Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound

Hung Le*, Lazar Milenkovic, Shay Solomon

*Corresponding author for this work

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

Abstract

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.

Original languageEnglish
Title of host publication38th International Symposium on Computational Geometry, SoCG 2022
EditorsXavier Goaoc, Michael Kerber
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772273
DOIs
StatePublished - 1 Jun 2022
Event38th International Symposium on Computational Geometry, SoCG 2022 - Berlin, Germany
Duration: 7 Jun 202210 Jun 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume224
ISSN (Print)1868-8969

Conference

Conference38th International Symposium on Computational Geometry, SoCG 2022
Country/TerritoryGermany
CityBerlin
Period7/06/2210/06/22

Funding

FundersFunder number
National Science FoundationCCF-2121952
Bloom's Syndrome Foundation
United States-Israel Binational Science Foundation
Israel Science Foundation1991/1

    Keywords

    • Euclidean spanners
    • hop-diameter
    • inverse-Ackermann
    • lower bounds
    • sparse spanners

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