Advertisement

SWAT 2008: Algorithm Theory – SWAT 2008 pp 378-389

# The Kinetic Facility Location Problem

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

## Abstract

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.

## Keywords

facility location kinetic data structure approximation

## Preview

Unable to display preview. Download preview PDF.

## References

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)
2. 2.
Agarwal, P., Har-Peled, S., Varadarajan, K.: Approximating Extent Measures of Points. Journal of the ACM 51(4), 606–635 (2004)
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)
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)
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)
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)
19. 19.
Mettu, R.R., Plaxton, C.G.: The Online Median Problem. SIAM J. Comput. 32(3), 816–832 (2003)

## 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