There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
On Euclidean spanner graphs with small degree
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
New sparseness results on graph spanners
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
A fast algorithm for constructing sparse Euclidean spanners
SCG '94 Proceedings of the tenth 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
Lower bounds for computing geometric spanners and approximate shortest paths
Discrete Applied Mathematics
Fast Greedy Algorithms for Constructing Sparse Geometric Spanners
SIAM Journal on Computing
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Lower bound for sparse Euclidean spanners
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On hierarchical routing in doubling metrics
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Small hop-diameter sparse spanners for doubling metrics
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Fast Construction of Nets in Low-Dimensional Metrics and Their Applications
SIAM Journal on Computing
Geometric Spanner Networks
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
Randomized and deterministic algorithms for geometric spanners of small diameter
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
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
The Weak Gap Property in Metric Spaces of Bounded Doubling Dimension
Efficient Algorithms
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
The traveling salesman problem: low-dimensionality implies a polynomial time approximation scheme
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
New doubling spanners: better and simpler
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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The degree, the (hop-)diameter, and the weight are the most basic and well-studied parameters of geometric spanners. In a seminal STOC'95 paper, titled "Euclidean spanners: short, thin and lanky", Arya et al. [2] devised a construction of Euclidean (1+ε)-spanners that achieves constant degree, diameter O(log n), weight O(log2 n) ⋅ ω(MST), and has running time O(n ⋅ log n). This construction applies to n-point constant-dimensional Euclidean spaces. Moreover, Arya et al. conjectured that the weight bound can be improved by a logarithmic factor, without increasing the degree and the diameter of the spanner, and within the same running time. This conjecture of Arya et al. became one of the most central open problems in the area of Euclidean spanners. Nevertheless, the only progress since 1995 towards its resolution was achieved in the lower bounds front: Any spanner with diameter O(log n) must incur weight Ω(log n) ⋅ ω(MST), and this lower bound holds regardless of the stretch or the degree of the spanner [12, 1]. In this paper we resolve the long-standing conjecture of Arya et al. in the affirmative. We present a spanner construction with the same stretch, degree, diameter, and running time, as in Arya et al.'s result, but with optimal weight O(log n) ⋅ ω(MST). So our spanners are as thin and lanky as those of Arya et al., but they are really short! Moreover, our result is more general in three ways. First, we demonstrate that the conjecture holds true not only in constant-dimensional Euclidean spaces, but also in doubling metrics. Second, we provide a general tradeoff between the three involved parameters, which is tight in the entire range. Third, we devise a transformation that decreases the lightness of spanners in general metrics, while keeping all their other parameters in check. Our main result is obtained as a corollary of this transformation.