Document Open Access Logo

Coresets for 1-Center in 𝓁₁ Metrics

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

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.ITCS.2025.28.pdf
  • Filesize: 0.87 MB
  • 20 pages
HTML5 Logo HTML (experimental)
LIPIcs.ITCS.2025.28.html

Document Identifiers

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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.

Cite As Get BibTex

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

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.

References

  1. Amir Abboud, MohammadHossein Bateni, Vincent Cohen-Addad, Karthik C. S., and Saeed Seddighin. On complexity of 1-center in various metrics. In APPROX/RANDOM, volume 275 of LIPIcs, pages 1:1-1:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.1.
  2. Pankaj K. Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Approximating extent measures of points. J. ACM, 51(4):606-635, 2004. URL: https://doi.org/10.1145/1008731.1008736.
  3. Pankaj K. Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Geometric approximation via coresets. In Combinatorial and computational geometry, volume 52 of MSRI Publications, chapter 1, pages 1-30. Cambridge University Press, 2005. Google Scholar
  4. Sepideh Aghamolaei and Mohammad Ghodsi. A composable coreset for k-center in doubling metrics. In CCCG, pages 165-171, 2018. URL: http://www.cs.umanitoba.ca/%7Ecccg2018/papers/session4A-p2.pdf.
  5. Christian Bachmaier, Franz J. Brandenburg, Andreas Gleißner, and Andreas Hofmeier. On the hardness of maximum rank aggregation problems. J. Discrete Algorithms, 31:2-13, 2015. URL: https://doi.org/10.1016/j.jda.2014.10.002.
  6. René Brandenberg and Stefan König. No dimension-independent core-sets for containment under homothetics. Discrete & Computational Geometry, 49(1):3-21, 2013. URL: https://doi.org/10.1007/s00454-012-9462-0.
  7. Mihai Bădoiu and Kenneth L. Clarkson. Optimal core-sets for balls. Comput. Geom., 40(1):14-22, 2008. URL: https://doi.org/10.1016/j.comgeo.2007.04.002.
  8. Mihai Bădoiu, Sariel Har-Peled, and Piotr Indyk. Approximate clustering via core-sets. In STOC, pages 250-257. ACM, 2002. URL: https://doi.org/10.1145/509907.509947.
  9. Marc Bury, Michele Gentili, Chris Schwiegelshohn, and Mara Sorella. Polynomial time approximation schemes for all 1-center problems on metric rational set similarities. Algorithmica, 83(5):1371-1392, 2021. URL: https://doi.org/10.1007/s00453-020-00787-3.
  10. Matteo Ceccarello, Andrea Pietracaprina, and Geppino Pucci. Solving k-center clustering (with outliers) in MapReduce and streaming, almost as accurately as sequentially. Proc. VLDB Endow., 12(7):766-778, 2019. URL: https://doi.org/10.14778/3317315.3317319.
  11. Diptarka Chakraborty, Debarati Das, and Robert Krauthgamer. Approximating the median under the ulam metric. In SODA, pages 761-775. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.48.
  12. Diptarka Chakraborty, Debarati Das, and Robert Krauthgamer. Clustering permutations: New techniques with streaming applications. In ITCS, volume 251 of LIPIcs, pages 31:1-31:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.31.
  13. Diptarka Chakraborty, Kshitij Gajjar, and Agastya Vibhuti Jha. Approximating the center ranking under ulam. In FSTTCS, volume 213 of LIPIcs, pages 12:1-12:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2021.12.
  14. Timothy M. Chan. Faster core-set constructions and data-stream algorithms in fixed dimensions. Computational Geometry, 35(1):20-35, 2006. URL: https://doi.org/10.1016/j.comgeo.2005.10.002.
  15. Ke Chen. On coresets for k-median and k-means clustering in metric and euclidean spaces and their applications. SIAM Journal on Computing, 39(3):923-947, 2009. URL: https://doi.org/10.1137/070699007.
  16. Flavio Chierichetti, Ravi Kumar, Sandeep Pandey, and Sergei Vassilvitskii. Finding the jaccard median. In SODA, pages 293-311. SIAM, 2010. URL: https://doi.org/10.1137/1.9781611973075.25.
  17. Michael B. Cohen, Yin Tat Lee, Gary L. Miller, Jakub Pachocki, and Aaron Sidford. Geometric median in nearly linear time. In STOC, pages 9-21. ACM, 2016. URL: https://doi.org/10.1145/2897518.2897647.
  18. Vincent Cohen-Addad, David Saulpic, and Chris Schwiegelshohn. A new coreset framework for clustering. In STOC, pages 169-182. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451022.
  19. Matan Danos. Coresets for clustering by uniform sampling and generalized rank aggregation. Master’s thesis, Weizmann Institute of Science, 2021. Google Scholar
  20. Mark de Berg, Leyla Biabani, and Morteza Monemizadeh. k-center clustering with outliers in the MPC and streaming model. In IPDPS, pages 853-863. IEEE, 2023. URL: https://doi.org/10.1109/IPDPS54959.2023.00090.
  21. Dan Feldman. Core-sets: An updated survey. WIREs Data Mining and Knowledge Discovery, 10(1):e1335, 2020. URL: https://doi.org/10.1002/widm.1335.
  22. Dan Feldman and Michael Langberg. A unified framework for approximating and clustering data. In STOC, pages 569-578. ACM, 2011. URL: https://doi.org/10.1145/1993636.1993712.
  23. Teofilo F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-306, 1985. URL: https://doi.org/10.1016/0304-3975(85)90224-5.
  24. Peter M Gruber and Jörg M Wills. Handbook of convex geometry, Volume B. North-Holland, 1993. Google Scholar
  25. Claire Kenyon-Mathieu and Warren Schudy. How to rank with few errors. In STOC, pages 95-103. ACM, 2007. URL: https://doi.org/10.1145/1250790.1250806.
  26. Ming Li, Bin Ma, and Lusheng Wang. On the closest string and substring problems. J. ACM, 49(2):157-171, 2002. URL: https://doi.org/10.1145/506147.506150.
  27. Bin Ma and Xiaoming Sun. More efficient algorithms for closest string and substring problems. SIAM Journal on Computing, 39(4):1432-1443, 2010. URL: https://doi.org/10.1137/080739069.
  28. Alexander Munteanu and Chris Schwiegelshohn. Coresets-methods and history: A theoreticians design pattern for approximation and streaming algorithms. Künstliche Intell., 32(1):37-53, 2018. URL: https://doi.org/10.1007/s13218-017-0519-3.
  29. François Nicolas and Eric Rivals. Hardness results for the center and median string problems under the weighted and unweighted edit distances. J. Discrete Algorithms, 3(2-4):390-415, 2005. URL: https://doi.org/10.1016/j.jda.2004.08.015.
  30. Rina Panigrahy. Minimum enclosing polytope in high dimensions. CoRR, cs.CG/0407020, 2004. URL: https://doi.org/10.48550/arXiv.CS/0407020.
  31. Jeff M. Phillips. Coresets and sketches. In Handbook of discrete and computational geometry, chapter 48, pages 1269-1288. Chapman and Hall/CRC, 3rd edition, 2017. URL: https://doi.org/10.1201/9781315119601.
  32. Alvin Yan Hong Yao and Diptarka Chakraborty. Approximate maximum rank aggregation: Beyond the worst-case. In FSTTCS, volume 284 of LIPIcs, pages 12:1-12:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2023.12.
  33. Hai Yu, Pankaj K. Agarwal, Raghunath Poreddy, and Kasturi R. Varadarajan. Practical methods for shape fitting and kinetic data structures using coresets. Algorithmica, 52(3):378-402, 2008. URL: https://doi.org/10.1007/s00453-007-9067-9.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact