Mobile Networks and Applications

, Volume 11, Issue 2, pp 177–186

Competitive Algorithms for Maintaining a Mobile Center

  • Sergey Bereg
  • Binay Bhattacharya
  • David Kirkpatrick
  • Michael Segal
Article

Abstract

In this paper we investigate the problem of locating a mobile facility at (or near) the center of a set of clients that move independently, continuously, and with bounded velocity. It is shown that the Euclidean 1-center of the clients may move with arbitrarily high velocity relative to the maximum client velocity. This motivates the search for strategies for moving a facility so as to closely approximate the Euclidean 1-center while guaranteeing low (relative) velocity.

We present lower bounds and efficient competitive algorithms for the exact and approximate maintenance of the Euclidean 1-center for a set of moving points in the plane. These results serve to accurately quantify the intrinsic velocity approximation quality tradeoff associated with the maintenance of the mobile Euclidean 1-center.

Keywords

approximation algorithms facility location online strategies 

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References

  1. [1]
    P.K. Agarwal, J. Gao and L.J. Guibas, Kinetic medians and kd-trees, in: Proc. 10th Annual European Symposium on Algorithms, LNCS 2461 (2002) pp. 5–16.Google Scholar
  2. [2]
    P. Agarwal, L. Guibas, J. Hershberger and E. Veach, Maintaining the extent of a moving point set, Discrete and Computational Geometry 26(3) (2001) 353–374.MathSciNetMATHGoogle Scholar
  3. [3]
    P.K. Agarwal and S. Har-Peled, Maintaining the approximate extent measures of moving points, in: Proc. 12th ACM-SIAM Sympos. Discrete Algorithms (2001) pp. 148–157.Google Scholar
  4. [4]
    P. Agarwal and M. Sharir, Planar geometric location problems, Algorithmica 11 (1994) 185–195.CrossRefMathSciNetGoogle Scholar
  5. [5]
    S. Arora, P. Raghavan and S. Rao, Polynomial time approximation schemes for Euclidean k-medians and related problems, in: Proc. 31st ACM Symp. on Theory of Comput. (1998) pp. 106–113.Google Scholar
  6. [6]
    C. Bajaj, Geometric optimization and computational complexity, Ph.D. thesis. Tech. Report TR-84-629. Cornell University (1984).Google Scholar
  7. [7]
    J. Basch, Kinetic data structures, Ph.D. thesis, Stanford University, USA (1999).Google Scholar
  8. [8]
    J. Basch, L. Guibas and J. Hershberger, Data structures for mobile data, Journal of Algorithms 31(1) (1999) 1–28.CrossRefMathSciNetMATHGoogle Scholar
  9. [9]
    S. Bespamyatnikh, B. Bhattacharya, D. Kirkpatrick and M. Segal, Mobile facility location, in: Proc. of 4th International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, DIAL M (2000) pp. 46–53.Google Scholar
  10. [10]
    S. Bespamyatnikh, K. Kedem and M. Segal, Optimal facility location under various distance functions, in Workshop on Algorithms and Data Structures, Lecture Notes in Computer Science 1663 (Springer-Verlag, 1999) pp. 318–329.Google Scholar
  11. [11]
    S. Bespamyatnikh and M. Segal, Rectilinear static and dynamic center problems, Workshop on Algorithms and Data Structures Lecture Notes in Computer Science 1663 (Springer-Verlag, 1999) pp. 276–287.Google Scholar
  12. [12]
    J. Brimberg and A. Mehrez, Multi-facility location using a maximin criterion and rectangular distances, Location Science 2 (1994) 11–19.MATHGoogle Scholar
  13. [13]
    M. Charikar and S. Guha, Improved combinatorial algorithms for the facility location and k-median problems, in: Proc. 40th Symp. on Found. of Comput. Science (1999) pp. 378–388.Google Scholar
  14. [14]
    M. Charikar, S. Guha, E. Tardos and D. Shmoys, A constant-factor approximation algorithm for the k-median problem, in: Proc. 32th ACM Symp. on Theory of Comput. (1999) pp. 106–113.Google Scholar
  15. [15]
    Z. Drezner, The p-center problem: Heuristic and optimal algorithms, Journal of Operational Research Society 35 (1984) 741–748.MATHGoogle Scholar
  16. [16]
    Z. Drezner, On the rectangular p-center problem, Naval Res. Logist. Q. 34 (1987) 229–234.MATHMathSciNetGoogle Scholar
  17. [17]
    S. Durocher and D. Kirkpatrick, The Gaussian center of a set of points with applications to mobile facility location, submitted for publication. (See Proc. 15th Canadian Conference on Computational Geometry, 2003, for a preliminary version.)Google Scholar
  18. [18]
    M. Dyer and M. Frieze, A simple heuristic for the p-center problem, Oper. Res. Lett. 3 (1985) 285–288.CrossRefMathSciNetMATHGoogle Scholar
  19. [19]
    H. Elgindy and M. Keil, Efficient algorithms for the capacitated 1-median problem, ORSA J. Comput. 4 (1982) 418–424.MathSciNetGoogle Scholar
  20. [20]
    D. Eppstein, Faster construction of planar two-centers, in: Proc. 8th ACM-SIAM Symp. on Discrete Algorithms (1997) pp. 131–138.Google Scholar
  21. [21]
    J. Gao, L. Guibas and A. Nguen, Deformable spanners and applications, ACM Sympos. Comp. Geom. (2004) 190–199.Google Scholar
  22. [22]
    L. Guibas, Kinetic data structures: A state of the art report, in: Proc. 1998 Workshop Algorithmic Found. Robot. (1998) pp. 191–209.Google Scholar
  23. [23]
    J. Gao, L. J. Guibas, J. Hershberger, L. Zhang and A. Zhu, Discrete mobile centers, Discrete Comput. Geom. 30 (2003) 45–63.MathSciNetMATHGoogle Scholar
  24. [24]
    S.L. Hakimi, M. Labbe and E. Schmeichel, Locations on time-varying networks, Networks 34(4) (1999) 250–257.CrossRefMathSciNetMATHGoogle Scholar
  25. [25]
    S. Har-Peled, Clustering motion, Discrete Comput. Geom. 31(4) (2004) 545–565.MATHMathSciNetGoogle Scholar
  26. [26]
    D. Hochbaum and D. Shmoys, A best possible approximation algorithm for the k-center probem, Math. Oper. Res. 10 (1985) 180–184.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    D. Hochbaum and A. Pathria, Locating centers in a dynamically changing network and related problems, Location Science 6 (1998) 243–256.CrossRefGoogle Scholar
  28. [28]
    K. Jain and V. Vazirani, Primal-dual approximation algorithms for metric facility location and k-median problems, in: Proc. 40th Symp. on Found. of Comput. Science (1999) pp. 2–13.Google Scholar
  29. [29]
    J. Jaromczyk and M. Kowaluk, An efficient algorithm for the Euclidian two-center problem, in: Proc. 10th ACM Sympos. Comput. Geom. (1994) pp. 303–311.Google Scholar
  30. [30]
    H. Kuhn, A note on Fermat’s problem, Mathematical Programming 4 (1973) 98–107.CrossRefMATHMathSciNetGoogle Scholar
  31. [31]
    J. Lin and J. Vitter, Approximation algorithm for geometric median problems, Inform. Proc. Lett. 44 (1992) 245–249.CrossRefMathSciNetMATHGoogle Scholar
  32. [32]
    N. Megiddo, Linear time algorithms for linear programming in R 3 and related problems, SIAM J. Comput. 12 (1983) 759–776.CrossRefMATHMathSciNetGoogle Scholar
  33. [33]
    M. Manasse, L. McGeoch and D. Sleator, Competitive algorithms for server problems, Journal of Algorithms 11 (1990) 208–230.CrossRefMathSciNetMATHGoogle Scholar
  34. [34]
    N. Megiddo and A. Tamir, New results on the complexity of p-center problems, SIAM J. Comput. 12(4) (1983) 751–758.CrossRefMathSciNetMATHGoogle Scholar
  35. [35]
    D. Sleator and R. Tarjan, Amortized efficiency of list update and paging rules, Communications of the A.C.M. 28(2) (1985) 202–208.MathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Sergey Bereg
    • 1
  • Binay Bhattacharya
    • 2
  • David Kirkpatrick
    • 3
  • Michael Segal
    • 4
  1. 1.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA
  2. 2.School of Computing ScienceSimon Fraser UniversityBurnabyCanada
  3. 3.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada
  4. 4.Communication Systems Engineering DepartmentBen-Gurion University of the NegevBeer-ShevaIsrael

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