Skip to main content
SpringerLink
Log in

Kinetic Facility Location

  • Published:
Algorithmica Aims and scope Submit manuscript

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. In our scenario, 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 in total, where \(R:=\frac{\max_{p_{i}\in\mathcal{P}}{f_{i}}\cdot\max_{p_{i}\in\mathcal{P}}{d_{i}}}{\min_{p_{i}\in\mathcal {P}}{f_{i}}\cdot\min_{p_{i}\in\mathcal{P}}{d_{i}}}\) , ℘={p 1,p 2,…,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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abam, M., de Berg, M., Poon, S., Speckmann, B.: Kinetic collision detection for convex fat objects. In: Proceedings of the 14th Annual European Symposium on Algorithms (ESA), pp. 4–15 (2006)

  2. Abam, M., de Berg, M., Speckmann, B.: Kinetic kd-trees and longest-side kd-trees. In: Proceedings of the 23rd Annual Symposium on Computational Geometry, pp. 364–372 (2007)

  3. Agarwal, P., Erickson, J., Guibas, L.: Kinetic binary space partitions for intersecting segments and disjoint triangles. In: Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 107–116 (1998)

  4. Agarwal, P., Guibas, L., Murali, T., Vitter, J.: Cylindrical static and kinetic binary space partitions. Comput. Geom. Theory Appl. 16(2), 103–127 (2000)

    MATH  MathSciNet  Google Scholar 

  5. Agarwal, P., Basch, J., de Berg, M., Guibas, L., Hershberger, J.: Lower bounds for kinetic planar subdivisions. Discrete Comput. Geom. 24(4), 721–733 (2000)

    MATH  MathSciNet  Google Scholar 

  6. Agarwal, P., Guibas, L., Hershberger, J., Veach, E.: Maintaining the extent of a moving point set. Discrete Comput. Geom. 26(3), 353–374 (2001)

    MATH  MathSciNet  Google Scholar 

  7. Agarwal, P., Gao, J., Guibas, L.: Kinetic medians and kd-trees. In: Proceedings of the 10th Annual European Symposium on Algorithms (ESA), pp. 5–16 (2002)

  8. Agarwal, P., Arge, L., Erickson, J.: Indexing moving points. J. Comput. Syst. Sci. 66(1), 207–243 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Agarwal, P., Har-Peled, S., Varadarajan, K.: Approximating extent measures of points. J. ACM 51(4), 606–635 (2004)

    MathSciNet  Google Scholar 

  10. Bădoiu, M., Czumaj, A., Indyk, P., Sohler, C.: Facility location in sublinear time. In: Proceedings of the 32nd Annual International Colloquium on Automata, Languages and Programming (ICALP), vol. 3580, pp. 866–877 (2005)

  11. Basch, J., Guibas, L., Zhang, L.: Proximity problems on moving points. In: Proceedings of the 13th Annual Symposium on Computational Geometry, pp. 344–351 (1997)

  12. Basch, J., Guibas, L., Hershberger, J.: Data structures for mobile data. J. Algorithms 31(1), 1–28 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Basch, J., Erickson, J., Guibas, L., Hershberger, J., Zhang, L.: Kinetic collision detection between two simple polygons. Comput. Geom. Theory Appl. 27(3), 211–235 (2004)

    MATH  MathSciNet  Google Scholar 

  14. Bespamyatnikh, S., Bhattacharya, B., Kirkpatrick, D., Segal, M.: Mobile facility location. In: Proceedings of the 4th International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, pp. 46–53 (2000)

  15. Charikar, M., Guha, S.: Improved combinatorial algorithms for facility location problems. SIAM J. Comput. 34(4), 803–824 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Czumaj, A., Frahling, G., Sohler, C.: Efficient kinetic data structures for MaxCut. In: Proceedings of the 19th Canadian Conference on Computional Geometry (CCCG), pp. 157–160 (2007)

  17. Degener, B., Gehweiler, J., Lammersen, C.: The kinetic facility location problem. In: Proceedings of the 11th Scandinavian Workshop on Algorithm Theory (SWAT) (2008)

  18. Gao, J., Guibas, L., Hershberger, J., Zhang, L., Zhu, A.: Discrete mobile centers. Discrete Comput. Geom. 30(1), 45–63 (2003)

    MATH  MathSciNet  Google Scholar 

  19. Gao, J., Guibas, L., Nguyen, A.: Deformable spanners and applications. Comput. Geom. 35(1–2), 2–19 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gehweiler, J., Lammersen, C., Sohler, C.: A distributed O(1)-approximation algorithm for the uniform facility location problem. In: Proceedings of the 18th ACM Symposium on Parallelism in Algorithms and Architectures, pp. 237–243 (2006)

  21. Guha, S., Khuller, S.: Greedy strikes back: improved facility location algorithms. J. Algorithms 31(1), 228–248 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Guibas, L.: Kinetic data structures: a state of the art report. In: Proceedings of the 3rd Workshop on Algorithmic Foundations of Robotics, pp. 191–209 (1998)

  23. Guibas, L., Hershberger, J., Suri, S., Zhang, L.: Kinetic connectivity for unit disks. Discrete Comput. Geom. 25(4), 591–610 (2001)

    MATH  MathSciNet  Google Scholar 

  24. Har-Peled, S.: Clustering motion. Discrete Comput. Geom. 31(4), 545–565 (2004)

    MATH  MathSciNet  Google Scholar 

  25. Hershberger, J.: Kinetic collision detection with fast flight plan changes. Inf. Process. Lett. 92(6), 287–291 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hershberger, J.: Smooth kinetic maintenance of clusters. Comput. Geom. Theory Appl. 31(1–2), 3–30 (2005)

    MATH  MathSciNet  Google Scholar 

  27. Hershberger, J., Suri, S.: Simplified kinetic connectivity for rectangles and hypercubes. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 158–167 (2001)

  28. Indyk, P.: Algorithms for dynamic geometric problems over data streams. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pp. 373–380 (2004)

  29. Jain, K., Vazirani, V.: Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. J. ACM 48(2), 274–296 (2001)

    MATH  MathSciNet  Google Scholar 

  30. Jain, K., Mahdian, M., Saberi, A.: A new greedy approach for facility location problems. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), pp. 731–740 (2002)

  31. Kirkpatrick, D., Snoeyink, J., Speckmann, B.: Kinetic collision detection for two simple polygons. Comput. Geom. Theory Appl. 27(3), 211–235 (2004)

    Google Scholar 

  32. Kolliopoulos, S., Rao, S.: A nearly linear-time approximation scheme for the Euclidean k-median problem. SIAM J. Comput. 37(3), 757–782 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  33. Lammersen, C., Sohler, C.: Facility location in dynamic geometric data streams. In: Proceedings of the 16th Annual European Symposium on Algorithms (ESA), pp. 660–671 (2008)

  34. Mahdian, M., Ye, Y., Zhang, J.: Improved approximation algorithms for metric facility location problems. In: Proceedings of the 5th International Workshop on Approximation Algorithms of Combinatorial Optimization Problems (APPROX), pp. 229–242 (2002)

  35. Mettu, R.R., Plaxton, C.G.: The online median problem. SIAM J. Comput. 32(3), 816–832 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  36. Moscibroda, T., Wattenhofer, R.: Facility location: distributed approximation. In: Proceedings of the 24th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, pp. 108–117 (2005)

  37. Shmoys, D.B., Tardos, E., Aardal, K.: Approximation algorithms for facility location problems. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC), pp. 265–274 (1997)

Download references

Author information

Authors and Affiliations

  1. International Graduate School Dynamic Intelligent Systems, Paderborn University, 33095, Paderborn, Germany

    Bastian Degener

  2. Heinz Nixdorf Institute, Computer Science Department, Paderborn University, 33095, Paderborn, Germany

    Bastian Degener & Joachim Gehweiler

  3. Computer Science Department I, University of Bonn, 53117, Bonn, Germany

    Christiane Lammersen

Authors
  1. Bastian Degener
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joachim Gehweiler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christiane Lammersen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christiane Lammersen.

Additional information

Partially supported by the EU within FP7-ICT-2007-1 under contract no. 215270 (FRONTS), DFG-project “Smart Teams” within the SPP 1183 “Organic Computing”, and DFG grant Me 872/8-3.

A preliminary version of this paper appeared in the Proceedings of the 11th Scandinavian Workshop on Algorithm Theory (SWAT), LNCS, vol. 5124, pp. 378–389, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Degener, B., Gehweiler, J. & Lammersen, C. Kinetic Facility Location. Algorithmica 57, 562–584 (2010). https://doi.org/10.1007/s00453-008-9250-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-008-9250-7

Keywords

Search

Navigation