Streaming and Dynamic Algorithms for Minimum Enclosing Balls in High Dimensions

  • Timothy M. Chan
  • Vinayak Pathak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6844)


At SODA’10, Agarwal and Sharathkumar presented a streaming algorithm for approximating the minimum enclosing ball of a set of points in d-dimensional Euclidean space. Their algorithm requires one pass, uses O(d) space, and was shown to have approximation factor at most \((1+\sqrt{3})/2 +\varepsilon \approx 1.3661\). We prove that the same algorithm has approximation factor less than 1.22, which brings us much closer to a \((1+\sqrt{2})/2 \approx 1.207\) lower bound given by Agarwal and Sharathkumar.

We also apply this technique to the dynamic version of the minimum enclosing ball problem (in the non-streaming setting). We give an O(dn)-space data structure that can maintain a 1.22-approximate minimum enclosing ball in O(d logn) expected amortized time per insertion/deletion.


Anchor Point Priority Queue Approximation Factor Dynamic Algorithm Quadratic Inequality 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. Journal of the ACM 51, 606–635 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agarwal, P.K., Sharathkumar, R.: Streaming algorithms for extent problems in high dimensions. In: Proc. 21st ACM–SIAM Sympos. Discrete Algorithms, pp. 1481–1489 (2010)Google Scholar
  3. 3.
    Agarwal, P.K., Yu, H.: A space-optimal data-stream algorithm for coresets in the plane. In: Proc. 23rd Sympos. Comput. Geom., pp. 1–10 (2007)Google Scholar
  4. 4.
    Barequet, G., Har-Peled, S.: Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. J. Algorithms 38, 91–109 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bentley, J.L., Saxe, J.B.: Decomposable searching problems I: Static-to-dynamic transformations. J. Algorithms 1, 301–358 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bădoiu, M., Clarkson, K.L.: Smaller core-sets for balls. In: Proc. 14th ACM-SIAM Sympos. Discrete Algorithms, pp. 801–802 (2003)Google Scholar
  7. 7.
    Bădoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. In: Proc. 34th ACM Sympos. Theory Comput., pp. 250–257 (2002)Google Scholar
  8. 8.
    Chan, T.M.: Faster core-set constructions and data stream algorithms in fixed dimensions. Comput. Geom. Theory Appl. 35, 20–35 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chan, T.M.: Dynamic coresets. Discrete Comput. Geom. 42, 469–488 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kumar, P., Mitchell, J.S.B., Yildirim, E.A.: Approximating minimum enclosing balls in high dimensions using core-sets. ACM J. Experimental Algorithmics 8, 1.1 (2003)zbMATHGoogle Scholar
  11. 11.
    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
  12. 12.
    Zarrabi-Zadeh, H., Chan, T.M.: A simple streaming algorithm for minimum enclosing balls. In: Proc. 18th Canad. Conf. Comput. Geom., pp. 139–142 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Timothy M. Chan
    • 1
  • Vinayak Pathak
    • 1
  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations