The weighted region problem: finding shortest paths through a weighted planar subdivision
Journal of the ACM (JACM)
Preprocessing an undirected planar network to enable fast approximate distance queries
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
On-Line Algorithms for Shortest Path Problems on Planar Digraphs
WG '96 Proceedings of the 22nd International Workshop on Graph-Theoretic Concepts in Computer Science
Compact oracles for reachability and approximate distances in planar digraphs
Journal of the ACM (JACM)
Journal of the ACM (JACM)
Determining approximate shortest paths on weighted polyhedral surfaces
Journal of the ACM (JACM)
Many distances in planar graphs
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Object location using path separators
Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing
Planar graphs, negative weight edges, shortest paths, and near linear time
Journal of Computer and System Sciences - Special issue on FOCS 2001
On finding approximate optimal paths in weighted regions
Journal of Algorithms
Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces
Discrete & Computational Geometry
Linear-space approximate distance oracles for planar, bounded-genus and minor-free graphs
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Exact distance oracles for planar graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Fast query structures in anisotropic media
Theoretical Computer Science
Shortest-path queries in static networks
ACM Computing Surveys (CSUR)
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Let P be a planar polyhedral surface consisting of n triangular faces, each assigned with a positive weight. The weight of a path p on P is defined as the weighted sum of the Euclidean lengths of the portions of p in each face multiplied by the corresponding face weights. We show that, for every ε ∈ (0, 1), there exists a data structure, termed distance oracle, computable in time O(nε-2 log3(n/ε) log2(1/ε)) and of size O(nε-3/2 log2(n/ε) log(1/ε)), such that (1+ε)-approximate distance queries in P can be answered in time O(ε-1 log(1/ε) + log log n). As in previous work (Aleksandrov, Maheshwari, and Sack (J. ACM 2005) and others), the big-O notation hides constants depending logarithmically on the ratio of the largest and smallest face weights and reciprocally on the sine of the smallest angle of P. The tradeoff between space and query time of our distance oracle is a significant improvement in terms of n over the previous best tradeoff obtained by a distance oracle of Aleksandrov, Djidjev, Guo, Maheshwari, Nussbaum, and Sack (Discrete Comput. Geom. 2010), which requires space roughly quadratic in n for a comparable query time.