Sparse euclidean spanners with tiny diameter

In STOC'95, Arya et al. [1995] showed that for any set of n points in ℝ^d, a (1 + ε)-spanner with diameter at most 2 (respectively, 3) and O(n log n) edges (respectively, O(n log log n) edges) can be built in O(n log n) time. Moreover, it was shown in Arya et al. [1995] and Narasimhan and Smid [2007] that for any k ≥ 4, one can build in O(n(log n)2^kα_k(n)) time a (1+ε)-spanner with diameter at most 2k and O(n^kα_k (n)) edges. The function α_k is the inverse of a certain 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 [Narasimhan and Smid 2007] 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(nkα_k(n)) edges can be constructed in O(n²) time [Narasimhan and Smid 2007]. A major open question in this area is whether one can construct within time O(n log n+nkα_k(n) ) a (1+ε)- spanner with diameter at most k and O(nkα_k(n)) edges. In this article, we answer this 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(nα_k(n)) edges can be built in optimal time O(n log n).