Computing k-center over Streaming Data for Small k

  • Hee-Kap Ahn
  • Hyo-Sil Kim
  • Sang-Sub Kim
  • Wanbin Son
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7676)

Abstract

The Euclidean k-center problem is to compute k congruent balls covering a given set of points in ℝd such that the radius is minimized. We consider the k-center problem in ℝd for k = 2,3 in a single-pass streaming model, where data is allowed to be examined once and only a small amount of information can be stored in a device. We present two approximation algorithms whose space complexity does not depend on the size of the input data. The first algorithm guarantees a (2 + ε)-factor using O(d/ε) space in arbitrary dimensions, and the second algorithm guarantees a (1 + ε)-factor using O(1/εd) space in constant dimensions. The same algorithms can be used to compute a k-center under any Lp metric for k = 2,3.

References

  1. 1.
    Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. Journal of the ACM 51(4), 606–635 (2004)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Agarwal, P.K., Procopiuc, C.M.: Exact and approximation algorithms for clustering. Algorithmica 33, 201–226 (2002)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Agarwal, P.K., Sharathkumar, R.: Streaming algorithms for extent problems in high dimensions. In: Proc. of the 21st ACM-SIAM Sympos. Discrete Algorithms, pp. 1481–1489 (2010)Google Scholar
  4. 4.
    Aggarwal, C.C.: Data streams: models and algorithms. Springer (2007)Google Scholar
  5. 5.
    Bern, M., Eppstein, D.: Approximation algorithms for geometric problems. In: Approximation Algorithms for NP-Hard Problems. PWS Publishing Co. (1996)Google Scholar
  6. 6.
    Chan, T.M.: More planar two-center algorithms. Computational Geometry 13(3), 189–198 (1999)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Chan, T.M.: Faster core-set constructions and data-stream algorithms in fixed dimensions. Computational Geometry 35, 20–35 (2006)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Chan, T.M., Pathak, V.: Streaming and Dynamic Algorithms for Minimum Enclosing Balls in High Dimensions. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 195–206. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Charikar, M., Chekuri, C., Feder, T., Motwani, R.: Incremental clustering and dynamic information retrieval. SIAM J. Comput. 33(6), 1417–1440 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Chazelle, B., Matoušek, J.: On linear-time deterministic algorithms for optimization problems in fixed dimension. Journal of Algorithms 21, 579–597 (1996)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)MATHGoogle Scholar
  12. 12.
    Guha, S.: Tight results for clustering and summarizing data streams. In: Proc. of the 12th Int. Conf. on Database Theory, pp. 268–275. ACM (2009)Google Scholar
  13. 13.
    Hershberger, J., Suri, S.: Adaptive sampling for geometric problems over data streams. Computational Geometry 39(3), 191–208 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Zarrabi-Zadeh, H.: Core-preserving algorithms. In: Proc. of 20th Canadian Conf. on Comput. Geom. (CCCG), pp. 159–162 (2008)Google Scholar
  15. 15.
    McCutchen, R.M., Khuller, S.: Streaming Algorithms for k-Center Clustering with Outliers and with Anonymity. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX and RANDOM 2008. LNCS, vol. 5171, pp. 165–178. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Megiddo, M.: On the complexity of some geometric problems in unbounded dimension. J. Symbolic Comput. 10, 327–334 (1990)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Megiddo, M., Supowit, K.J.: On the complexity of some common geometric location problems. SIAM J. Comput. 13(1), 182–196 (1984)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Poon, C.K., Zhu, B.: Streaming with Minimum Space: An Algorithm for Covering by Two Congruent Balls. In: Lin, G. (ed.) COCOA 2012. LNCS, vol. 7402, pp. 269–280. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  19. 19.
    Zarrabi-Zadeh, H.: An almost space-optimal streaming algorithm for coresets in fixed dimensions. Algorithmica 60, 46–59 (2011)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Zarrabi-Zadeh, H., Chan, T.M.: A simple streaming algorithm for minimum enclosing balls. In: Proc. of 18th CCCG, pp. 139–142 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Hee-Kap Ahn
    • 1
  • Hyo-Sil Kim
    • 1
  • Sang-Sub Kim
    • 1
  • Wanbin Son
    • 1
  1. 1.Pohang University of Science and TechnologyKorea

Personalised recommendations