Data Stability in Clustering: A Closer Look

  • Lev Reyzin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7568)


We consider the model introduced by Bilu and Linial [12],, who study problems for which the optimal clustering does not change when distances are perturbed. They show that even when a problem is NP-hard, it is sometimes possible to obtain efficient algorithms for instances resilient to certain multiplicative perturbations, e.g. on the order of \(O(\sqrt{n})\) for max-cut clustering. Awasthi et al. [6], consider center-based objectives, and Balcan and Liang [9], analyze the k-median and min-sum objectives, giving efficient algorithms for instances resilient to certain constant multiplicative perturbations.

Here, we are motivated by the question of to what extent these assumptions can be relaxed while allowing for efficient algorithms. We show there is little room to improve these results by giving NP-hardness lower bounds for both the k-median and min-sum objectives. On the other hand, we show that multiplicative resilience parameters, even only on the order of Θ(1), can be so strong as to make the clustering problem trivial, and we exploit these assumptions to present a simple one-pass streaming algorithm for the k-median objective. We also consider a model of additive perturbations and give a correspondence between additive and multiplicative notions of stability. Our results provide a close examination of the consequences of assuming, even constant, stability in data.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ackerman, M., Ben-David, S.: Clusterability: A theoretical study. Journal of Machine Learning Research - Proceedings Track 5, 1–8 (2009)Google Scholar
  2. 2.
    Ailon, N., Jaiswal, R., Monteleoni, C.: Streaming k-means approximation. In: NIPS (2009)Google Scholar
  3. 3.
    Arora, S., Raghavan, P., Rao, S.: Approximation schemes for euclidean-medians and related problems. In: STOC, pp. 106–113 (1998)Google Scholar
  4. 4.
    Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k-median and facility location problems. SIAM J. Comput. 33(3), 544–562 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Awasthi, P., Blum, A., Sheffet, O.: Stability yields a ptas for k-median and k-means clustering. In: FOCS, pp. 309–318 (2010)Google Scholar
  6. 6.
    Awasthi, P., Blum, A., Sheffet, O.: Center-based clustering under perturbation stability. Inf. Process. Lett. 112(1-2), 49–54 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Balcan, M.F., Blum, A., Gupta, A.: Approximate clustering without the approximation. In: SODA (2009)Google Scholar
  8. 8.
    Balcan, M.F., Blum, A., Vempala, S.: A discriminative framework for clustering via similarity functions. In: STOC, pp. 671–680 (2008)Google Scholar
  9. 9.
    Balcan, M.F., Liang, Y.: Clustering under Perturbation Resilience. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 63–74. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  10. 10.
    Bartal, Y., Charikar, M., Raz, D.: Approximating min-sum -clustering in metric spaces. In: STOC, pp. 11–20 (2001)Google Scholar
  11. 11.
    Ben-David, S.: Alternative Measures of Computational Complexity with Applications to Agnostic Learning. In: Cai, J.-Y., Cooper, S.B., Li, A. (eds.) TAMC 2006. LNCS, vol. 3959, pp. 231–235. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Bilu, Y., Linial, N.: Are stable instances easy? In: Innovations in Computer Science, pp. 332–341 (2010)Google Scholar
  13. 13.
    Charikar, M., Guha, S., Tardos, É., Shmoys, D.B.: A constant-factor approximation algorithm for the k-median problem. J. Comput. Syst. Sci. 65(1), 129–149 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Charikar, M., O’Callaghan, L., Panigrahy, R.: Better streaming algorithms for clustering problems. In: STOC, pp. 30–39 (2003)Google Scholar
  15. 15.
    de la Vega, W.F., Karpinski, M., Kenyon, C., Rabani, Y.: Approximation schemes for clustering problems. In: STOC, pp. 50–58 (2003)Google Scholar
  16. 16.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)zbMATHGoogle Scholar
  17. 17.
    Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. J. Algorithms 31(1), 228–248 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Guha, S., Meyerson, A., Mishra, N., Motwani, R., O’Callaghan, L.: Clustering data streams: Theory and practice. IEEE Trans. Knowl. Data Eng. 15(3), 515–528 (2003)CrossRefGoogle Scholar
  19. 19.
    Jain, K., Mahdian, M., Saberi, A.: A new greedy approach for facility location problems. In: STOC, pp. 731–740. ACM (2002)Google Scholar
  20. 20.
    Kumar, A., Sabharwal, Y., Sen, S.: Linear-time approximation schemes for clustering problems in any dimensions. J. ACM 57(2) (2010)Google Scholar
  21. 21.
    Lloyd, S.: Least squares quantization in pcm. IEEE Transactions on Information Theory 28(2), 129–137 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Muthukrishnan, S.: Data streams: algorithms and applications. In: SODA, p. 413 (2003)Google Scholar
  23. 23.
    Ostrovsky, R., Rabani, Y., Schulman, L.J., Swamy, C.: The effectiveness of lloyd-type methods for the k-means problem. In: FOCS, pp. 165–176 (2006)Google Scholar
  24. 24.
    Schalekamp, F., Yu, M., van Zuylen, A.: Clustering with or without the Approximation. In: Thai, M.T., Sahni, S. (eds.) COCOON 2010. LNCS, vol. 6196, pp. 70–79. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  25. 25.
    van Rooij, J.M.M., van Kooten Niekerk, M.E., Bodlaender, H.L.: Partition into Triangles on Bounded Degree Graphs. In: Černá, I., Gyimóthy, T., Hromkovič, J., Jefferey, K., Králović, R., Vukolić, M., Wolf, S. (eds.) SOFSEM 2011. LNCS, vol. 6543, pp. 558–569. Springer, Heidelberg (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Lev Reyzin
    • 1
  1. 1.School of Computer ScienceGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations