Low dimensional embeddings of ultrametrics

Yair Bartal; Nathan Linial; Manor Mendel; Assaf Naor

doi:10.1016/j.ejc.2003.08.003

Low dimensional embeddings of ultrametrics

Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor

Research output: Contribution to journal › Article › peer-review

Abstract

In this note we show that every n-point ultrametric embeds with constant distortion in ℓ_p^O(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 ℓ_p ^O(k2^logn). These facts have implications to embeddings of finite metric spaces in low dimensional ℓ_p spaces in the context of metric Ramsey-type theorems.

Original language	English
Pages (from-to)	87-92
Number of pages	6
Journal	European Journal of Combinatorics
Volume	25
Issue number	1
DOIs	https://doi.org/10.1016/j.ejc.2003.08.003
State	Published - Jan 2004
Externally published	Yes

Bibliographical note

Funding Information:
Y. Bartal and N. Linial are supported in part by a grant from the Israeli National Science Foundation. M. Mendel is supported in part by the Landau Center.

Keywords

Metric embeddings
Ultrametrics

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10.1016/j.ejc.2003.08.003

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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 ℓp O(k2logn). These facts have implications to embeddings of finite metric spaces in low dimensional ℓp spaces in the context of metric Ramsey-type theorems.",

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author = "Yair Bartal and Nathan Linial and Manor Mendel and Assaf Naor",

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AU - Bartal, Yair

AU - Linial, Nathan

AU - Mendel, Manor

AU - Naor, Assaf

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N2 - 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 ℓp O(k2logn). These facts have implications to embeddings of finite metric spaces in low dimensional ℓp spaces in the context of metric Ramsey-type theorems.

AB - 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 ℓp O(k2logn). These facts have implications to embeddings of finite metric spaces in low dimensional ℓp spaces in the context of metric Ramsey-type theorems.

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JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

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ER -

Low dimensional embeddings of ultrametrics

Abstract

Bibliographical note

Keywords

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