An Almost Space-Optimal Streaming Algorithm for Coresets in Fixed Dimensions

  • Hamid Zarrabi-Zadeh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5193)


We present a new streaming algorithm for maintaining an ε-kernel of a point set in ℝ d using O((1/ε (d − 1)/2) log(1/ε)) space. The space used by our algorithm is optimal up to a small logarithmic factor. This substantially improves (for any fixed dimension \(d \geqslant 3\)) the best previous algorithm for this problem that uses O(1/ε d − (3/2)) space, presented by Agarwal and Yu at SoCG’07. Our algorithm immediately improves the space complexity of the best previous streaming algorithms for a number of fundamental geometric optimization problems in fixed dimensions, including width, minimum enclosing cylinder, minimum-width enclosing annulus, minimum-width enclosing cylindrical shell, etc.


Cylindrical Shell Space Complexity Input Point Compression Step Maximum Span Tree 
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.: Maintaining approximate extent measures of moving points. In: Proc. 12th ACM-SIAM Sympos. Discrete Algorithms, pp. 148–157 (2001)Google Scholar
  2. 2.
    Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. J. ACM 51(4), 606–635 (2004)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Geometric approximation via coresets. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry, Math. Sci. Research Inst. Pub., Cambridge (2005)Google Scholar
  4. 4.
    Agarwal, P.K., Har-Peled, S., Yu, H.: Robust shape fitting via peeling and grating coresets. In: Proc. 17th ACM-SIAM Sympos. Discrete Algorithms, pp. 182–191 (2006)Google Scholar
  5. 5.
    Agarwal, P.K., Matoušek, J., Suri, S.: Farthest neighbors, maximum spanning trees and related problems in higher dimensions. Comput. Geom. Theory Appl. 1(4), 189–201 (1992)zbMATHGoogle Scholar
  6. 6.
    Agarwal, P.K., Yu, H.: A space-optimal data-stream algorithm for coresets in the plane. In: Proc. 23rd Annu. ACM Sympos. Comput. Geom., pp. 1–10 (2007)Google Scholar
  7. 7.
    Barequet, G., Har-Peled, S.: Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. J. Algorithms 38(1), 91–109 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bentley, J.L., Saxe, J.B.: Decomposable searching problems I: Static-to-dynamic transformations. J. Algorithms 1, 301–358 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chan, T.M.: Faster core-set constructions and data stream algorithms in fixed dimensions. Comput. Geom. Theory Appl. 35(1–2), 20–35 (2006)zbMATHGoogle Scholar
  10. 10.
    Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory 10, 227–236 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Frahling, G., Sohler, C.: Coresets in dynamic geometric data streams. In: Proc. 37th Annu. ACM Sympos. Theory Comput., pp. 209–217 (2005)Google Scholar
  12. 12.
    Har-Peled, S., Mazumdar, S.: On coresets for k-means and k-median clustering. In: Proc. 36th Annu. ACM Sympos. Theory Comput., pp. 291–300 (2004)Google Scholar
  13. 13.
    Har-Peled, S., Wang, Y.: Shape fitting with outliers. SIAM J. Comput. 33(2), 269–285 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Yu, H., Agarwal, P.K., Poreddy, R., Varadarajan, K.R.: Practical methods for shape fitting and kinetic data structures using core sets. In: Proc. 20th Annu. ACM Sympos. Comput. Geom., pp. 263–272 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Hamid Zarrabi-Zadeh
    • 1
  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations