Sparse Euclidean Spanners with Optimal Diameter: A General and Robust Lower Bound via a Concave Inverse-Ackermann Function

In STOC'95 [6] Arya et al. showed that any set of n points in R^d admits a (1 + ϵ)-spanner with hop-diameter at most 2 (respectively, 3) and O(n log n) edges (resp., O(n log log n) edges). They also gave a general upper bound tradeoff of hop-diameter k with O(nα_k(n)) edges, for any k ≥ 2. The function α_k is the inverse of a certain Ackermann-style function, where (Equation presented), (Equation presented), (Equation presented), .... Roughly speaking, for k ≥ 2 the function α_k is close to (Equation presented)-iterated log-star function, i.e., log with (Equation presented) stars. Despite a large body of work on spanners of bounded hop-diameter, the fundamental question of whether this tradeoff between size and hop-diameter of Euclidean (1 + ϵ)-spanners is optimal has remained open, even in one-dimensional spaces. Three lower bound tradeoffs are known: An optimal k versus (Equation presented) by Alon and Schieber [4], but it applies to stretch 1 (not 1 + ϵ). A suboptimal k versus (Equation presented) by Chan and Gupta [13]. A suboptimal k versus (Equation presented) by Le et al. [38]. This paper establishes the optimal k versus (Equation presented) lower bound tradeoff for stretch 1 + ϵ, for any ϵ > 0, and for any k. An important conceptual contribution of this work is in achieving optimality by shaving off an extremely slowly growing term, namely (Equation presented); such a fine-grained optimization (that achieves optimality) is very rare in the literature. To shave off the (Equation presented) term from the previous bound of Le et al., our argument has to drill much deeper. In particular, we propose a new way of analyzing recurrences that involve inverse-Ackermann style functions, and our key technical contribution is in presenting the first explicit construction of concave versions of these functions. An important advantage of our approach over previous ones is its robustness: While all previous lower bounds are applicable only to restricted 1-dimensional point sets, ours applies even to random point sets in constant-dimensional spaces.