A Linear Time Algorithm for the r-Gathering Problem on the Line (Extended Abstract)

  • Anik SarkerEmail author
  • Wing-kin SungEmail author
  • M. Sohel RahmanEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)


In this paper, we revisit the r-gathering problem. Given sets C and F of points on the plane and distance d(cf) for each \(c \in C\) and \(f\in F\), an r-gathering of C to F is an assignment A of C to open facilities \(F' \subseteq F\) such that r or more members of C are assigned to each open facility. The cost of an r-gathering is \(\max _{c \in C}{d(c, A(c))}\). The r-gathering problem computes the r-gathering minimizing the cost. In this paper we study the r-gathering problem when C and F are on a line and present a \(O(|C| + |F|)\)-time algorithm to solve the problem. Our solution is optimal since any algorithm needs to read C and F at least once.



First and second authors are partially supported through a grant from Pubali Bank Ltd.


  1. 1.
    Aggarwal, G., et al.: Achieving anonymity via clustering. ACM Trans. Algorithms 6(3), 49:1–49:19 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Akagi, T., Nakano, S.: On r-gatherings on the line. In: Wang, J., Yap, C. (eds.) FAW 2015. LNCS, vol. 9130, pp. 25–32. Springer, Cham (2015). Scholar
  3. 3.
    Armon, A.: On min-max r-gatherings. Theor. Comput. Sci. 412(7), 573–582 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Drezner, Z.: Facility Location: A Survey of Applications and Methods. Springer, New York (1995)CrossRefGoogle Scholar
  5. 5.
    Drezner, Z., Hamacher, H.W.: Facility Location: Applications and Theory. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  6. 6.
    Gabow, H.N., Tarjan, R.E.: Algorithms for two bottleneck optimization problems. J. Algorithms 9(3), 411–417 (1988)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Nakano, S.: A simple algorithm for r-gatherings on the line. In: Rahman, M.S., Sung, W.-K., Uehara, R. (eds.) WALCOM 2018. LNCS, vol. 10755, pp. 1–7. Springer, Cham (2018). Scholar
  8. 8.
    Han, Y., Nakano, S.: On r-gatherings on the line. In: Proceedings of FCS 2016, pp. 99–104 (2016)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of CSEBUETDhakaBangladesh
  2. 2.School of ComputingNational University of SingaporeSingaporeSingapore
  3. 3.Genome Institute of SingaporeSingaporeSingapore

Personalised recommendations