r-Gatherings on a Star

  • Shareef AhmedEmail author
  • Shin-ichi Nakano
  • Md. Saidur Rahman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)


Let C be a set of n customers and F be a set of m facilities. An r-gather clustering of C is a partition of the points in clusters such that each cluster contains at least r points. The r-gather clustering problem asks to find an r-gather clustering which minimizes the maximum distance between any two points in a cluster. An r-gathering of C is an assignment of each customer \(c \in C\) to a facility \(f \in F\) such that each open facility has zero or at least r customers. The r-gathering problem asks to find an r-gathering that minimizes the maximum distance between a customer and its facility. In this work we consider the r-gather clustering and r-gathering problems when the customers and the facilities are lying on a “star”. We show that the r-gather clustering problem and the r-gathering problem with points on a star with d rays can be solved in \(O(rn+(r+1)^ddr)\) and \(O(n+r^2 m + d^2r^2 (d+\log m)+(r+1)^d2^d(r +d)d)\) time respectively.


r-Gathering Clustering Facility location problem 



We thank CodeCrafters International and Investortools, Inc. for supporting the first author under the grant “CodeCrafters-Investortools Research Grant”.


  1. 1.
    Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Comput. Surv. 30(4), 412–458 (1998)CrossRefGoogle Scholar
  2. 2.
    Aggarwal, G., et al.: Achieving anonymity via clustering. ACM Trans. Algorithms 6(3), 49:1–49:19 (2010)Google Scholar
  3. 3.
    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
  4. 4.
    Armon, A.: On min-max \(r\)-gatherings. Theor. Comput. Sci. 412(7), 573–582 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Frederickson, G.N., Johnson, D.B.: Generalized selection and ranking: sorted matrices. SIAM J. Comput. 13(1), 14–30 (1984)MathSciNetCrossRefGoogle 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.
    Guha, S., Meyerson, A., Munagala, K.: Hierarchical placement and network design problems. In: Proceedings 41st Annual Symposium on Foundations of Computer Science, pp. 603–612 (2000)Google Scholar
  8. 8.
    Han, Y., Nakano, S.: On \(r\)-gatherings on the line. In: Proceedings of the FCS 2016, pp. 99–104 (2016)Google Scholar
  9. 9.
    Karget, D.R., Minkoff, M.: Building steiner trees with incomplete global knowledge. In: Proceedings 41st Annual Symposium on Foundations of Computer Science, pp. 613–623 (2000)Google Scholar
  10. 10.
    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
  11. 11.
    Zeng, J., et al.: Mobile \(r\)-gather: distributed and geographic clustering for location anonymity. In: Proceedings of the 18th ACM International Symposium on Mobile Ad Hoc Networking and Computing, Mobihoc 2017, pp. 7:1–7:10. ACM (2017)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Shareef Ahmed
    • 1
    Email author
  • Shin-ichi Nakano
    • 2
  • Md. Saidur Rahman
    • 1
  1. 1.Graph Drawing and Information Visualization Laboratory, Department of Computer Science and Engineering, Bangladesh University of Engineering and TechnologyDhakaBangladesh
  2. 2.Gunma UniversityKiryuJapan

Personalised recommendations