Light spanners for snowflake metrics

Lee Ad Gottlieb, Shay Solomon

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

Abstract

A classic result in the study of spanners is the existence of light low-stretch spanners for Euclidean spaces. These spanners have arbitrary low stretch, and weight only a constant factor greater than that of the minimum spanning tree of the points (with dependence on the stretch and Euclidean dimension). A central open problem in this field asks whether other spaces admit low weight spanners as well - for example metric space with low intrinsic dimension - yet only a handful of results of this type are known. In this paper, we consider snowflake metric spaces of low intrinsic dimension. The α-snowflake of a metric (X, δ) is the metric (X, δα), for 0 < α < 1. By utilizing an approach completely different than those used for Euclidean spaces, we demonstrate that snowflake metrics admit light spanners. Further, we show that the spanner is of diameter O(log n), a result not possible for Euclidean spaces. As an immediate corollary to our spanner, we obtain dramatic improvements in algorithms for the traveling salesman problem in this setting, achieving a polynomial-time approximation scheme with near-linear runtime. Along the way, we also show that all ℓp spaces admit light spanners, a result of interest in its own right.

Original languageEnglish
Title of host publicationProceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014
PublisherAssociation for Computing Machinery
Pages387-395
Number of pages9
ISBN (Print)9781450325943
DOIs
StatePublished - 2014
Externally publishedYes
Event30th Annual Symposium on Computational Geometry, SoCG 2014 - Kyoto, Japan
Duration: 8 Jun 201411 Jun 2014

Publication series

NameProceedings of the Annual Symposium on Computational Geometry

Conference

Conference30th Annual Symposium on Computational Geometry, SoCG 2014
Country/TerritoryJapan
CityKyoto
Period8/06/1411/06/14

Keywords

  • Metric spanners
  • Snowflake metrics

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