Low dimensional embeddings of ultrametrics

Authors:
Yair Bartal;Nathan Linial;Manor Mendel;Assaf Naor
Affiliations:
School of Computer Science, Hebrew University, Jerusalem 91904, Israel;School of Computer Science, Hebrew University, Jerusalem 91904, Israel;School of Computer Science, Hebrew University, Jerusalem 91904, Israel;Theory Group, Microsoft Research, One Microsoft Way 113/2131, Redmond, WA
Venue:
European Journal of Combinatorics
Year:
2004

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Abstract

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