There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
Euclidean spanners: short, thin, and lanky
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Deformable spanners and applications
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
On hierarchical routing in doubling metrics
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Fully dynamic geometric spanners
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Improved algorithms for fully dynamic geometric spanners and geometric routing
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
An Optimal Dynamic Spanner for Doubling Metric Spaces
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Shallow-Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Small Hop-diameter Sparse Spanners for Doubling Metrics
Discrete & Computational Geometry
Balancing degree, diameter and weight in Euclidean spanners
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Sparse fault-tolerant spanners for doubling metrics with bounded hop-diameter or degree
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Optimal euclidean spanners: really short, thin and lanky
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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In a seminal STOC'95 paper, Arya et al. conjectured that spanners for low-dimensional Euclidean spaces with constant maximum degree, hop-diameter O(logn) and lightness O(logn) (i.e., weight $O(\log n) \cdot w(\textsf{MST}))$ can be constructed in O(n logn) time. This conjecture, which became a central open question in this area, was resolved in the affirmative by Elkin and Solomon in STOC'13 (even for doubling metrics). In this work we present a simpler construction of spanners for doubling metrics with the above guarantees. Moreover, our construction extends in a simple and natural way to provide k-fault tolerant spanners with maximum degree O(k2), hop-diameter O(logn) and lightness O(k2 logn).