Skip to main content
SpringerLink
Log in

Geometric complexity of some location problems

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Given a set ofn demand points with weightW i ,i = 1,2,...,n, in the plane, we consider several geometric facility location problems. Specifically we study the complexity of the Euclidean 1-line center problem, discrete 1-point center problem and a competitive location problem. The Euclidean 1-line center problem is to locate a line which minimizes the maximum weighted distance from the line (or the center) to the demand points. The discrete 1-point center problem is to locate one of the demand points so as to minimize the maximum unweighted distance from the point to other demand points. The competitive location problem studied is to locate a new facility point to compete against an existing facility so that a certain objective function is optimized. An Ω(n logn) lower bound is proved for these problems under appropriate models of computation. Efficient algorithms for these problems that achieve the lower bound and other related problems are also given.

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. A. V. Aho, J. E. Hopcroft, and J. D. Ullman.The Design and Analysis of Computer Algorithms. Addison-Wesley; Reading, MA, 1974.

    MATH  Google Scholar 

  2. M. Ben-Or. Lower bounds for algebraic computation trees,Proc. 15th Annual Symp. on Theory of Computing, 1983, pp. 80–86.

  3. P. Dobkin and R. J. Lipton. On the complexity of computations under varying sets of primitives,J. Comput. System Sci. 18 (1979), 86–91.

    Article  MATH  MathSciNet  Google Scholar 

  4. Z. Drezner. Competitive location strategies for two facilities. In: tRegional Science and Urban Economics, Vol. 12, Elsevier: North-Holland, 1982, pp. 485–493.

    Google Scholar 

  5. H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision.SIAM J. Comput. (to appear).

  6. J. Elzinga and D. W. Hearn. Geometrical solutions for some minimax location problems.Transportation Sci.,6 (1972), 379–394.

    Article  MathSciNet  Google Scholar 

  7. M. L. Fredman and B. Weide. On the complexity of computing the measure of U[a i,b i],Commun. ACM,21, 7 (July 1978), 540–544.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. L. Graham. An efficient algorithm for determining the convex hull of a finite set.Inform. Process. Lett,1 (1972), 132–133.

    Article  MATH  Google Scholar 

  9. S. L. Hakimi. “On locating new facilities in a competitive environment,”ISOLDE II, June 1981, Skodsberg, Denmark.

    Google Scholar 

  10. M. E. Houle and G. T. Toussaint. Computing the width of a set,ACM Symp. on Computational Geometry, Baltimore, Maryland, 1985, pp. 1–7.

  11. D. G. Kirkpatrick. Optimal search in planar subdivisions.SIAM J. Comput. 12, 1 (Feb. 1983), 28–35.

    Article  MATH  MathSciNet  Google Scholar 

  12. D. T. Lee. Farthest neighbor Voronoi diagrams and applications. Tech. Rep. #80-11-FC-04, Dept. EE/CS, Northwestern University, Nov. 1980.

  13. D. T. Lee and C. C. Yang. Location of multiple points in a planar subdivision,Inform. Process. Lett. 9, 4 (Nov. 1979), 190–192.

    Article  MathSciNet  Google Scholar 

  14. U. Manber and M. Tompa. The complexity of problems on probabilistic, nondeterministic and alternating decision trees.J. ACM 32, 3 (1985), 720–732.

    Article  MATH  MathSciNet  Google Scholar 

  15. N. Megiddo. Linear-time algorithms for linear programming in R3 and related problems,SIAM J. Comput.,12, 4 (1983), 759–776.

    Article  MATH  MathSciNet  Google Scholar 

  16. N. Megiddo and A. Tamir. On the complexity of locating linear facilities in the plane,Oper. Res. Lett. 1 (1982), 194–197.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. G. Morris and J. P. Norback. Linear facility location-Solving extensions of the basic problem.European J. Oper. Res.,12, (1983), 90–94.

    Article  MATH  MathSciNet  Google Scholar 

  18. F. P. Preparata. A note on locating a set of points in a planar subdivision.SIAM J. Comput.,8, 4 (1979), 542–545.

    Article  MATH  MathSciNet  Google Scholar 

  19. F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions.Commun. ACM,20, 2 (1977), 87–93.

    Article  MATH  MathSciNet  Google Scholar 

  20. E. M. Reingold. On the optimality of some set algorithms,J. ACM,19, 4 (1972), 649–659.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. I. Shamos. Problems in computational geometry, Dept. Comput. Sci., Yale University: New Haven, CO, May 1975.

    Google Scholar 

  22. M. I. Shamos and D. Hoey. Closest-point problems.Proc. IEEE 16th Symp. Foundations of Comput. Sci., 1975, pp. 151–162.

  23. G. T. Toussaint. Solving geometric problems with the rotating calipers.Proc. of IEEE MELECON, Athens, Greece, 1983.

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, 60201, Evanston, IL, USA

    D. T. Lee & Y. F. Wu

Authors
  1. D. T. Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. F. Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Leonidas. J. Guibus.

Supported in part by the National Science Foundation under Grants ECS 83-40031 and DCR 84-20814.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lee, D.T., Wu, Y.F. Geometric complexity of some location problems. Algorithmica 1, 193–211 (1986). https://doi.org/10.1007/BF01840442

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01840442

Key words

Search

Navigation