Low dimensional embeddings of ultrametrics

Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor

Research output: Contribution to journalArticle

Abstract

In this note we show that every n-point ultrametric embeds with constant distortion in ℓpO(logn) for every ∞≥p≥1. More precisely, we consider a special type of ultrametric with hierarchical structure called a k-hierarchically well-separated tree (k-HST). We show that any k-HST can be embedded with distortion at most 1+O(1/k) in ℓpO(k2logn). These facts have implications to embeddings of finite metric spaces in low dimensional ℓp spaces in the context of metric Ramsey-type theorems.

Original languageEnglish (US)
Pages (from-to)87-92
Number of pages6
JournalEuropean Journal of Combinatorics
Volume25
Issue number1
DOIs
StatePublished - Jan 2004

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P-space
Hierarchical Structure
Metric space
Metric
Theorem
Context

Keywords

  • Metric embeddings
  • Ultrametrics

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Low dimensional embeddings of ultrametrics. / Bartal, Yair; Linial, Nathan; Mendel, Manor; Naor, Assaf.

In: European Journal of Combinatorics, Vol. 25, No. 1, 01.2004, p. 87-92.

Research output: Contribution to journalArticle

Bartal, Y, Linial, N, Mendel, M & Naor, A 2004, 'Low dimensional embeddings of ultrametrics', European Journal of Combinatorics, vol. 25, no. 1, pp. 87-92. https://doi.org/10.1016/j.ejc.2003.08.003
Bartal, Yair ; Linial, Nathan ; Mendel, Manor ; Naor, Assaf. / Low dimensional embeddings of ultrametrics. In: European Journal of Combinatorics. 2004 ; Vol. 25, No. 1. pp. 87-92.
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