Lipschitz Decompositions of Finite $ell_{p}$ Metrics

2025-02-03

AI Summary

This paper addresses the Lipschitz decomposition problem in finite ℓₚ metric spaces. For an n-point set, it establishes— for the first time—the optimal splitting parameter bound β = O(log^{1−1/p} n) for p > 2, fully resolving a long-standing open question posed by Naor (SODA 2017); it also significantly improves the best-known bounds for 1 < p ≤ 2. Methodologically, the work integrates probabilistic arguments, geometric functional analysis, and metric embedding theory, introducing novel extensions of hierarchical random partitions and sparse covers. The results advance high-dimensional geometric spanner construction efficiency and yield the first deterministic, logarithmically near-optimal guarantee on compression rate for distance labeling schemes—bridging deep theoretical insights with practical algorithmic impact.

Technology Category

Constraint Satisfaction and Optimization: Distributed CSP/OptimizationMachine Learning: Learning with ManifoldsPlanning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Large-scale security measurementsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $eta=O(log^{1-1/p} n)$. 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

Problem

• Lipschitz condition
• ell_p metric spaces
• optimal partition parameters

Innovation

• Lipschitz Decomposition
• High-dimensional Spaces
• Metric Embeddings