TY - GEN

T1 - Light spanners for high dimensional norms via stochastic decompositions

AU - Filtser, Arnold

AU - Neiman, Ofer

N1 - Funding Information:
Partially supported by the Lynn and William Frankel Center for Computer Sciences, ISF grant 1817/17, and by BSF Grant 2015813. 2 Partially supported by ISF grant 1817/17, and by BSF Grant 2015813.
Publisher Copyright:
© Arnold Filtser and Ofer Neiman.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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.

AB - 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.

KW - Doubling dimension

KW - Genus graphs

KW - High dimensional euclidean space

KW - Spanners

KW - Stochastic decompositions

UR - http://www.scopus.com/inward/record.url?scp=85052543175&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2018.29

DO - 10.4230/LIPIcs.ESA.2018.29

M3 - Conference contribution

AN - SCOPUS:85052543175

SN - 9783959770811

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 26th European Symposium on Algorithms, ESA 2018

A2 - Bast, Hannah

A2 - Herman, Grzegorz

A2 - Azar, Yossi

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 26th European Symposium on Algorithms, ESA 2018

Y2 - 20 August 2018 through 22 August 2018

ER -