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Lipschitz Decompositions of Finite 𝓁_{p} Metrics

Authors Robert Krauthgamer , Nir Petruschka

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Subject Classification

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.

References

  1. Ittai Abraham, Cyril Gavoille, Anupam Gupta, Ofer Neiman, and Kunal Talwar. Cops, robbers, and threatening skeletons: padded decomposition for minor-free graphs. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC '14, pages 79-88. Association for Computing Machinery, 2014. URL: https://doi.org/10.1145/2591796.2591849.
  2. Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen Kobourov, and Richard Spence. Graph spanners: A tutorial review. Computer Science Review, 37:100253, 2020. URL: https://doi.org/10.1016/j.cosrev.2020.100253.
  3. A. Andoni and P. Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. In 47th Annual IEEE Symposium on Foundations of Computer Science, pages 459-468. IEEE, 2006. URL: https://doi.org/10.1109/FOCS.2006.49.
  4. A. Andoni and P. Indyk. Nearest neighbors in high-dimensional spaces. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 43, pages 1135-1150. CRC Press, 3rd edition, 2017. URL: https://doi.org/10.1201/9781315119601.
  5. Alexandr Andoni, Assaf Naor, Aleksandar Nikolov, Ilya P. Razenshteyn, and Erik Waingarten. Data-dependent hashing via nonlinear spectral gaps. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 787-800. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188846.
  6. S. Arora, J. R. Lee, and A. Naor. Euclidean distortion and the sparsest cut. J. Amer. Math. Soc., 21(1):1-21, 2008. Google Scholar
  7. Y. Bartal. Probabilistic approximation of metric spaces and its algorithmic applications. In 37th Annual Symposium on Foundations of Computer Science, pages 184-193. IEEE, 1996. Google Scholar
  8. Y. Bartal. Graph decomposition lemmas and their role in metric embedding methods. In 12th Annual European Symposium on Algorithms, volume 3221 of LNCS, pages 89-97. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-30140-0_10.
  9. Yair Bartal and Lee-Ad Gottlieb. Dimension reduction techniques for 𝓁_p (1 < p < 2) with applications. In 32nd International Symposium on Computational Geometry, SoCG 2016, volume 51 of LIPIcs, pages 16:1-16:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.SOCG.2016.16.
  10. Yair Bartal and Lee-Ad Gottlieb. Approximate nearest neighbor search for 𝓁_p-spaces (2 < p < ∞) via embeddings. Theoretical Computer Science, 757:27-35, 2019. URL: https://doi.org/10.1016/j.tcs.2018.07.011.
  11. Yoav Benyamini and Joram Lindenstrauss. Geometric nonlinear functional analysis, volume 48. American Mathematical Soc., 1998. Google Scholar
  12. Glencora Borradaile, Hung Le, and Christian Wulff-Nilsen. Greedy spanners are optimal in doubling metrics. In Proceedings of the 2019 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2371-2379, 2019. URL: https://doi.org/10.1137/1.9781611975482.145.
  13. M. Charikar, C. Chekuri, A. Goel, S. Guha, and S. Plotkin. Approximating a finite metric by a small number of tree metrics. In 39th Annual Symposium on Foundations of Computer Science, pages 379-388, 1998. Google Scholar
  14. Moses Charikar and Amit Sahai. Dimension reduction in the 𝓁₁ norm. In 43rd Symposium on Foundations of Computer Science (FOCS 2002), pages 551-560. IEEE Computer Society, 2002. URL: https://doi.org/10.1109/SFCS.2002.1181979.
  15. A. Dvoretzky. Some results on convex bodies and Banach spaces. In Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pages 123-160. Jerusalem Academic Press, Jerusalem, 1961. Google Scholar
  16. David Eppstein. Spanning trees and spanners. In Handbook of Computational Geometry, pages 425-461. North Holland / Elsevier, 2000. URL: https://doi.org/10.1016/B978-044482537-7/50010-3.
  17. J. Fakcharoenphol, S. Rao, and K. Talwar. A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci., 69(3):485-497, 2004. URL: https://doi.org/10.1016/j.jcss.2004.04.011.
  18. Arnold Filtser. On strong diameter padded decompositions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019, volume 145 of LIPIcs, pages 6:1-6:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPICS.APPROX-RANDOM.2019.6.
  19. Arnold Filtser. Scattering and sparse partitions, and their applications. ACM Trans. Algorithms, 20(4):30:1-30:42, 2024. URL: https://doi.org/10.1145/3672562.
  20. Arnold Filtser, Tobias Friedrich, Davis Issac, Nikhil Kumar, Hung Le, Nadym Mallek, and Ziena Zeif. Optimal padded decomposition for bounded treewidth graphs. CoRR, abs/2407.12230, 2024. URL: https://doi.org/10.48550/arXiv.2407.12230.
  21. Arnold Filtser, Lee-Ad Gottlieb, and Robert Krauthgamer. Labelings vs. embeddings: On distributed and prioritized representations of distances. Discrete & Computational Geometry, 71:849-871, 2024. URL: https://doi.org/10.1007/s00454-023-00565-2.
  22. Arnold Filtser and Ofer Neiman. Light spanners for high dimensional norms via stochastic decompositions. Algorithmica, 84(10):2987-3007, 2022. URL: https://doi.org/10.1007/s00453-022-00994-0.
  23. C. Gavoille, D. Peleg, S. Pérennes, and R. Raz. Distance labeling in graphs. Journal of Algorithms, 53(1):85-112, 2004. URL: https://doi.org/10.1016/J.JALGOR.2004.05.002.
  24. A. Gupta, R. Krauthgamer, and J. R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In 44th Annual IEEE Symposium on Foundations of Computer Science, pages 534-543, 2003. URL: https://doi.org/10.1109/SFCS.2003.1238226.
  25. Sariel Har-Peled, Piotr Indyk, and Anastasios Sidiropoulos. Euclidean spanners in high dimensions. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, pages 804-809. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.57.
  26. P. Indyk and R. Motwani. Approximate nearest neighbors: towards removing the curse of dimensionality. In 30th Annual ACM Symposium on Theory of Computing, pages 604-613, 1998. URL: https://doi.org/10.1145/276698.276876.
  27. W. B. Johnson and J. Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. In Conference in modern analysis and probability (New Haven, Conn., 1982), pages 189-206. Amer. Math. Soc., Providence, RI, 1984. Google Scholar
  28. Nigel J. Kalton. The nonlinear geometry of Banach spaces. Revista Matematica Complutense, 21(1):7-60, 2008. URL: http://eudml.org/doc/42299.
  29. R. Krauthgamer and J. R. Lee. The intrinsic dimensionality of graphs. Combinatorica, 27(5):551-585, 2007. URL: https://doi.org/10.1007/s00493-007-2183-y.
  30. R. Krauthgamer, J. R. Lee, M. Mendel, and A. Naor. Measured descent: A new embedding method for finite metrics. Geometric And Functional Analysis, 15(4):839-858, 2005. URL: https://doi.org/10.1007/s00039-005-0527-6.
  31. Robert Krauthgamer, Nir Petruschka, and Shay Sapir. The power of recursive embeddings for 𝓁_p metrics, 2025. URL: https://doi.org/10.48550/arXiv.2503.18508.
  32. Robert Krauthgamer and Tim Roughgarden. Metric clustering via consistent labeling. Theory of Computing, 7(5):49-74, 2011. URL: https://doi.org/10.4086/toc.2011.v007a005.
  33. Deepanshu Kush, Aleksandar Nikolov, and Haohua Tang. Near neighbor search via efficient average distortion embeddings. In 37th International Symposium on Computational Geometry, SoCG 2021, volume 189 of LIPIcs, pages 50:1-50:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.SOCG.2021.50.
  34. Hung Le and Shay Solomon. A unified framework for light spanners. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 295-308. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585185.
  35. J. R. Lee and A. Naor. Metric decomposition, smooth measures, and clustering. Unpublished manuscript, 2003. URL: https://web.math.princeton.edu/~naor/homepage files/cluster.pdf.
  36. James R. Lee and Anastasios Sidiropoulos. Genus and the geometry of the cut graph. In 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pages 193-201. SIAM, 2010. URL: https://doi.org/10.1137/1.9781611973075.18.
  37. N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215-245, 1995. URL: https://doi.org/10.1007/BF01200757.
  38. J. Matoušek. On embedding expanders into l_p spaces. Israel J. Math., 102:189-197, 1997. Google Scholar
  39. M. Mendel and A. Naor. Ramsey partitions and proximity data structures. J. Eur. Math. Soc., 9(2):253-275, 2007. Google Scholar
  40. Assaf Naor. Probabilistic clustering of high dimensional norms. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, pages 690-709. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.44.
  41. Assaf Naor. Extension, separation and isomorphic reverse isoperimetry, volume 11 of Mem. Eur. Math. Soc. European Mathematical Society (EMS), 2024. URL: https://doi.org/10.4171/MEMS/11.
  42. Assaf Naor and Kevin Ren. 𝓁_p has nontrivial Euclidean distortion growth when 2 < p < 4, 2025. URL: https://arxiv.org/abs/2502.10543.
  43. Huy L. Nguyen. Approximate nearest neighbor search in 𝓁_p. CoRR, abs/1306.3601, 2013. URL: https://arxiv.org/abs/1306.3601.
  44. R. Ostrovsky and Y. Rabani. Polynomial-time approximation schemes for geometric min-sum median clustering. J. ACM, 49(2):139-156, 2002. URL: https://doi.org/10.1145/506147.506149.
  45. D. Peleg. Proximity-preserving labeling schemes. Journal of Graph Theory, 33(3):167-176, 2000. URL: https://doi.org/10.1002/(SICI)1097-0118(200003)33:3%3C167::AID-JGT7%3E3.0.CO;2-5.
  46. D. Peleg and A. A. Schäffer. Graph spanners. J. Graph Theory, 13(1):99-116, 1989. URL: https://doi.org/10.1002/jgt.3190130114.
  47. David Peleg and Jeffrey D. Ullman. An optimal synchronizer for the hypercube. SIAM J. Comput., 18(4):740-747, 1989. URL: https://doi.org/10.1137/0218050.
  48. S. Rao. Small distortion and volume preserving embeddings for planar and Euclidean metrics. In Proceedings of the 15th Annual Symposium on Computational Geometry, pages 300-306. ACM, 1999. URL: https://doi.org/10.1145/304893.304983.
  49. Uri Zwick. Exact and approximate distances in graphs - A survey. In Algorithms - ESA 2001, 9th Annual European Symposium, volume 2161 of Lecture Notes in Computer Science, pages 33-48. Springer, 2001. URL: https://doi.org/10.1007/3-540-44676-1_3.
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