Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound

In STOC'95 [ADMSS95] Arya et al. showed that any set of n points in R admits a (1 + ?)spanner with hop-diameter at most 2 (respectively, 3) and O(nlog n) edges (resp., O(nlog log n) edges). They also gave a general upper bound tradeoff of hop-diameter at most k and O(nak(n)) edges, for any k = 2. The function ak is the inverse of a certain Ackermann-style function at the ?k/2?th level of the primitive recursive hierarchy, where a0(n) = ?n/2?, a1(n) = [vn], a2(n) = ?log n?, a3(n) = ?log log n?, a4(n) = log^* n, a5(n) = ?¹₂ log^* n?,.... Roughly speaking, for k = 2 the function ak is close to ?^k-₂² ?-iterated log-star function, i.e., log with ?^k-₂² ? stars. Also, a₂a(n₎₊₄(n) = 4, where a(n) is the one-parameter inverse Ackermann function, which is an extremely slowly growing function. Whether or not this tradeoff is tight has remained open, even for the cases k = 2 and k = 3. Two lower bounds are known: The first applies only to spanners with stretch 1 and the second is sub-optimal and applies only to sufficiently large (constant) values of k. In this paper we prove a tight lower bound for any constant k: For any fixed ? > 0, any (1 + ?)-spanner for the uniform line metric with hop-diameter at most k must have at least ?(nak(n)) edges.