Advertisement

Online Unit Covering in Euclidean Space

  • Adrian Dumitrescu
  • Anirban Ghosh
  • Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

Abstract

We revisit the online Unit Covering problem in higher dimensions: Given a set of n points in \(\mathbb {R}^d\), that arrive one by one, cover the points by balls of unit radius, so as to minimize the number of balls used. In this paper, we work in \(\mathbb {R}^d\) using Euclidean distance. The current best competitive ratio of an online algorithm, \(O(2^d d \log {d})\), is due to Charikar et al. (2004); their algorithm is deterministic.

  1. (I)

    We give an online deterministic algorithm with competitive ratio \(O(1.321^d)\), thereby improving on the earlier record by an exponential factor. In particular, the competitive ratios are 5 for the plane and 12 for 3-space (the previous ratios were 7 and 21, respectively). For \(d=3\), the ratio of our online algorithm matches the ratio of the current best offline algorithm for the same problem due to Biniaz et al. (2017), which is remarkable (and rather unusual).

     
  2. (II)

    We show that the competitive ratio of every deterministic online algorithm (with an adaptive deterministic adversary) for Unit Covering in \(\mathbb {R}^d\) under the \(L_{2}\) norm is at least \(d+1\) for every \(d \ge 1\). This greatly improves upon the previous best lower bound, \(\varOmega (\log {d} / \log {\log {\log {d}}})\), due to Charikar et al. (2004).

     
  3. (III)

    We obtain lower bounds of 4 and 5 for the competitive ratio of any deterministic algorithm for online Unit Covering in \(\mathbb {R}^2\) and respectively \(\mathbb {R}^3\); the previous best lower bounds were both 3.

     
  4. (IV)

    When the input points are taken from the square or hexagonal lattices in \(\mathbb {R}^2\), we give deterministic online algorithms for Unit Covering with an optimal competitive ratio of 3.

     

Keywords

Online algorithm Unit covering Unit clustering Competitive ratio Lower bound Newton number 

References

  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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Biniaz, A., Liu, P., Maheshwari, A., Smid, M.H.M.: Approximation algorithms for the unit disk cover problem in 2D and 3D. Comput. Geom. 60, 8–18 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  4. 4.
    Boyvalenkov, P., Dodunekov, S., Musin, O.R.: A survey on the kissing numbers. Serdica Math. J. 38, 507–522 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brass, P., Moser, W.O.J., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005).  https://doi.org/10.1007/0-387-29929-7CrossRefzbMATHGoogle Scholar
  6. 6.
    Buchbinder, N., Naor, J.: Online primal-dual algorithms for covering and packing. Math. Oper. Res. 34(2), 270–286 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chan, T.M., Zarrabi-Zadeh, H.: A randomized algorithm for online unit clustering. Theory Comput. Syst. 45(3), 486–496 (2009)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dumitrescu, A., Tóth, C.D.: Online unit clustering in higher dimensions. In: Solis-Oba, R., Fleischer, R. (eds.) WAOA 2017. LNCS, vol. 10787, pp. 238–252. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-89441-6_18CrossRefzbMATHGoogle Scholar
  10. 10.
    Edel, Y., Rains, E.M., Sloane, N.J.A.: On kissing numbers in dimensions 32 to 128. Electr. J. Comb. 5, R22 (1998)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ehmsen, M.R., Larsen, K.S.: Better bounds on online unit clustering. Theor. Comput. Sci. 500, 1–24 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Epstein, L., van Stee, R.: On the online unit clustering problem. ACM Trans. Algorithms 7(1), 7:1–7:18 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Feder, T., Greene, D.H.: Optimal algorithms for approximate clustering. In: Proceedings of 20th ACM Symposium on Theory of Computing (STOC), pp. 434–444 (1988)Google Scholar
  14. 14.
    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)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci. 38, 293–306 (1985)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hadwiger, H., Debrunner, H.: Combinatorial Geometry in the Plane. Holt, Rinehart and Winston, New York (1964). (English translation by Victor Klee)zbMATHGoogle Scholar
  17. 17.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32(1), 130–136 (1985)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jenssen, M., Joos, F., Perkins, W.: On kissing numbers and spherical codes in high dimensions. Adv. Math. 335, 307–321 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kabatiansky, G.A., Levenshtein, V.I.: On bounds for packings on the sphere and in space. Probl. Inform. Transm. 14(1), 1–17 (1978)MathSciNetGoogle Scholar
  20. 20.
    Kawahara, J., Kobayashi, K.M.: An improved lower bound for one-dimensional online unit clustering. Theor. Comput. Sci. 600, 171–173 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Liao, C., Hu, S.: Polynomial time approximation schemes for minimum disk cover problems. J. Comb. Optim. 20(4), 399–412 (2010)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discrete Comput. Geom. 44(4), 883–895 (2010)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Rankin, R.A.: The closest packing of spherical caps in \(n\) dimensions. Proc. Glasgow Math. Assoc. 2(3), 139–144 (1955)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
  26. 26.
    Wyner, A.D.: Capabilities of bounded discrepancy decoding. Bell Syst. Tech. J. 44(6), 1061–1122 (1965)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wynn, E.: Covering a unit ball with balls half the radius (2012). https://mathoverflow.net/questions/98007/covering-a-unit-ball-with-balls-half-the-radius
  28. 28.
    Zarrabi-Zadeh, H., Chan, T.M.: An improved algorithm for online unit clustering. Algorithmica 54(4), 490–500 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Anirban Ghosh
    • 2
  • Csaba D. Tóth
    • 3
    • 4
  1. 1.University of Wisconsin–MilwaukeeMilwaukeeUSA
  2. 2.University of North FloridaJacksonvilleUSA
  3. 3.California State University NorthridgeLos AngelesUSA
  4. 4.Tufts UniversityMedfordUSA

Personalised recommendations