Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound
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In STOC'95 [ADMSS'95] 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 loglog n) edges). They also gave a general upper bound tradeoff of hop-diameter at most k and O(n α_k(n)) edges, for any k ≥ 2. The function α_k is the inverse of a certain Ackermann-style function at the ⌊ k/2 ⌋th level of the primitive recursive hierarchy, where α_0(n) = ⌈ n/2 ⌉, α_1(n) = ⌈√(n)⌉, α_2(n) = ⌈logn⌉, α_3(n) = ⌈loglogn⌉, α_4(n) = log^* n, α_5(n) = ⌊1/2log^*n ⌋, …. Roughly speaking, for k ≥ 2 the function α_k is close to ⌊k-2/2⌋-iterated log-star function, i.e., log with ⌊k-2/2⌋ stars. Also, α_2α(n)+4(n) ≤ 4, where α(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 ϵ > 0, any (1+ϵ)-spanner for the uniform line metric with hop-diameter at most k must have at least Ω(n α_k(n)) edges.
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