Lipschitz Decompositions of Finite $\ell_{p}$ Metrics
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908841701015552 |
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| author | Krauthgamer, Robert Petruschka, Nir |
| author_facet | Krauthgamer, Robert Petruschka, Nir |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01120 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lipschitz Decompositions of Finite $\ell_{p}$ Metrics Krauthgamer, Robert Petruschka, Nir Computational Geometry Metric Geometry 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]. |
| title | Lipschitz Decompositions of Finite $\ell_{p}$ Metrics |
| topic | Computational Geometry Metric Geometry |
| url | https://arxiv.org/abs/2502.01120 |