Advertisement

Kinetic Maintenance of Mobile k-Centres on Trees

  • Stephane Durocher
  • Christophe Paul
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4835)

Abstract

Let C denote a set of n mobile clients, each of which follows a continuous trajectory on a weighted tree T. We establish tight bounds on the maximum relative velocity of the 1-centre and 2-centre of C. When each client in C moves with linear motion along a path on T we derive a tight bound of Θ(n) on the complexity of the motion of the 1-centre and corresponding bounds of O(n 2 α(n)) and Ω(n 2) for a 2-centre, where α(n) denotes the inverse Ackermann function. We describe efficient algorithms for calculating the trajectories of the 1-centre and 2-centre of C: the 1-centre can be found in optimal time O(n logn) when the distance function between mobile clients is known or O(n 2) when the function must be calculated, and a 2-centre can be found in time O(n 2 logn). These algorithms lend themselves to implementation within the framework of kinetic data structures, resulting in structures that are compact, efficient, responsive, and local.

Keywords

Convex Hull Linear Motion Graph Distance Mobile Client Weighted Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal, P.K., Guibas, L.J., Hershberger, J., Veach, E.: Maintaining the extent of a moving point set. Disc. & Comp. Geom. 26, 353–374 (2001)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Agarwal, P.K., Har-Peled, S.: Maintaining approximate extent measures of moving points. In: SODA, pp. 148–157 (2001)Google Scholar
  3. 3.
    Basch, J.: Kinetic Data Structures. PhD thesis, Stanford U. (1999)Google Scholar
  4. 4.
    Basch, J., Guibas, L., Hershberger, J.: Data structures for mobile data. J. Alg. 31, 1–28 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Basch, J., Guibas, L.J., Silverstein, C., Zhang, L.: A practical evaluation of kinetic data structures. In: SOCG, pp. 388–390 (1997)Google Scholar
  6. 6.
    Bespamyatnikh, S., Bhattacharya, B., Kirkpatrick, D., Segal, M.: Mobile facility location. In: Int. ACM Work. on Disc. Alg. & Meth. for Mob. Comp. & Comm, pp. 46–53 (2000)Google Scholar
  7. 7.
    Bruno, G., Ghiani, G., Improta, G.: Dynamic positioning of idle automated guided vehicles. J. Int. Man. 11, 209–215 (2000)CrossRefGoogle Scholar
  8. 8.
    Dearing, P.M., Francis, R.L.: A minimax location problem on a network. Trans. Sci. 8, 333–343 (1974)MathSciNetGoogle Scholar
  9. 9.
    Durocher, S.: Geometric Facility Location under Continuous Motion. PhD thesis, U. of British Columbia (2006)Google Scholar
  10. 10.
    Durocher, S., Kirkpatrick, D.: The Steiner centre: Stability, eccentricity, and applications to mobile facility location. Int. J. Comp. Geom. & App. 16, 345–371 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Durocher, S., Kirkpatrick, D.: Bounded-velocity approximations of mobile Euclidean 2-centres. Int. J. Comp. Geom. & App. (to appear, 2007)Google Scholar
  12. 12.
    Dvir, D., Handler, G.Y.: The absolute center of a network. Networks 43, 109–118 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Frederickson, G.N.: Parametric search and locating supply centers in trees. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1991. LNCS, vol. 519, pp. 299–319. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  14. 14.
    Gao, J., Guibas, L.J., Hershberger, J., Zhang, L., Zhu, A.: Discrete mobile centers. Disc. & Comp. Geom. 30, 45–65 (2003)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Gao, J., Guibas, L.J., Nguyen, A.: Deformable spanners and applications. Comp. Geom.: Th. & App. 35, 2–19 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Guibas, L.J.: Kinetic data structures: a state of the art report. In: Work. Alg. Found. Rob, pp. 191–209. A. K. Peters, Ltd (1998)Google Scholar
  17. 17.
    Halpern, J., Maimon, O.: Algorithms for the m-center problems: A survey. Eur. J. Oper. Res. 10, 90–99 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Handler, G.Y.: Minimax location of a facility in an undirected tree graph. Trans. Sci. 7, 287–293 (1973)MathSciNetGoogle Scholar
  19. 19.
    Handler, G.Y.: Finding two-centers of a tree: The continuous case. Trans. Sci. 12, 93–106 (1978)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hansen, P., Labbé, M., Peeters, D., Thisse, J.F.: Single facility location on networks. An. Disc. Math. 31, 113–146 (1987)Google Scholar
  21. 21.
    Hershberger, J.: Finding the upper envelope of n line segments in O(n logn) time. Inf. Proc. Let. 33, 169–174 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hershberger, J.: Smooth kinetic maintenance of clusters. Comp. Geom.: Th. & App. 31, 3–30 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems. I: The p-centers. SIAM J. App. Math. 37, 513–538 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Tansel, B.C., Francis, R.L., Lowe, T.J.: Location on networks: A survey. part I: The p-center and p-median problems. Man. Sci. 29, 482–497 (1983)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Tansel, B.C., Francis, R.L., Lowe, T.J.: Location on networks: A survey. part II: Exploiting tree network structure. Man. Sci. 29, 498–511 (1983)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Stephane Durocher
    • 1
  • Christophe Paul
    • 2
  1. 1.School of Computer Science, McGill University, MontréalCanada
  2. 2.CNRS & LIRMM, MontpellierFrance

Personalised recommendations

Cite paper

Buy options