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Coresets for 1-Center in 𝓁₁ Metrics

Authors Amir Carmel , Chengzhi Guo , Shaofeng H.-C. Jiang , Robert Krauthgamer

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Author Details

Amir Carmel
  • Weizmann Institute of Science, Rehovot, Israel
Chengzhi Guo
  • Peking University, China
Shaofeng H.-C. Jiang
  • Peking University, China
Robert Krauthgamer
  • Weizmann Institute of Science, Rehovot, Israel

Funding

  • Jiang, Shaofeng H.-C.: Research partially supported by a national key R&D program of China No. 2021YFA1000900 and a startup fund from Peking University.
  • 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. Part of this work was done while visiting the Simons Institute for the Theory of Computing.

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Amir Carmel, Chengzhi Guo, Shaofeng H.-C. Jiang, and Robert Krauthgamer. Coresets for 1-Center in 𝓁₁ Metrics. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.28

Abstract

We explore the applicability of coresets - a small subset of the input dataset that approximates a predefined set of queries - to the 1-center problem in 𝓁₁ spaces. This approach could potentially extend to solving the 1-center problem in related metric spaces, and has implications for streaming and dynamic algorithms. 
We show that in 𝓁₁, unlike in Euclidean space, even weak coresets exhibit exponential dependency on the underlying dimension. Moreover, while inputs with a unique optimal center admit better bounds, they are not dimension independent. We then relax the guarantee of the coreset further, to merely approximate the value (optimal cost of 1-center), and obtain a dimension-independent coreset for every desired accuracy ε > 0. Finally, we discuss the broader implications of our findings to related metric spaces, and show explicit implications to Jaccard and Kendall’s tau distances.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sketching and sampling
  • Theory of computation → Streaming models
Keywords
  • clustering
  • k-center
  • minimum enclosing balls
  • coresets
  • 𝓁₁ norm
  • Kendall’s tau
  • Jaccard metric

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