Stability of ε-Kernels

  • Pankaj K. Agarwal
  • Jeff M. Phillips
  • Hai Yu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6346)


Given a set P of n points in ℝ d , an ε-kernel K ⊆ P approximates the directional width of P in every direction within a relative (1 − ε) factor. In this paper we study the stability of ε-kernels under dynamic insertion and deletion of points to P and by changing the approximation factor ε. In the first case, we say an algorithm for dynamically maintaining a ε-kernel is stable if at most O(1) points change in K as one point is inserted or deleted from P. We describe an algorithm to maintain an ε-kernel of size O(1/ε (d − 1)/2) in O(1/ε (d − 1)/2 + logn) time per update. Not only does our algorithm maintain a stable ε-kernel, its update time is faster than any known algorithm that maintains an ε-kernel of size O(1/ε (d − 1)/2). Next, we show that if there is an ε-kernel of P of size κ, which may be dramatically less than O(1/ε (d − 1)/2), then there is an (ε/2)-kernel of P of size \(O(\min\{ 1/\varepsilon^{(d-1)/2}, \kappa^{\lfloor d/2 \rfloor} \log^{d-2} (1/\varepsilon)\})\). Moreover, there exists a point set P in ℝ d and a parameter ε> 0 such that if every ε-kernel of P has size at least κ, then any (ε/2)-kernel of P has size \(\Omega(\kappa^{\lfloor d/2 \rfloor})\).


Anchor Point Edge Region Sparse Grid Point Region Triangle Region 
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.: Approximating extent measure of points. Journal of ACM 51(4), 606–635 (2004)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Agarwal, P.K., Har-Peled, S., Varadarajan, K.: Geometric approximations via coresets. In: Combinatorial and Computational Geometry, pp. 1–31 (2005)Google Scholar
  3. 3.
    Agarwal, P.K., Phillips, J.M., Yu, H.: Stability of ε-kernels. arXiv:1003.5874Google Scholar
  4. 4.
    Agarwal, P.K., Yu, H.: A space-optimal data-stream algorithm for coresets in the plane. In: SoCG, pp. 1–10 (2007)Google Scholar
  5. 5.
    Barequet, G., Har-Peled, S.: Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. Journ. of Algs. 38, 91–109 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chan, T.: Faster core-set constructions and data-stream algorithms in fixed dimensions. Computational Geometry: Theory and Applications 35, 20–35 (2006)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Chan, T.: Dynamic coresets. In: SoCG, pp. 1–9 (2008)Google Scholar
  8. 8.
    Har-Peled, S.: Approximation Algorithm in Geometry, ch. 22 (2010),
  9. 9.
    Hershberger, J., Suri, S.: Adaptive sampling for geometric problems over data streams. Computational Geometry: Theory and Applications 39, 191–208 (2008)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Yu, H., Agarwal, P.K., Poreddy, R., Varadarajan, K.: Practical methods for shape fitting and kinetic data structures using coresets. Algorithmica 52, 378–402 (2008)zbMATHCrossRefMathSciNetGoogle 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)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Jeff M. Phillips
    • 1
  • Hai Yu
    • 1
  1. 1.Duke University, University of Utah, and Google 

Personalised recommendations