of the path. The data structure uses $O(\frac{\rho^2n^3}{\varepsilon^2}\log\frac{\rho n}{\varepsilon})$ space and can be built in $O(\frac{\rho^2n^3}{\varepsilon^2}(\log\frac{\rho n}{\varepsilon})^2)$ time. Our time and space bounds do not depend on any other parameter; in particular, they do not depend on any geometric parameter of the subdivision such as the minimum angle.