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Better Bounds on Online Unit Clustering

  • Martin R. Ehmsen
  • Kim S. Larsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6139)

Abstract

Unit Clustering is the problem of dividing a set of points from a metric space into a minimal number of subsets such that the points in each subset are enclosable by a unit ball. We continue work initiated by Chan and Zarrabi-Zadeh on determining the competitive ratio of the online version of this problem. For the one-dimensional case, we develop a deterministic algorithm, improving the best known upper bound of 7/4 by Epstein and van Stee to 5/3. This narrows the gap to the best known lower bound of 8/5 to only 1/15. Our algorithm automatically leads to improvements in all higher dimensions as well. Finally, we strengthen the deterministic lower bound in two dimensions and higher from 2 to 13/6.

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References

  1. 1.
    Chan, T.M., Zarrabi-Zadeh, H.: A randomized algorithm for online unit clustering. Theory of Computing Systems 45(3), 486–496 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Charikar, M., Chekuri, C., Feder, T., Motwani, R.: Incremental clustering and dynamic information retrieval. SIAM Journal on Computing 33(6), 1417–1440 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ehmsen, M.R., Larsen, K.S.: Better Bounds on Online Unit Clustering. Preprint 8, Department of Mathematics and Computer Science, University of Southern Denmark (2009)Google Scholar
  4. 4.
    Epstein, L., Levin, A., van Stee, R.: Online unit clustering: Variations on a theme. Theoretical Computer Science 407, 85–96 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Epstein, L., van Stee, R.: On the online unit clustering problem. In: Proceedings of the 5th International Workshop on Approximation and Online Algorithms, pp. 193–206 (2007)Google Scholar
  6. 6.
    Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Systems Technical Journal 45, 1563–1581 (1966)Google Scholar
  7. 7.
    Karlin, A.R., Manasse, M.S., Rudolph, L., Sleator, D.D.: Competitive snoopy caching. Algorithmica 3, 79–119 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Communications of the ACM 28(2), 202–208 (1985)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Zarrabi-Zadeh, H., Chan, T.M.: An improved algorithm for online unit clustering. Algorithmica 54(4), 490–500 (2009)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Martin R. Ehmsen
    • 1
  • Kim S. Larsen
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark

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