License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2022.54
URN: urn:nbn:de:0030-drops-160629
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16062/
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Le, Hung ; Milenković, Lazar ; Solomon, Shay

Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound

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Abstract

In STOC'95 [ADMSS95] 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 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 α₀(n)=⌈n/2⌉, α₁(n)=⌈√n⌉, α₂(n)=⌈log n⌉, α₃(n)=⌈log log n⌉, α₄(n)=log^* n, α₅(n)=⌊1/2 log^*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.

BibTeX - Entry

@InProceedings{le_et_al:LIPIcs.SoCG.2022.54,
  author =	{Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay},
  title =	{{Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{54:1--54:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16062},
  URN =		{urn:nbn:de:0030-drops-160629},
  doi =		{10.4230/LIPIcs.SoCG.2022.54},
  annote =	{Keywords: Euclidean spanners, hop-diameter, inverse-Ackermann, lower bounds, sparse spanners}
}

Keywords: Euclidean spanners, hop-diameter, inverse-Ackermann, lower bounds, sparse spanners
Collection: 38th International Symposium on Computational Geometry (SoCG 2022)
Issue Date: 2022
Date of publication: 01.06.2022

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