Discrete & Computational Geometry

, Volume 50, Issue 4, pp 977–1032 | Cite as

Dimension Reduction for Finite Trees in \(\varvec{\ell _1}\)

  • James R. Lee
  • Arnaud de Mesmay
  • Mohammad Moharrami
Article

Abstract

We show that every \(n\)-point tree metric admits a \((1+\varepsilon )\)-embedding into \(\ell _1^{C(\varepsilon ) \log n}\), for every \(\varepsilon > 0\), where \(C(\varepsilon ) \le O\big ((\frac{1}{\varepsilon })^4 \log \frac{1}{\varepsilon })\big )\). This matches the natural volume lower bound up to a factor depending only on \(\varepsilon \). Previously, it was unknown whether even complete binary trees on \(n\) nodes could be embedded in \(\ell _1^{O(\log n)}\) with \(O(1)\) distortion. For complete \(d\)-ary trees, our construction achieves \(C(\varepsilon ) \le O\big (\frac{1}{\varepsilon ^2}\big )\).

Keywords

Dimension reduction Metric embeddings Bi-Lipschitz distortion 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • James R. Lee
    • 1
  • Arnaud de Mesmay
    • 2
  • Mohammad Moharrami
    • 1
  1. 1.University of WashingtonSeattleUSA
  2. 2.École Normale SupérieureParisFrance

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