Max-Min Dispersion on a Line

  • Tetsuya Araki
  • Shin-ichi NakanoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)


Given a set P of n locations on which facilities can be placed and an integer k, we want to place k facilities on some locations so that a designated objective function is maximized. The problem is called the k-dispersion problem.

In this paper we give a simple O(n) time algorithm to solve the max-min version of the k-dispersion problem if P is a set of points on a line. This is the first O(n) time algorithm to solve the max-min k-dispersion problem for the set of “unsorted” points on a line.

If P is a set of sorted points on a line, and the input is given as an array in which the coordinates of the points are stored in the sorted order, then by slightly modifying the algorithm above one can solve the dispersion problem in \(O(\log n)\) time. This is the first sublinear time algorithm to solve the max-min k-dispersion problem for the set of sorted points on a line.


Dispersion problem Algorithm 


  1. 1.
    Agarwal, P., Sharir, M.: Efficient algorithms for geometric optimization. Comput. Surv. 30, 412–458 (1998)CrossRefGoogle Scholar
  2. 2.
    Akagi, T., Nakano, S.: Dispersion on the line. IPSJ SIG Technical reports, 2016-AL-158-3 (2016)Google Scholar
  3. 3.
    Akagi, T., et al.: Exact algorithms for the max-min dispersion problem. In: Chen, J., Lu, P. (eds.) FAW 2018. LNCS, vol. 10823, pp. 263–272. Springer, Cham (2018). Scholar
  4. 4.
    Baur, C., Fekete, S.P.: Approximation of geometric dispersion problems. In: Jansen, K., Rolim, J. (eds.) APPROX 1998. LNCS, vol. 1444, pp. 63–75. Springer, Heidelberg (1998). Scholar
  5. 5.
    Birnbaum, B., Goldman, K.J.: An improved analysis for a greedy remote-clique algorithm using factor-revealing LPs. Algorithmica 50, 42–59 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cevallos, A., Eisenbrand, F., Zenklusen, R.: Max-sum diversity via convex programming. In: Proceedings of SoCG 2016, pp. 26:1–26:14 (2016)Google Scholar
  7. 7.
    Cevallos, A., Eisenbrand, F., Zenklusen, R.: Local search for max-sum diversification. In: Proceedings of SODA 2017, pp. 130–142 (2017)Google Scholar
  8. 8.
    Chandra, B., Halldorsson, M.M.: Approximation algorithms for dispersion problems. J. Algorithms 38, 438–465 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Drezner, Z. (ed.): Facility Location: A Survey of Applications and Methods. Springer, New York (1995)Google Scholar
  10. 10.
    Drezner, Z., Hamacher, H.W. (eds.): Facility Location: Applications and Theory. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  11. 11.
    Erkut, E.: The discrete \(p\)-dispersion problem. Eur. J. Oper. Res. 46, 48–60 (1990)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fekete, S.P., Meijer, H.: Maximum dispersion and geometric maximum weight cliques. Algorithmica 38, 501–511 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Frederickson, G.: Optimal algorithms for tree partitioning. In: Proceedings of SODA 1991, pp. 168–177 (1991)Google Scholar
  14. 14.
    Hassin, R., Rubinstein, S., Tamir, A.: Approximation algorithms for maximum dispersion. Oper. Res. Lett. 21, 133–137 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ravi, S.S., Rosenkrantz, D.J., Tayi, G.K.: Heuristic and special case algorithms for dispersion problems. Oper. Res. 42, 299–310 (1994)CrossRefGoogle Scholar
  16. 16.
    Sydow, M.: Approximation guarantees for max sum and max min facility dispersion with parameterised triangle inequality and applications in result diversification. Mathematica Applicanda 42, 241–257 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Tsai, K.-H., Wang, D.-W.: Optimal algorithms for circle partitioning. In: Jiang, T., Lee, D.T. (eds.) COCOON 1997. LNCS, vol. 1276, pp. 304–310. Springer, Heidelberg (1997). Scholar
  18. 18.
    Wang, D.W., Kuo, Y.-S.: A study on two geometric location problems. Inf. Process. Lett. 28, 281–286 (1988)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Tokyo Metropolitan UniversityHachiojiJapan
  2. 2.Gunma UniversityKiryuJapan

Personalised recommendations