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An Improved Algorithm for Online Unit Clustering

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Abstract

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.

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References

  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

    Chapter  Google Scholar 

  2. Charikar, M., Chekuri, C., Feder, T., Motwani, R.: Incremental clustering and dynamic information retrieval. SIAM J. Comput. 33(6), 1417–1440 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gonzalez, T.: Covering a set of points in multidimensional space. Inf. Process. Lett. 40, 181–188 (1991)

    Article  MATH  Google Scholar 

  8. Gyárfás, A., Lehel, J.: On-line and First-Fit colorings of graphs. J. Graph Theory 12, 217–227 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kierstead, H.A., Qin, J.: Coloring interval graphs with First-Fit. SIAM J. Discrete Math. 8, 47–57 (1995)

    Article  MathSciNet  Google Scholar 

  11. Meyerson, A.: Online facility location. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, pp. 426–433 (2001)

  12. Nielsen, F.: Fast stabbing of boxes in high dimensions. Theor. Comput. Sci. 246, 53–72 (2000)

    Article  MATH  Google Scholar 

  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)

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Correspondence to Hamid Zarrabi-Zadeh.

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A preliminary version of this paper appeared in the Proceedings of the 13th Annual International Computing and Combinatorics Conference (COCOON 2007), LNCS 4598, pp. 383–393.

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Zarrabi-Zadeh, H., Chan, T.M. An Improved Algorithm for Online Unit Clustering. Algorithmica 54, 490–500 (2009). https://doi.org/10.1007/s00453-008-9208-9

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  • DOI: https://doi.org/10.1007/s00453-008-9208-9

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