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Lipschitz Decompositions of Finite $\ell_{p}$ Metrics

Robert Krauthgamer, Nir Petruschka

Abstract

Lipschitz decomposition is a useful tool in the design of efficient algorithms involving metric spaces. While many bounds are known for different families of finite metrics, the optimal parameters for $n$-point subsets of $\ell_p$, for $p > 2$, remained open, see e.g. [Naor, SODA 2017]. We make significant progress on this question and establish the bound $β=O(\log^{1-1/p} n)$. Building on prior work, we demonstrate applications of this result to two problems, high-dimensional geometric spanners and distance labeling schemes. In addition, we sharpen a related decomposition bound for $1<p<2$, due to Filtser and Neiman [Algorithmica 2022].

Lipschitz Decompositions of Finite $\ell_{p}$ Metrics

Abstract

Lipschitz decomposition is a useful tool in the design of efficient algorithms involving metric spaces. While many bounds are known for different families of finite metrics, the optimal parameters for -point subsets of , for , remained open, see e.g. [Naor, SODA 2017]. We make significant progress on this question and establish the bound . Building on prior work, we demonstrate applications of this result to two problems, high-dimensional geometric spanners and distance labeling schemes. In addition, we sharpen a related decomposition bound for , due to Filtser and Neiman [Algorithmica 2022].

Paper Structure

This paper contains 12 sections, 16 theorems, 9 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. Decompositions and Spanners in $\ell_p$ for $p>2$
  5. Lipschitz Decomposition in $\ell_{p}$ for $p\in(2,\infty)$
  6. Capped Decomposition in $\ell_{p}$ for $p\in(2,\infty)$
  7. Spanners in $\ell_{p}$ for $p\in(2,\infty)$
  8. Spanners in $\ell_{p}$ for $p \in (1,2)$
  9. Spanners of Near-Linear Size
  10. Spanners with Stretch-Size Tradeoff
  11. Distance Labeling
  12. Future Directions

Key Result

Theorem 1.3

Let $p \in [2,\infty]$. Then $\beta^*_n(\ell_p) = O(\log^{1-1/p}{n})$. That is, for every $n$-point metric $X\subset \ell_{p}$ and $\Delta > 0$, there exists an $(O(\log^{1-1/p}{n}), \Delta)$-Lipschitz decomposition of $X$.

Theorems & Definitions (30)

  • Definition 1.1: Lipschitz decomposition Bartal96
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1: Bartal96
  • Theorem 2.2: benyamini1998geometricBG19
  • Corollary 2.3
  • proof : Proof of \ref{['thm:(Lipschitz-Decompositions-in-lp)']}
  • Definition 2.4
  • ...and 20 more