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

James R. Lee, Manor Mendel, Assaf Naor

Research output: Chapter in Book/Report/Conference proceedingChapter

4 Scopus citations

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. 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 in [1] about the impossibility of linear dimension reduction in the above cases, and we show that the example of [2,3] 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)
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
EditorsMartin Farach-Colton
PublisherSpringer Verlag
Pages401-412
Number of pages12
ISBN (Print)3540212582, 9783540212584
DOIs
StatePublished - 2004
Externally publishedYes

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume2976
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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    Lee, J. R., Mendel, M., & Naor, A. (2004). Metric structures in L1: Dimension, snowflakes, and average distortion. In M. Farach-Colton (Ed.), Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (pp. 401-412). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2976). Springer Verlag. https://doi.org/10.1007/978-3-540-24698-5_44