There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
Classes of graphs which approximate the complete Euclidean graph
Discrete & Computational Geometry
On sparse spanners of weighted graphs
Discrete & 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
Parallel shortcutting of rooted trees
Journal of Algorithms
Efficient algorithms for constructing fault-tolerant geometric spanners
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Approximating geometrical graphs via “spanners” and “banyans”
STOC '98 Proceedings of the thirtieth 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
Faster algorithms for some geometric graph problems in higher dimensions
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
New techniques for the union-find problem
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Efficiency of a Good But Not Linear Set Union Algorithm
Journal of the ACM (JACM)
Applications of Path Compression on Balanced Trees
Journal of the ACM (JACM)
Sparse communication networks and efficient routing in the plane (extended abstract)
Proceedings of the nineteenth annual ACM symposium on Principles of distributed computing
Approximate distance oracles for geometric graphs
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Trade-offs in non-reversing diameter
Nordic Journal of Computing
Space-time tradeoff for answering range queries (Extended Abstract)
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Tight bounds for the partial-sums problem
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Compact routing on euclidian metrics
Proceedings of the twenty-third annual ACM symposium on Principles of distributed computing
Lower bound for sparse Euclidean spanners
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
Geometric Spanner Networks
Approximate distance oracles for geometric spanners
ACM Transactions on Algorithms (TALG)
Randomized and deterministic algorithms for geometric spanners of small diameter
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
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
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Balancing degree, diameter and weight in Euclidean spanners
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Fast pruning of geometric spanners
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
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 STOC'95 [5] Arya et al. showed that for any set of n points in Rd, a (1 + ε)-spanner with diameter at most 2 (respectively 3) and O(n log n) edges (resp., O(n log log n) edges) can be built in O(n log n) time. Moreover, Arya et al. [5] conjectured that one can build in O(n log n) time a (1 + ε)-spanner with diameter at most 4 and O(n log* n) edges. Since then, this conjecture became a central open problem in this area. Nevertheless, very little progress on this problem was reported up to this date. In particular, the previous state-of-the-art subquadratic-time construction of (1 + ε)-spanners with o(n log log n) edges due to Arya et al. [5] produces spanners with diameter 8. In addition, general tradeoffs between the diameter and number of edges were established [5, 26]. Specifically it was shown in [5, 26] that for any k ≥ 4, one can build in O(n(log n)2kαk(n)) time a (1 + ε)-spanner with diameter at most 2k and O(n2kαk(n)) edges. The function αk is the inverse of a certain Ackermann-style function at the ⌊k/2⌋th level of the primitive recursive hierarchy, where α0(n) = ⌈n/2⌉, α1(n) = ⌈√n⌉, α2(n) = ⌈log n⌉, α3(n) = ⌈log log n⌉, α4(n) = log* n, α5(n) = ⌈1/2 log* n⌉,..., etc. It is also known [26] that if one allows quadratic time then these bounds can be improved. Specifically, for any k ≥ 4, a (1 + ε)-spanner with diameter at most k and O(nkαk(n)) edges can be constructed in O(n2) time [26]. A major open question in this area is whether one can construct within time O(n log n + nkαk(n)) a (1 + ε)-spanner with diameter at most k and O(nkαk(n)) edges. This question in the particular case of k = 4 coincides with the aforementioned conjecture of Arya et al. [5]. In this paper we answer this long-standing question in the affirmative. Moreover, in fact, we provide a stronger result. Specifically we show that for any k ≥ 4, a (1 + ε)-spanner with diameter at most k and O(nαk(n)) edges can be built in optimal time O(n log n). In particular, our tradeoff for k = 4 provides an O(n log n)-time construction of (1 + ε)-spanners with diameter at most 4 and O(n log* n) edges, thus settling the conjecture of Arya et al. [5]. The tradeoff between the diameter and number of edges of our spanner construction is tight up to constant factors in the entire range of parameters, even if one allows the spanner to use (arbitrarily many) Steiner points.