Light spanners for high dimensional norms via stochastic decompositions

Arnold Filtser, Ofer Neiman

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

6 Scopus citations


Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with Õ(n1+1/t2) edges, little is known. In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of ℓp with 1 < p < 2. Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of ℓp for 1 < p ≤ 2 has an O(t)-spanner with n1+Õ(1/tp) edges and lightness nÕ(1/tp). In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an O(t)-spanner with lightness O(n1/t). We exhibit the following tradeoff: metrics with decomposability parameter ν = ν(t) admit an O(t)-spanner with lightness Õ(ν1/t). For example, n-point Euclidean metrics have ν η n1/t, metrics with doubling constant λ have ν ≤ λ, and graphs of genus g have ν ≤ g. While these families do admit a (1 + ϵ)-spanner, its lightness depend exponentially on the dimension (resp. log g). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.

Original languageEnglish
Title of host publication26th European Symposium on Algorithms, ESA 2018
EditorsHannah Bast, Grzegorz Herman, Yossi Azar
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Print)9783959770811
StatePublished - 1 Aug 2018
Event26th European Symposium on Algorithms, ESA 2018 - Helsinki, Finland
Duration: 20 Aug 201822 Aug 2018

Publication series

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


Conference26th European Symposium on Algorithms, ESA 2018


  • Doubling dimension
  • Genus graphs
  • High dimensional euclidean space
  • Spanners
  • Stochastic decompositions

ASJC Scopus subject areas

  • Software


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