The Kinetic Facility Location Problem

  • Bastian Degener
  • Joachim Gehweiler
  • Christiane Lammersen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5124)


We present a deterministic kinetic data structure for the facility location problem that maintains a subset of the moving points as facilities such that, at any point of time, the accumulated cost for the whole point set is at most a constant factor larger than the optimal cost. Each point can change its status between client and facility and moves continuously along a known trajectory in a d-dimensional Euclidean space, where d is a constant.

Our kinetic data structure requires \(\mathcal{O}(n (\log^{d}(n)+\log(nR)))\) space, where \(R:=\frac{\max_{p_i \in \mathcal{P}}{f_i} \,\cdot\, \max_{p_i\in \mathcal{P}}{d_i}}{\min_{p_i \in \mathcal{P}}{f_i} \,\cdot\, \min_{p_i\in \mathcal{P}}{d_i}}\), \(\mathcal{P} = \{ p_1, p_2, \ldots , p_n \}\) is the set of given points, and f i , d i are the maintenance cost and the demand of a point p i , respectively. In case that each trajectory can be described by a bounded degree polynomial, we process \(\mathcal{O}(n^2 \log^2(nR))\) events, each requiring \(\mathcal{O}(\log^{d+1}(n) \cdot \log(nR))\) time and \(\mathcal{O}(\log(nR))\) status changes. To the best of our knowledge, this is the first kinetic data structure for the facility location problem.


facility location kinetic data structure approximation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agarwal, P., Guibas, L., Hershberger, J., Veach, E.: Maintaining the Extent of a Moving Point Set. Discrete & Computational Geometry 26(3), 353–374 (2001)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Agarwal, P., Har-Peled, S., Varadarajan, K.: Approximating Extent Measures of Points. Journal of the ACM 51(4), 606–635 (2004)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bădoiu, M., Czumaj, A., Indyk, P., Sohler, C.: Facility Location in Sublinear Time. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 866–877. Springer, Heidelberg (2005)Google Scholar
  4. 4.
    Basch, J., Guibas, L., Hershberger, J.: Data Structures for Mobile Data. Journal of Algorithms 31(1), 1–28 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Basch, J., Guibas, L., Zhang, L.: Proximity Problems on Moving Points. In: Proc. 13th Symposium on Computational Geometry, pp. 344–351 (1997)Google Scholar
  6. 6.
    Bespamyatnikh, S., Bhattacharya, B., Kirkpatrick, D., Segal, M.: Mobile Facility Location. In: Proc 4th International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, pp. 46–53 (2000)Google Scholar
  7. 7.
    Czumaj, A., Frahling, G., Sohler, C.: Efficient Kinetic Data Structures for MaxCut. In: Proc. 19th Canadian Conference on Computational Geometry, pp. 157–160 (2007)Google Scholar
  8. 8.
    Degener, B., Gehweiler, J., Lammersen, C.: The Kinetic Facility Location Problem. Technical Report tr-ri-08-2880, University of Paderborn (2008)Google Scholar
  9. 9.
    Gao, J., Guibas, L., Nguyen, A.: Deformable Spanners and Applications. In: Proc. 20th Symposium on Computational Geometry, pp. 190–199 (2004)Google Scholar
  10. 10.
    Gao, J., Guibas, L., Hershberger, J., Zhang, L., Zhu, A.: Discrete Mobile Centers. Journal of Discrete and Computational Geometry 30(1), 45–63 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gehweiler, J., Lammersen, C., Sohler, C.: A Distributed O(1)-Approximation Algorithm for the Uniform Facility Location Problem. In: Proc. 18th ACM Symposium on Parallelism in Algorithms and Architectures, pp. 237–243 (2006)Google Scholar
  12. 12.
    Guibas, L.: Kinetic Data Structures: A State of the Art Report. In: Proc. 3rd Workshop on Algorithmic Foundations of Robotics, pp. 191–209 (1998)Google Scholar
  13. 13.
    Har-Peled, S.: Clustering Motion. In: Proc. 42nd IEEE Symposium on Foundations of Computer Science, pp. 84–93 (2001)Google Scholar
  14. 14.
    Hershberger, J.: Smooth Kinetic Maintenance of Clusters. In: Proc. Symposium on Computational Geometry, pp. 48–57 (2003)Google Scholar
  15. 15.
    Indyk, P.: Algorithms for Dynamic Geometric Problems over Data Streams. In: Proc. 36th ACM Symposium on Theory of Computing, pp. 373–380 (2004)Google Scholar
  16. 16.
    Jain, K., Mahdian, M., Saberi, A.: A New Greedy Approach for Facility Location Problems. In: Proc. 34th ACM Symposium on Theory of Computing, pp. 731–740 (2002)Google Scholar
  17. 17.
    Jain, K., Vazirani, V.: Approximation Algorithms for Metric Facility Location and k-Median Problems Using the Primal-Dual Schema and Lagrangian Relaxation. Journal of the ACM 48(2), 274–296 (2001)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Kolliopoulos, S., Rao, S.: A Nearly Linear-Time Approximation Scheme for the Euclidean k-Median Problem. SIAM Journal on Computing 37(3), 757–782 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Mettu, R.R., Plaxton, C.G.: The Online Median Problem. SIAM J. Comput. 32(3), 816–832 (2003)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Bastian Degener
    • 1
    • 2
  • Joachim Gehweiler
    • 2
  • Christiane Lammersen
    • 2
  1. 1.International Graduate School Dynamic Intelligent Systems 
  2. 2.Heinz Nixdorf Institute, Computer Science DepartmentPaderborn UniversityPaderbornGermany

Personalised recommendations