Metric structures in L1: Dimension, snowflakes, and average distortion

James R. Lee, Manor Mendel, Assaf Naor

Research output: Contribution to journalArticle

Abstract

We study the metric properties of finite subsets of L1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L1. We present some new observations concerning the relation of L1 to dimension, topology, and Euclidean distortion. We show that every n-point subset of L1 embeds into L2 with average distortion O(√log n), yielding the first evidence that the conjectured worst-case bound of O(√log n) is valid. We also address the issue of dimension reduction in Lp for p ∈. (1, 2). We resolve a question left open by M. Charikar and A. Sahai [Dimension reduction in the ℓ1 norm, in: Proceedings of the 43rd Annual IEEE Conference on Foundations of Computer Science, ACM, 2002, pp. 251-260] concerning the impossibility of dimension reduction with a linear map in the above cases, and we show that a natural variant of the recent example of Brinkman and Charikar [On the impossibility of dimension reduction in ℓ1, in: Proceedings of the 44th Annual IEEE Conference on Foundations of Computer Science, ACM, 2003, pp. 514-523], cannot be used to prove a lower bound for the non-linear case. This is accomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space.

Original languageEnglish (US)
Pages (from-to)1180-1190
Number of pages11
JournalEuropean Journal of Combinatorics
Volume26
Issue number8
DOIs
StatePublished - Nov 2005

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Dimension Reduction
Metric
Computer science
Annual
Subset
Computer Science
Functional analysis
Linear map
Functional Analysis
Approximation algorithms
Weighted Graph
Set theory
Nonlinear Analysis
Euclidean space
Approximation Algorithms
Euclidean
Open Problems
Resolve
Topology
Valid

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Metric structures in L1 : Dimension, snowflakes, and average distortion. / Lee, James R.; Mendel, Manor; Naor, Assaf.

In: European Journal of Combinatorics, Vol. 26, No. 8, 11.2005, p. 1180-1190.

Research output: Contribution to journalArticle

Lee, James R. ; Mendel, Manor ; Naor, Assaf. / Metric structures in L1 : Dimension, snowflakes, and average distortion. In: European Journal of Combinatorics. 2005 ; Vol. 26, No. 8. pp. 1180-1190.
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