Skip to main content
SpringerLink
Log in

A Randomized Algorithm for Online Unit Clustering

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

In this paper, we consider the online version of the following problem: partition a set of input points into subsets, each enclosable by a unit ball, so as to minimize the number of subsets used. In the one-dimensional case, we show that surprisingly the naïve upper bound of 2 on the competitive ratio can be beaten: we present a new randomized 15/8-competitive online algorithm. We also provide some lower bounds and an extension to higher dimensions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adamy, U., Erlebach, T.: Online coloring of intervals with bandwidth. In: Proc. 1st Workshop Approx. Online Algorithms. Lecture Notes in Computer Science, vol. 2909, pp. 1–12 (2003)

  2. Agarwal, P.K., van Kreveld, M., Suri, S.: Label placement by maximum independent set in rectangles. Comput. Geom. Theory Appl. 11, 209–218 (1998)

    MATH  Google Scholar 

  3. Chan, T.M.: Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms 46, 178–189 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Charikar, M., O’Callaghan, L., Panigrahy, R.: Better streaming algorithms for clustering problems. In: Proc. 35th ACM Sympos. Theory Comput., pp. 30–39 (2003)

  5. 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 

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

    MATH  Google Scholar 

  7. Epstein, L., Levy, M.: Online interval coloring and variants. In: Proc. 32nd International Colloquium on Automata, Languages, and Programming (ICALP). Lecture Notes in Computer Science, vol. 3580, pp. 602–613 (2005)

  8. Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. Comput. 34, 1302–1323 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Feder, T., Greene, D.H.: Optimal algorithms for approximate clustering. In: Proc. 20th ACM Sympos. Theory Comput., pp. 434–444 (1988)

  10. 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 

  11. Gonzalez, T.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci. 38, 293–306 (1985)

    Article  MATH  Google Scholar 

  12. Guha, S., Mishra, N., Motwani, R., O’Callaghan, L.: Clustering data streams. In: Proc. 41st IEEE Sympos. Found. Comput. Sci., pp. 359–366 (2000)

  13. Gyárfás, A., Lehel, J.: On-line and first-fit colorings of graphs. J. Graph Theory 12, 217–227 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Kierstead, H.A., Trotter, W.A.: An extremal problem in recursive combinatorics. Congr. Numer. 33, 143–153 (1981)

    MathSciNet  Google Scholar 

  17. Lipton, R.J., Tomkins, A.: Online interval scheduling. In: Proc. 5th Sympos. Discrete Algorithms, pp. 302–311 (1994)

  18. Marathe, M.V., Breu, H., Hunt III, H.B., Ravi, S.S., Rosenkrantz, D.J.: Simple heuristics for unit disk graphs. Networks 25, 59–68 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Megiddo, N., Supowit, K.J.: On the complexity of some common geometric location problems. SIAM J. Comput. 13(1), 182–196 (1984)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Timothy M. Chan & Hamid Zarrabi-Zadeh

Authors
  1. Timothy M. Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hamid Zarrabi-Zadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hamid Zarrabi-Zadeh.

Additional information

Work of the first author has been supported in part by NSERC.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chan, T.M., Zarrabi-Zadeh, H. A Randomized Algorithm for Online Unit Clustering. Theory Comput Syst 45, 486–496 (2009). https://doi.org/10.1007/s00224-007-9085-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-007-9085-7

Keywords

Search

Navigation