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Lipschitz Decompositions of Finite 𝓁_{p} Metrics

Authors Robert Krauthgamer , Nir Petruschka

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Sparsification and spanners
Keywords
  • Lipschitz decompositions
  • metric embeddings
  • geometric spanners

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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 𝓁_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].

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Robert Krauthgamer and Nir Petruschka. Lipschitz Decompositions of Finite 𝓁_{p} Metrics. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.66

Author Details

Robert Krauthgamer
  • Weizmann Institute of Science, Rehovot, Israel
Nir Petruschka
  • Weizmann Institute of Science, Rehovot, Israel

Funding

  • Krauthgamer, Robert: Work partially supported by the Israel Science Foundation grant #1336/23, by the Israeli Council for Higher Education (CHE) via the Weizmann Data Science Research Center, and by a research grant from the Estate of Harry Schutzman.

Acknowledgements

We thank Arnold Filtser and Ofer Neiman for helpful discussions.

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