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Light Spanners for High Dimensional Norms via Stochastic Decompositions

Authors Arnold Filtser, Ofer Neiman

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Author Details

Arnold Filtser
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Ofer Neiman
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Filtser, Arnold: Partially supported by the Lynn and William Frankel Center for Computer Sciences, ISF grant 1817/17, and by BSF Grant 2015813.
  • Neiman, Ofer: Partially supported by ISF grant 1817/17, and by BSF Grant 2015813.

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Arnold Filtser and Ofer Neiman. Light Spanners for High Dimensional Norms via Stochastic Decompositions. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ESA.2018.29

Abstract

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 O~(n^{1+1/t^2}) 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 l_p 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 l_p for 1<p <=2 has an O(t)-spanner with n^{1+O~(1/t^p)} edges and lightness n^{O~(1/t^p)}.
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(n^{1/t}). We exhibit the following tradeoff: metrics with decomposability parameter nu=nu(t) admit an O(t)-spanner with lightness O~(nu^{1/t}). For example, n-point Euclidean metrics have nu <=n^{1/t}, metrics with doubling constant lambda have nu <=lambda, and graphs of genus g have nu <=g. While these families do admit a (1+epsilon)-spanner, its lightness depend exponentially on the dimension (resp. log g). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
Keywords
  • Spanners
  • Stochastic Decompositions
  • High Dimensional Euclidean Space
  • Doubling Dimension
  • Genus Graphs

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