Approximating the complete Euclidean graph
No. 318 on SWAT 88: 1st Scandinavian workshop on algorithm theory
A sparse graph almost as good as the complete graph on points in K dimensions
Discrete & Computational Geometry
On sparse spanners of weighted graphs
Discrete & Computational Geometry
Euclidean spanners: short, thin, and lanky
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
A new way to weigh Malnourished Euclidean graphs
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
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
Computational Geometry: Theory and Applications - Special issue on the 14th Canadian conference on computational geometry — CCCG02
Randomized and deterministic algorithms for geometric spanners of small diameter
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Finding the best shortcut in a geometric network
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Approximate distance oracles for graphs with dense clusters
Computational Geometry: Theory and Applications
Approximate distance oracles for geometric spanners
ACM Transactions on Algorithms (TALG)
Journal of Discrete Algorithms
On spanners of geometric graphs
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
An optimal-time construction of sparse Euclidean spanners with tiny diameter
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
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Let S be a set of points in ℝd. Given a geometric spanner graph, G = (S,E), with constant stretch factor t, and a positive constant ε, we show how to construct a (1+ε)-spanner of G with $\mathcal{O}(|S|)$ edges in time $\mathcal{O}(|E|+|S|{\rm log}|S|)$. Previous algorithms require a preliminary step in which the edges are sorted in non-decreasing order of their lengths and, thus, have running time Ω(|E| log |S|). We obtain our result by designing a new algorithm that finds the pair in a well-separated pair decomposition separating two given query points. Previously, it was known how to answer such a query in $\mathcal{O}({\rm log}|S|)$ time. We show how a sequence of such queries can be answered in $\mathcal{O}(1)$ amortized time per query, provided all query pairs are from a polynomially bounded range.