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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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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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