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

TL;DR

This paper advances the understanding of Lipschitz decompositions in finite ll_p metrics, establishing new bounds for p > 2 and applying these to geometric spanners and labeling schemes, while also refining bounds for 1 < p < 2.

Contribution

It provides the first optimal bound eta=O(^{1-1/p} n) for ll_p metrics with p > 2 and improves bounds for 1 < p < 2, with applications to algorithms.

Findings

Established eta=O(^{1-1/p} n) bound for p > 2.

Applied bounds to high-dimensional geometric spanners.

Refined decomposition bounds for 1 < p < 2.

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 $\beta=O(\log^{1-1/p} n)$. 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].