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

Hung Le, Lazar Milenković, Shay Solomon

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

In STOC'95 [6] Arya et al. showed that any set of n points in Rd 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(nαk(n)) edges, for any k ≥ 2. The function αk is the inverse of a certain Ackermann-style function, where (Equation presented), (Equation presented), (Equation presented), .... Roughly speaking, for k ≥ 2 the function αk 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.

Original languageEnglish
Title of host publication39th International Symposium on Computational Geometry, SoCG 2023
EditorsErin W. Chambers, Joachim Gudmundsson
ISBN (Electronic)9783959772730
DOIs
StatePublished - 1 Jun 2023
Event39th International Symposium on Computational Geometry, SoCG 2023 - Dallas, United States
Duration: 12 Jun 202315 Jun 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume258

Conference

Conference39th International Symposium on Computational Geometry, SoCG 2023
Country/TerritoryUnited States
CityDallas
Period12/06/2315/06/23

Keywords

  • Ackermann functions
  • Euclidean spanners
  • convex functions
  • hop-diameter

All Science Journal Classification (ASJC) codes

  • Software

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