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Fast Metric Decompositions in High Dimension

Authors Robert Krauthgamer , Asaf Petruschka , Nir Petruschka

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LIPIcs.ESA.2026.95.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Sparsification and spanners
Keywords
  • Metric Spaces
  • Separating Decomposition
  • Padded Decomposition

Metrics

Abstract

Metric decompositions are a fundamental tool in the design of algorithms involving distances. We study fast algorithms for sampling from probabilistic metric decompositions of n-point sets in 𝓁_∞ and 𝓁₂ spaces of high dimension d. For 𝓁_∞, we design a padded-decomposition algorithm that runs in time Õ(nd²), which is near-linear in n, and achieves padding parameter Õ(log n). Our algorithm constructs a new sparse neighborhood cover that is based on geometric properties of 𝓁_∞ [Indyk, JCSS'01], and utilizes recent reductions between covers and decompositions [Conroy and Filtser, STOC'25]. For 𝓁₂, we design a separating-decomposition algorithm that achieves near optimal separation Õ(√{log n}) in almost-linear time n^{1+o(1)}. Our bounds improve over known algorithms with similar running time by a factor Ω(√{log n}), and the techniques have additional applications to spanners and nearest-neighbor search.

Cite As Get BibTex

Robert Krauthgamer, Asaf Petruschka, and Nir Petruschka. Fast Metric Decompositions in High Dimension. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 95:1-95:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.95

Author Details

Robert Krauthgamer
  • The Harry Weinrebe Professorial Chair of Computer Science, Weizmann Institute of Science, Rehovot, Israel
Asaf Petruschka
  • 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.
  • Petruschka, Asaf: Supported by an Azrieli Foundation fellowship.

Acknowledgements

We thank an anonymous reviewer for useful comments.

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