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Streaming with Minimum Space: An Algorithm for Covering by Two Congruent Balls

  • Chung Keung Poon
  • Binhai Zhu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7402)

Abstract

In this paper we design a simple streaming algorithm for maintaining two smallest balls (of equal radius) in d-dimension to cover a set of points in an on-line fashion. Different from most of the traditional streaming models, at any step we use the minimum amount of space by only storing the locations and the (common) radius of the balls. Previously, such a geometric algorithm is only investigated for covering with one ball (one-center) by Zarrabi-Zadeh and Chan. We give an analysis of our algorithm, which is significantly different from the one-center algorithm due to the obvious possibility of grouping points wrongly under this streaming model. We obtain upper bounds of 2 and 5.708 for the case of d = 1 and d > 1 respectively. We also present some lower bounds for the corresponding problems.

Keywords

Convex Hull Approximation Ratio Input Point Minimum Space Optimal Radius 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agarwal, P., Har-Peled, S., Varadarajan, K.: Approximating extent measures of points. J. ACM 51(4), 606–635 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Agarwal, P., Sharathkumar, R.: Streaming algorithms for extent problems in high dimensions. In: Proc. 21st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 1481–1489 (2010)Google Scholar
  3. 3.
    Bǎdoiu, M., Clarkson, K.: Smaller core-sets for balls. In: Proc. 14th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2003, pp. 801–802 (2003)Google Scholar
  4. 4.
    Bǎdoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. In: Proc. 34th Annual ACM Symposium on Theory of Computing, STOC 2002, pp. 250–257 (2002)Google Scholar
  5. 5.
    Chan, T.: More planar two-center algorithms. Comput. Geom. Theory Appls. 13, 189–198 (1999)zbMATHCrossRefGoogle Scholar
  6. 6.
    Chan, T.: Dynamic coresets. In: Proc. 24th Annual ACM Symposium on Computational Geometry, SoCG 2008, pp. 1–9 (2008)Google Scholar
  7. 7.
    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
  8. 8.
    Charikar, M., Chekuri, C., Feder, T., Motwani, R.: Incremental clustering and dynamic information retrieval. SIAM Journal on Computing 33(6), 1417–1440 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Eppstein, D.: Faster construction of planar two-centers. In: Proc. 8th Annu. ACM-SIAM Sympos. Discrete Algo., pp. 131–138 (1997)Google Scholar
  10. 10.
    Guha, S.: Tight results for clusering and summarizing data streams. In: Proc. of 12th International Conference on Database Theory, pp. 268–275. ACM (2009)Google Scholar
  11. 11.
    Har-Peled, S., Kushal, A.: Smaller coresets for k-median and k-means clustering. Discrete Computational Geometry 37, 3–19 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Har-Peled, S., Mazumdar, S.: Coresets for k-means and k-median clustering and their applications. In: Proc. 36th Annual ACM Symposium on Theory of Computing, STOC 2004, pp. 291–300 (2004)Google Scholar
  13. 13.
    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
  14. 14.
    Sharir, M.: A near-linear algorithm for the planar 2-center problem. Discrete Comput. Geom. 4, 125–134 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zarrabi-Zadeh, H.: An Almost Space-Optimal Streaming Algorithm for Coresets in Fixed Dimensions. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 817–829. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Zarrabi-Zadeh, H., Chan, T.: A simple streaming algorithm for minimum enclosing balls. In: Proc. 18th Canadian Conference on Computational Geometry, CCCG 2006, pp. 139–142 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Chung Keung Poon
    • 1
  • Binhai Zhu
    • 2
  1. 1.Department of Computer ScienceCity University of Hong KongKowloonHong Kong
  2. 2.Department of Computer ScienceMontana State UniversityBozemanUSA

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