, Volume 54, Issue 4, pp 490–500 | Cite as

An Improved Algorithm for Online Unit Clustering

  • Hamid Zarrabi-Zadeh
  • Timothy M. Chan


We revisit the online unit clustering problem in one dimension which we recently introduced at WAOA’06: given a sequence of n points on the line, the objective is to partition the points into a minimum number of subsets, each enclosable by a unit interval. We present a new randomized online algorithm that achieves expected competitive ratio 11/6 against oblivious adversaries, improving the previous ratio of 15/8. This immediately leads to improved upper bounds for the problem in two and higher dimensions as well.


Online algorithms Randomized algorithms Unit clustering 


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  1. 1.
    Chan, T.M., Zarrabi-Zadeh, H.: A randomized algorithm for online unit clustering. In: Proceedings of the 4th Workshop on Approximation and Online Algorithms. Lecture Notes in Computer Science, vol. 4368, pp. 121–131. Springer, Berlin (2006). To appear in Theory of Computing Systems CrossRefGoogle Scholar
  2. 2.
    Charikar, M., Chekuri, C., Feder, T., Motwani, R.: Incremental clustering and dynamic information retrieval. SIAM J. Comput. 33(6), 1417–1440 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001) zbMATHGoogle Scholar
  4. 4.
    Epstein, L., van Stee, R.: On the online unit clustering problem. In: Proceedings of the 5th Workshop on Approximation and Online Algorithms. Lecture Notes in Computer Science, vol. 4927, pp. 193–206. Springer, Berlin (2007) CrossRefGoogle Scholar
  5. 5.
    Fotakis, D.: Incremental algorithms for facility location and k-median. In: Proceedings of the 12th Annual European Symposium on Algorithms. Lecture Notes in Computer Science, vol. 3221, pp. 347–358. Springer, Berlin (2004) Google Scholar
  6. 6.
    Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Inf. Process. Lett. 12(3), 133–137 (1981) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gonzalez, T.: Covering a set of points in multidimensional space. Inf. Process. Lett. 40, 181–188 (1991) zbMATHCrossRefGoogle Scholar
  8. 8.
    Gyárfás, A., Lehel, J.: On-line and First-Fit colorings of graphs. J. Graph Theory 12, 217–227 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kierstead, H.A., Qin, J.: Coloring interval graphs with First-Fit. SIAM J. Discrete Math. 8, 47–57 (1995) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Meyerson, A.: Online facility location. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, pp. 426–433 (2001) Google Scholar
  12. 12.
    Nielsen, F.: Fast stabbing of boxes in high dimensions. Theor. Comput. Sci. 246, 53–72 (2000) zbMATHCrossRefGoogle Scholar
  13. 13.
    Tanimoto, S.L., Fowler, R.J.: Covering image subsets with patches. In: Proceedings of the 5th International Conference on Pattern Recognition, pp. 835–839 (1980) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada

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