Light, Reliable Spanners

Authors Arnold Filtser, Yuval Gitlitz, Ofer Neiman



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Arnold Filtser
  • Bar-Ilan University, Ramat-Gan, Israel
Yuval Gitlitz
  • Ben-Gurion University of the Negev, Be'er Sheva, Israel
Ofer Neiman
  • Ben-Gurion University of the Negev, Be'er Sheva, Israel

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Arnold Filtser, Yuval Gitlitz, and Ofer Neiman. Light, Reliable Spanners. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.56

Abstract

A ν-reliable spanner of a metric space (X,d), is a (dominating) graph H, such that for any possible failure set B ⊆ X, there is a set B^+ just slightly larger |B^+| ≤ (1+ν)⋅|B|, and all distances between pairs in X⧵B^+ are (approximately) preserved in H⧵B. Recently, there have been several works on sparse reliable spanners in various settings, but so far, the weight of such spanners has not been analyzed at all. In this work, we initiate the study of light reliable spanners, whose weight is proportional to that of the Minimum Spanning Tree (MST) of X. 
We first observe that unlike sparsity, the lightness of any deterministic reliable spanner is huge, even for the metric of the simple path graph. Therefore, randomness must be used: an oblivious reliable spanner is a distribution over spanners, and the bound on |B^+| holds in expectation.
We devise an oblivious ν-reliable (2+2/(k-1))-spanner for any k-HST, whose lightness is ≈ ν^{-2}. We demonstrate a matching Ω(ν^{-2}) lower bound on the lightness (for any finite stretch). We also note that any stretch below 2 must incur linear lightness.
For general metrics, doubling metrics, and metrics arising from minor-free graphs, we construct light tree covers, in which every tree is a k-HST of low weight. Combining these covers with our results for k-HSTs, we obtain oblivious reliable light spanners for these metric spaces, with nearly optimal parameters. In particular, for doubling metrics we get an oblivious ν-reliable (1+ε)-spanner with lightness ε^{-O(ddim)} ⋅ Õ(ν^{-2}⋅log n), which is best possible (up to lower order terms).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Sparsification and spanners
Keywords
  • light spanner
  • reliable spanner
  • HST cover
  • doubling metric
  • minor free graphs

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