Approximating shortest path for the skew lines problem in time doubly logarithmic in 1/epsilon
Theoretical Computer Science - Algebraic and numerical algorithm
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
Shortest paths on realistic polyhedra
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Querying approximate shortest paths in anisotropic regions
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Automated antenna positioning algorithms for wireless fixed-access networks
Journal of Heuristics
Efficiently determining a locally exact shortest path on polyhedral surfaces
Computer-Aided Design
Approximate Euclidean shortest paths amid convex obstacles
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Improving Chen and Han's algorithm on the discrete geodesic problem
ACM Transactions on Graphics (TOG)
Approximation algorithms for shortest descending paths in terrains
Journal of Discrete Algorithms
Querying Approximate Shortest Paths in Anisotropic Regions
SIAM Journal on Computing
A survey of geodesic paths on 3D surfaces
Computational Geometry: Theory and Applications
Approximate shortest path queries on weighted polyhedral surfaces
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
Expansion-Based algorithms for finding single pair shortest path on surface
W2GIS'04 Proceedings of the 4th international conference on Web and Wireless Geographical Information Systems
Exact geodesic metric in 2-manifold triangle meshes using edge-based data structures
Computer-Aided Design
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We present a new technique for constructing a data structure that approximates shortest path maps in $\Re^d$. By applying this technique, we get the following two results on approximate shortest path maps in $\Re^3$.(i) Given a polyhedral surface or a convex polytope $\P$ with n edges in $\Re^3$, a source point s on $\P$, and a real parameter $0 t on $\P$, one can compute, in $O(\log{(n/\eps)})$ time, a distance $\Delta_{\P,s}(t)$, such that $d_{\P,s}(t) \leq \Delta_{\P,s}(t) \leq (1 + \eps)d_{\P,s}(t)$, where $d_{\P,s}(t)$ is the length of a shortest path between s and t on $\P$. The map can be computed in $O(n^2 \log{n} + (n/\eps) \log{(1/\eps)} \log{(n/\eps)})$ time, for the case of a polyhedral surface, and in $O((n/\eps^3) \log ( 1/\eps ) + (n/\eps^{1.5}) \log{(1/\eps)} \log{n})$ time if $\P$ is a convex polytope. (ii) Given a set of polyhedral obstacles $\O$ with a total of n edges in $\Re^3$, a source point {\it s} in $\Re^3 {\rm \setminus \inter} \cup\scriptstyle{_{O \in \O}} O$, and a real parameter $0