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Analysis of Agglomerative Clustering

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Abstract

The diameter k-clustering problem is the problem of partitioning a finite subset of ℝd into k subsets called clusters such that the maximum diameter of the clusters is minimized. One early clustering algorithm that computes a hierarchy of approximate solutions to this problem (for all values of k) is the agglomerative clustering algorithm with the complete linkage strategy. For decades, this algorithm has been widely used by practitioners. However, it is not well studied theoretically. In this paper, we analyze the agglomerative complete linkage clustering algorithm. Assuming that the dimension d is a constant, we show that for any k the solution computed by this algorithm is an O(logk)-approximation to the diameter k-clustering problem. Our analysis does not only hold for the Euclidean distance but for any metric that is based on a norm. Furthermore, we analyze the closely related k-center and discrete k-center problem. For the corresponding agglomerative algorithms, we deduce an approximation factor of O(logk) as well.

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References

  1. Bādoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. In: Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC’02, pp. 250–257. ACM, New York (2002)

    Chapter  Google Scholar 

  2. Broder, A.Z., Glassman, S.C., Manasse, M.S., Zweig, G.: Syntactic clustering of the web. Comput. Netw. ISDN Syst. 29(8–13), 1157–1166 (1997). Papers from the Sixth International World Wide Web Conference

    Article  Google Scholar 

  3. Charikar, M., Chekuri, C., Feder, T., Motwani, R.: Incremental clustering and dynamic information retrieval. In: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, STOC’97, pp. 626–635. ACM, New York (1997)

    Chapter  Google Scholar 

  4. Dasgupta, S., Long, P.M.: Performance guarantees for hierarchical clustering. J. Comput. Syst. Sci. 70(4), 555–569 (2005). Special Issue on COLT 2002

    Article  MATH  MathSciNet  Google Scholar 

  5. Eisen, M.B., Spellman, P.T., Brown, P.O., Botstein, D.: Cluster analysis and display of genome-wide expression patterns. Proc. Natl. Acad. Sci. 95(25), 14863–14868 (1998)

    Article  Google Scholar 

  6. Feder, T., Greene, D.: Optimal algorithms for approximate clustering. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, STOC’88, pp. 434–444. ACM, New York (1988)

    Chapter  Google Scholar 

  7. Florek, K., Lukaszewicz, J., Perkal, J., Steinhaus, H., Zubrzycki, S.: Sur la liaison et la division des points d’un ensemble fini. Colloq. Math. 2, 282–285 (1951)

    Google Scholar 

  8. Fréchet, M.: Les dimensions d’un ensemble abstrait. Math. Ann. 68(2), 145–168 (1910)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci. 38(0), 293–306 (1985)

    Article  MATH  Google Scholar 

  10. Johnson, W.B., Joram, L.: Extensions of Lipschitz mappings into a Hilbert space. In: Conference in Modern Analysis and Probability. Contemporary Mathematics, vol. 26, pp. 189–206. Am. Math. Soc., Providence (1984)

    Chapter  Google Scholar 

  11. Lee, K., Kim, J., Hoon Kwon, K., Han, Y., Kim, S.: DDoS attack detection method using cluster analysis. Expert Syst. Appl. 34(3), 1659–1665 (2008)

    Article  Google Scholar 

  12. McQuitty, L.L.: Elementary linkage analysis for isolating orthogonal and oblique types and typal relevancies. Educ. Psychol. Meas. 17, 207–209 (1957)

    Article  Google Scholar 

  13. Naszódi, M.: Covering a set with homothets of a convex body. Positivity 14, 69–74 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Pereira, F., Tishby, N., Lee, L.: Distributional clustering of English words. In: Proceedings of the 31st Annual Meeting on Association for Computational Linguistics, ACL’93, pp. 183–190. Association for Computational Linguistics, Stroudsburg (1993)

    Chapter  Google Scholar 

  15. Sneath, P.H.A., Sokal, R.R.: Numerical Taxonomy: the Principles and Practice of Numerical Classification. Freeman, New York (1973)

    MATH  Google Scholar 

  16. Webster, R.: Convexity. Oxford Science Publications. Oxford University Press, London (1994)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Schloss Dagstuhl—Leibniz Center for Informatics, 66687, Wadern, Germany

    Marcel R. Ackermann

  2. Department of Computer Science, University of Paderborn, 33098, Paderborn, Germany

    Johannes Blömer & Daniel Kuntze

  3. Department of Computer Science, TU Dortmund, 44221, Dortmund, Germany

    Christian Sohler

Authors
  1. Marcel R. Ackermann
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  2. Johannes Blömer
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  3. Daniel Kuntze
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  4. Christian Sohler
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Corresponding author

Correspondence to Daniel Kuntze.

Additional information

A preliminary version of this article appeared in Proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS’11), March 2011, pp. 308–319.

Work was done while M.R. Ackermann was at Department of Computer Science, University of Paderborn, Germany.

For all four authors this research was supported by the German Research Foundation (DFG), grants BL 314/6-2 and SO 514/4-2.

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Ackermann, M.R., Blömer, J., Kuntze, D. et al. Analysis of Agglomerative Clustering. Algorithmica 69, 184–215 (2014). https://doi.org/10.1007/s00453-012-9717-4

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  • DOI: https://doi.org/10.1007/s00453-012-9717-4

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