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Rectilinear Static and Dynamic Discrete 2-center Problems

  • Sergei Bespamyatnikh
  • Michael Segal
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1663)

Abstract

In this paper we consider several variants of the discrete 2-center problem. The problem is: Given a set S of n demand points and a set C of m supply points, find two “minimal” axis-parallel squares (or rectangles) centered at the points of C that cover all the points of S. We present efficient solutions for both the static and dynamic versions of the problem (i.e. points of S are allowed to be inserted or deleted) and also consider the problem in fixed d; d ≥ 3 dimensional space. For the static version in the plane we give an optimal algorithm.

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References

  1. 1.
    Agarwal P., Sharir M., Welzl E.: The discrete 2-center problem, Proc. 13th ACM Symp. on Computational Geometry (1997) 147–155Google Scholar
  2. 2.
    Agarwal P., Erickson J.: Geometric range searching and its relatives. TR CS-1997-11, Duke University (1997)Google Scholar
  3. 3.
    Bajaj C.: Geometric optimization and computational complexity. PhD thesis, TR 84-629, Cornell University (1984)Google Scholar
  4. 4.
    Bespamyatnikh S., Segal M.: Covering the set of points by boxes. Proc. 9th Canadian Conference on Computational Geometry (1997) 33–38Google Scholar
  5. 5.
    Cormen T,, Leiserson C., Rivest R.: Introduction to algorithms. The MIT Press (1990)Google Scholar
  6. 6.
    Chazelle B.: A functional approach to data structures and its use in multidimensional searching. SIAM J. Computing 17 (1988) 427–462MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    de Berg M., van Kreveld M., Overmars M., Schwartzkopf O.: Computational Geometry, Algorithms and Applications. Springer Verlag (1997)Google Scholar
  8. 8.
    Drezner Z.: The p-center problem: heuristic and optimal algorithms. Journal Operational Research Society 35 (1984) 741–748zbMATHGoogle Scholar
  9. 9.
    Drezner Z.: On the rectangular p-center problem. Naval Research Logist. Quart. bf 34 (1987) 229–234MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Eppstein D.: Faster construction of planar two-centers. Proc. 8th ACM-SIAM Sympos. Discrete Algorithms (1997) 131–138Google Scholar
  11. 11.
    Frederickson G. N., Johnson D. B.: Generalized selection and ranking: sorted matrices. SIAM J. Computing 13 (1994) 14–30MathSciNetCrossRefGoogle Scholar
  12. 12.
    Glozman A., Kedem K., Shpitalnik G.: Efficient solution of the two-line center problem and other geometric problems via sorted matrices. Proc. 4th Workshop Algorithms Data Struct., Lecture Notes in Computer Science 955 (1995) 26–37Google Scholar
  13. 13.
    Hershberger J., Suri S.: Finding Tailored Partitions. J. Algorithms 12 (1991) 431–463MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jaromczyk J. W., Kowaluk M.: Orientation independent covering of point sets in R2 with pairs of rectangles or optimal squares. Proc. European Workshop on Computational Geometry. Lecture Notes in Computer Science 871 (1996) 71–78Google Scholar
  15. 15.
    Jaromczyk J. W., Kowaluk M.: An efficient algorithm for the euclidean two center problem. Proc. 10th ACM Symposium on Computational Geometry (1994) 303–311Google Scholar
  16. 16.
    Jaromczyk J. W., Kowaluk M.: The two-line center problem from a polar view: A new algorithm and data structure. Proc. 4th Workshop Algorithms Data Struct., Lecture Notes in Computer Science 955 (1995) 13–25Google Scholar
  17. 17.
    Katz M. J., Kedem K., Segal M.: Constrained square-center problems. In 6th Scan-dinavian Workshop on Algorithm Theory (1998) 95–106Google Scholar
  18. 18.
    Katz M. J., Sharir M.: An expander-based approach to geometric optimization. SIAM J. Computing 26 (1997) 1384–1408MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Nussbaum D.: Rectilinear p-Piercing Problems. Proc. of the Intern. Symposium on Symbolic and Algebraic Computation (1997) 316–323Google Scholar
  20. 20.
    Segal M.: On the piercing of axis-parallel rectangles and rings. Int. Journal of Comp. Geom. and Appls., to appearGoogle Scholar
  21. 21.
    Sharir M.: A near-linear algorithm for the planar 2-center problem. Proc. 12th ACM Symp. On Computational Geometry (1996) 106–112Google Scholar
  22. 22.
    Sharir M., Welzl E.: Rectilinear and polygonal p-piercing and p-center problems. Proc. 12th ACM Symp. on Comput. Geometry (1996) 122–132Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Sergei Bespamyatnikh
    • 1
  • Michael Segal
    • 2
  1. 1.University of British ColumbiaVancouverCanada
  2. 2.Ben-Gurion University of the NegevBeer-ShevaIsrael

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