Optimization and Engineering

, Volume 14, Issue 4, pp 575–593 | Cite as

An online 2-dimensional clustering problem with variable sized clusters



In the online clustering problems, the classification of points into sets (called clusters) is done in an online fashion. Points arrive one by one at arbitrary locations, and we have to assign them to clusters at the time of arrival without any information about the further points. A point can be assigned to an existing cluster, or a new cluster can be opened for it. We study two-dimensional variants in the l norm, thus clusters are actually squares. The cost of a cluster is the sum of a fixed setup cost and the area of the square. The goal is to minimize the sum of the costs of the clusters used by the algorithm.

We study two variants, both maintaining the properties that a point which was assigned to a given cluster must remain assigned to this cluster, and clusters cannot be merged. In the strict variant, the size and the exact location of the cluster must be fixed when it is initialized. In the flexible variant, the algorithm can shift the cluster or expand it, as long as it contains all points assigned to it.

We present a 7-competitive algorithm in the strict model and an approximately 5.22-competitive algorithm for the flexible variant. We also give lower bounds on the possible competitive ratio, this bound is 2.768 in the strict model and 1.743 in the flexible model.


Online algorithms Competitive analysis Clustering problems 



The authors would like to thank a lot the anonymous referees for their very useful comments which helped to improve the results and their presentation. Csanád Imreh completed some of this work during his Humboldt scholarship at the Humboldt University at Berlin and he is grateful to Prof. Suzanne Albers for the kind hospitality.


  1. Borodin A, El-Yaniv R (1998) Online computation and competitive analysis. Cambridge University Press, Cambridge MATHGoogle Scholar
  2. Chan TM, Zarrabi-Zadeh H (2009) A randomized algorithm for online unit clustering. Theory Comput Syst 45(3):486–496 MathSciNetCrossRefMATHGoogle Scholar
  3. Charikar M, Chekuri C, Feder T, Motwani R (2004) Incremental clustering and dynamic information retrieval. SIAM J Comput 33(6):1417–1440 MathSciNetCrossRefMATHGoogle Scholar
  4. Csirik J, Epstein L, Imreh Cs, Levin A (2013) Online clustering with variable sized clusters. Algorithmica 65(2):251–274 MathSciNetCrossRefMATHGoogle Scholar
  5. Divéki G, Imreh Cs (2011) Online facility location with facility movements. Cent Eur J Oper Res 19(2):191–200 MathSciNetCrossRefMATHGoogle Scholar
  6. Divéki G (2013) Online clustering on the line with square cost variable sized clusters. Acta Cybern 21(1):75–88 MATHGoogle Scholar
  7. Ehmsen MR, Larsen KS (2013) Better bounds on online unit clustering. Theor Comput Sci 500:1–24 MathSciNetCrossRefGoogle Scholar
  8. Epstein L, Levin A, van Stee R (2008) Online unit clustering: variations on a theme. Theor Comput Sci 407(1–3):85–96 CrossRefMATHGoogle Scholar
  9. Epstein L, van Stee R (2010) On the online unit clustering problem. ACM Trans Algorithms 7(1):7 MathSciNetGoogle Scholar
  10. Fiat A, Woeginger GJ (eds) (1998) Online algorithms: the state of the art. LNCS, vol 1442. Springer, Berlin MATHGoogle Scholar
  11. Fotakis D (2006a) Incremental algorithms for facility location and k-median. Theor Comput Sci 361:275–313 MathSciNetCrossRefMATHGoogle Scholar
  12. Fotakis D (2006b) A primal-dual algorithm for online non-uniform facility location. J Discrete Algorithms 5:141–148 MathSciNetCrossRefGoogle Scholar
  13. Fotakis D (2011) Memoryless facility location in one pass. ACM Trans Algorithms 7(4):49 MathSciNetGoogle Scholar
  14. Fotakis D (2008) On the competitive ratio for online facility location. Algorithmica 50(1):1–57 MathSciNetCrossRefMATHGoogle Scholar
  15. Fotakis D, Koutris P (2013) Online sum-radii clustering. Theor Comput Sci. doi: 10.1016/j.tcs.2013.03.010 MATHGoogle Scholar
  16. Imreh Cs (2007) Competitive analysis. In: Iványi A (ed) Algorithms of informatics, mondAt, Budapest, vol 1, pp 395–428 Google Scholar
  17. Meyerson A (2001) Online facility location. In: Proceedings of the 42nd annual symposium on foundations of computer science. IEEE Comput. Soc., Los Alamitos, pp 426–431 Google Scholar
  18. Zarrabi-Zadeh H, Chan TM (2009) An improved algorithm for online unit clustering. Algorithmica 54(4):490–500 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Subotica Tech—College of Applied SciencesSuboticaSerbia
  2. 2.Department of InformaticsUniversity of SzegedSzegedHungary

Personalised recommendations