Skip to main content
Springer Nature Link
Log in

Online Unit Clustering and Unit Covering in Higher Dimensions

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We revisit the online Unit Clustering and Unit Covering problems in higher dimensions: Given a set of n points in a metric space, that arrive one by one, Unit Clustering asks to partition the points into the minimum number of clusters (subsets) of diameter at most one; whereas Unit Covering asks to cover all points by the minimum number of balls of unit radius. In this paper, we work in \({\mathbb {R}}^d\) using the \(L_\infty \) norm. We show that the competitive ratio of any online algorithm (deterministic or randomized) for Unit Clustering is \(\Omega (d)\). In particular, it depends on the dimension d, and this resolves an open problem raised by Epstein and van Stee (Theor Comput Sci 407(1–3):85–96, 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 the \(L_{\infty }\) norm). We show that the competitive ratio of any deterministic online algorithm for Unit Covering is at least \(2^d\). This ratio is the best possible, as it can be attained by a simple deterministic algorithm that assigns points to a predefined set of unit hypercubes. We complement these results with some additional lower bounds for related problems in 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

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Problems 2 and 3 are equivalent under any norm in which every set of unit diameter is contained in a ball of unit diameter. This holds under the \(L_1\) and \(L_\infty \) norms, but not under the \(L_p\) norm for any \(1<p<\infty \) in \({\mathbb {R}}^d\), \(d\ge 2\).

  2. For a convex body \(C \subset {\mathbb {R}}^d\), the Newton number (a.k.a. kissing number) of C is the maximum number of nonoverlapping congruent copies of C that can be arranged around C so that they each touch C [8, Sec. 2.4].

References

  1. Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: The online set cover problem. SIAM J. Comput. 39(2), 361–370 (2009)

    Article  MathSciNet  Google Scholar 

  2. Azar, Y., Niv Buchbinder, T.-H., Chan, H., Chen, S., Cohen, I.R., Gupta, A., Huang, Z., Kang, N., Nagarajan, V., Naor, J., Debmalya, P.: 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)

  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)

  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)

  5. Ben-David, S., Borodin, A., Karp, R.M., Tardos, G., Wigderson, A.: On the power of randomization in on-line algorithms. Algorithmica 11(1), 2–14 (1994)

    Article  MathSciNet  Google Scholar 

  6. Biniaz, A., Liu, P., Maheshwari, A., Smid, M.: Approximation algorithms for the unit disk cover problem in 2D and 3D. Comput. Geom. 60, 8–18 (2017)

    Article  MathSciNet  Google Scholar 

  7. Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  8. Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)

    MATH  Google Scholar 

  9. Buchbinder, N., Naor, J.: Online primal-dual algorithms for covering and packing. Math. Oper. Res. 34(2), 270–286 (2009)

    Article  MathSciNet  Google Scholar 

  10. Chan, T.M., Zarrabi-Zadeh, H.: A randomized algorithm for online unit clustering. Theor. Comput. Syst. 45(3), 486–496 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Chrobak, M.: SIGACT news online algorithms column 13. SIGACT News Bull. 39(3), 96–121 (2008)

    Article  Google Scholar 

  13. Csirik, J., Epstein, L., Imreh, C., Levin, A.: Online clustering with variable sized clusters. Algorithmica 65(2), 251–274 (2013)

    Article  MathSciNet  Google Scholar 

  14. Divéki, G., Imreh, C.: An online 2-dimensional clustering problem with variable sized clusters. Optimization and Engineering 14(4), 575–593 (2013)

    Article  MathSciNet  Google Scholar 

  15. 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), pp. 159. IEEE (2014)

  16. Dumitrescu, A.: Computational geometry column 68. SIGACT News 49(4), 46–54 (2018)

    Article  MathSciNet  Google Scholar 

  17. Dumitrescu, A., Ghosh, A., Tóth, C.D.: Online unit covering in Euclidean space. Theor. Comput. Sci. 809, 218–230 (2020)

    Article  MathSciNet  Google Scholar 

  18. Ehmsen, M.R., Larsen, K.S.: Better bounds on online unit clustering. Theor. Comput. Sci. 500, 1–24 (2013)

    Article  MathSciNet  Google Scholar 

  19. Epstein, L., Levin, A., van Stee, R.: Online unit clustering: variations on a theme. Theor. Comput. Sci. 407(1–3), 85–96 (2008)

    Article  MathSciNet  Google Scholar 

  20. Epstein, L., van Stee, R.: On the online unit clustering problem. ACM Trans. Algorithms 7(1), 1–18 (2010)

    Article  MathSciNet  Google Scholar 

  21. Feder, T., Greene, D.H.: Optimal algorithms for approximate clustering. In: Proceedings of the 20th ACM Symposium on Theory of Computing (STOC), pp. 434–444 (1988)

  22. Fowler, R.J., Paterson, M., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett. 12(3), 133–137 (1981)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Gupta, A., Nagarajan, V.: Approximating sparse covering integer programs online. Math. Oper. Res. 39(4), 998–1011 (2014)

    Article  MathSciNet  Google Scholar 

  25. Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32(1), 130–136 (1985)

    Article  MathSciNet  Google Scholar 

  26. Kawahara, J., Kobayashi, K.M.: An improved lower bound for one-dimensional online unit clustering. Theor. Comput. Sci. 600, 171–173 (2015)

    Article  MathSciNet  Google Scholar 

  27. Liaw, C., Liu, P., Reiss, R.: Approximation schemes for covering and packing in the streaming model. In: Proceedings of the 30th Canadian Conference on Computational Geometry (CCCG), Winnipeg, MB, pp. 172–179 (2018)

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

    Article  MathSciNet  Google Scholar 

  29. Vazirani, V.: Approximation Algorithms. Springer, New York (2001)

    MATH  Google Scholar 

  30. Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)

    Book  Google Scholar 

  31. Zarrabi-Zadeh, H., Chan, T.M.: An improved algorithm for online unit clustering. Algorithmica 54(4), 490–500 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Research on this paper was supported in part by National Science Foundation awards CCF-1422311 and CCF-1423615.

Author information

Authors and Affiliations

  1. Algoresearch L.L.C., Milwaukee, WI, USA

    Adrian Dumitrescu

  2. Department of Mathematics, California State University Northridge, Los Angeles, CA, USA

    Csaba D. Tóth

  3. Department of Computer Science, Tufts University, Medford, MA, USA

    Csaba D. Tóth

Authors
  1. Adrian Dumitrescu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Csaba D. Tóth
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Csaba D. Tóth.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A preliminary version of this paper appeared in the Proceedings of the 15th Workshop on Approximation and Online Algorithms (WAOA), LNCS 10787, Springer, Cham, 2017, pp. 238–252.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dumitrescu, A., Tóth, C.D. Online Unit Clustering and Unit Covering in Higher Dimensions. Algorithmica 84, 1213–1231 (2022). https://doi.org/10.1007/s00453-021-00916-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-021-00916-6

Keywords