Fast algorithms for finding nearest common ancestors
SIAM Journal on Computing
Computational geometry: an introduction
Computational geometry: an introduction
Approximation algorithms for shortest path motion planning
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
On finding lowest common ancestors: simplification and parallelization
SIAM Journal on Computing
Optimal shortest path queries in a simple polygon
Journal of Computer and System Sciences
Classes of graphs which approximate the complete Euclidean graph
Discrete & Computational Geometry
Dealing with higher dimensions: the well-separated pair decomposition and its applications
Dealing with higher dimensions: the well-separated pair decomposition and its applications
Range searching and point location among fat objects
Journal of Algorithms
Algorithms on strings, trees, and sequences: computer science and computational biology
Algorithms on strings, trees, and sequences: computer science and computational biology
Approximate shortest paths and geodesic diameters on convex polytopes in three dimensions
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximating shortest paths on a convex polytope in three dimensions
Journal of the ACM (JACM)
Handbook of discrete and computational geometry
Handbook of discrete and computational geometry
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
Journal of the ACM (JACM)
Fast Algorithms for Constructing t-Spanners and Paths with Stretch t
SIAM Journal on Computing
Fast Estimation of Diameter and Shortest Paths (Without Matrix Multiplication)
SIAM Journal on Computing
On the all-pairs Euclidean short path problem
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Faster algorithms for some geometric graph problems in higher dimensions
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
SIAM Journal on Computing
Computing approximate shortest paths on convex polytopes
Proceedings of the sixteenth annual symposium on Computational geometry
Shortest Path Queries Among Weighted Obstacles in the Rectilinear Plane
SIAM Journal on Computing
All-Pairs Almost Shortest Paths
SIAM Journal on Computing
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Approximate distance oracles for geometric graphs
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Approximating the Stretch Factor of Euclidean Graphs
SIAM Journal on Computing
Fast Greedy Algorithms for Constructing Sparse Geometric Spanners
SIAM Journal on Computing
Approximate Distance Oracles Revisited
ISAAC '02 Proceedings of the 13th International Symposium on Algorithms and Computation
LATIN '00 Proceedings of the 4th Latin American Symposium on Theoretical Informatics
Which Triangulations Approximate the Complete Graph?
Proceedings of the International Symposium on Optimal Algorithms
Planar Spanners and Approximate Shortest Path Queries among Obstacles in the Plane
ESA '96 Proceedings of the Fourth Annual European Symposium on Algorithms
Nearly linear time approximation schemes for Euclidean TSP and other geometric problems
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Probabilistic approximation of metric spaces and its algorithmic applications
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Approximate distance oracles for unweighted graphs in Õ (n2) time
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Compact oracles for reachability and approximate distances in planar digraphs
Journal of the ACM (JACM)
Fast pruning of geometric spanners
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
Balancing degree, diameter and weight in Euclidean spanners
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
An optimal-time construction of sparse Euclidean spanners with tiny diameter
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Fast, precise and dynamic distance queries
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Sparse Euclidean Spanners with Tiny Diameter
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
Shortest-path queries in static networks
ACM Computing Surveys (CSUR)
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Given an arbitrary real constant ϵ 0, and a geometric graph G in d-dimensional Euclidean space with n points, O(n) edges, and constant dilation, our main result is a data structure that answers (1 + ϵ)-approximate shortest-path-length queries in constant time. The data structure can be constructed in O(n log n) time using O(n log n) space. This represents the first data structure that answers (1 + ϵ)-approximate shortest-path queries in constant time, and hence functions as an approximate distance oracle. The data structure is also applied to several other problems. In particular, we also show that approximate shortest-path queries between vertices in a planar polygonal domain with “rounded” obstacles can be answered in constant time. Other applications include query versions of closest-pair problems, and the efficient computation of the approximate dilations of geometric graphs. Finally, we show how to extend the main result to answer (1 + ϵ)-approximate shortest-path-length queries in constant time for geometric spanner graphs with m = ω(n) edges. The resulting data structure can be constructed in O(m + n log n) time using O(n log n) space.