Online Unit Clustering in Higher Dimensions

  • Adrian Dumitrescu
  • Csaba D. TóthEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10787)


We revisit the online Unit Clustering problem in higher dimensions: Given a set of n points in \(\mathbb {R}^d\), that arrive one by one, partition the points into clusters (subsets) of diameter at most one, so as to minimize the number of clusters used. In this paper, we work in \(\mathbb {R}^d\) using the \(L_\infty \) norm. We show that the competitive ratio of any algorithm (deterministic or randomized) for this problem must depend on the dimension d. This resolves an open problem raised by Epstein and van Stee (WAOA 2008). We also give a randomized online algorithm with competitive ratio \(O(d^2)\) for Unit Clustering of integer points (i.e., points in \(\mathbb {Z}^d\), \(d\in \mathbb {N}\), under \(L_{\infty }\) norm). We complement these results with some additional lower bounds for related problems in higher dimensions.


  1. 1.
    Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: The online set cover problem. SIAM J. Comput. 39(2), 361–370 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Azar, Y., Buchbinder, N., Hubert Chan, T.-H., Chen, S., Cohen, I.R., Gupta, A., Huang, Z., Kang, N., Nagarajan, V., Naor, J., Panigrahi, D.: Online algorithms for covering and packing problems with convex objectives. In: Proceedings of the 57th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 148–157. IEEE (2016)Google Scholar
  3. 3.
    Azar, Y., Bhaskar, U., Fleischer, L., Panigrahi, D.: Online mixed packing and covering. In: Proceedings of the 24th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 85–100. SIAM (2013)Google Scholar
  4. 4.
    Azar, Y., Cohen, I.R., Roytman, A.: Online lower bounds via duality. In: Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1038–1050. SIAM (2017)Google Scholar
  5. 5.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  6. 6.
    Buchbinder, N., Naor, J.: Online primal-dual algorithms for covering and packing. Math. Oper. Res. 34(2), 270–286 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chan, T.M., Zarrabi-Zadeh, H.: A randomized algorithm for online unit clustering. Theory Comput. Syst. 45(3), 486–496 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Charikar, M., Chekuri, C., Feder, T., Motwani, R.: Incremental clustering and dynamic information retrieval. SIAM J. Comput. 33(6), 1417–1440 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chrobak, M.: SIGACT news online algorithms column 13. SIGACT News Bull. 39(3), 96–121 (2008)CrossRefGoogle Scholar
  10. 10.
    Csirik, J., Epstein, L., Imreh, C., Levin, A.: Online clustering with variable sized clusters. Algorithmica 65(2), 251–274 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Divéki, G., Imreh, C.: An online 2-dimensional clustering problem with variable sized clusters. Optim. Eng. 14(4), 575–593 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Divéki, G., Imreh, C.: Grid based online algorithms for clustering problems. In. Proceedings of the 15th IEEE International Symposium on Computational Intelligence and Informatics (CINTI), p. 159. IEEE (2014)Google Scholar
  13. 13.
    Ehmsen, M.R., Larsen, K.S.: Better bounds on online unit clustering. Theor. Comput. Sci. 500, 1–24 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Epstein, L., Levin, A., van Stee, R.: Online unit clustering: variations on a theme. Theor. Comput. Sci. 407(1–3), 85–96 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Epstein, L., van Stee, R.: On the online unit clustering problem. ACM Trans. Algorithms 7(1), 1–18 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Feder, T., Greene, D.H.: Optimal algorithms for approximate clustering. In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC), pp. 434–444 (1988)Google Scholar
  17. 17.
    Fowler, R.J., Paterson, M., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Inf. Process. Lett. 12(3), 133–137 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci. 38, 293–306 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gupta, A., Nagarajan, V.: Approximating sparse covering integer programs online. Math. Oper. Res. 39(4), 998–1011 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32(1), 130–136 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kawahara, J., Kobayashi, K.M.: An improved lower bound for one-dimensional online unit clustering. Theor. Comput. Sci. 600, 171–173 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Megiddo, N., Supowit, K.J.: On the complexity of some common geometric location problems. SIAM J. Comput. 13(1), 182–196 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Vazirani, V.: Approximation Algorithms. Springer, New York (2001). Scholar
  24. 24.
    Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  25. 25.
    Zarrabi-Zadeh, H., Chan, T.M.: An improved algorithm for online unit clustering. Algorithmica 54(4), 490–500 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Wisconsin–MilwaukeeMilwaukeeUSA
  2. 2.California State University, NorthridgeLos AngelesUSA

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