Discrete & Computational Geometry

, Volume 1, Issue 3, pp 265–276 | Cite as

Diameter partitioning

  • David Avis
Article

Abstract

We discuss the problem of partitioning a set of points into two subsets with certain conditions on the diameters of the subsets and on their cardinalities. For example, we give anO(n2 logn) algorithm to find the smallestt such that the set can be split into two equal cardinality subsets each of which has diameter at mostt. We also give an algorithm that takes two pairs of points (x, y) and (s, t) and decides whether the set can be partitioned into two subsets with the respective pairs of points as diameters.

Keywords

Line Segment Discrete Comput Geom Auxiliary Graph Bipartite Subgraph Brute Force Approach 

References

  1. 1.
    A. V. Aho, J. E. Hopcroft, and J. D. Ullman,The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA 1974.MATHGoogle Scholar
  2. 2.
    D. Avis, Non-partitionable point sets,Inform. Proc. Lett. 19 (1984), 125–129.MathSciNetCrossRefGoogle Scholar
  3. 3.
    D. Avis, Space partitioning and its application to generalized retrieval problems,Proceedings of the International Conference on Foundations of Data Organization, 154–162, Kyoto, May 1985.Google Scholar
  4. 4.
    D. Avis and M. Doskas, Algorithms for high dimensional stabbing problems, (to appear).Google Scholar
  5. 5.
    D. P. Dobkin and H. Edelsbrunner, Space searching for intersecting objects,Proceedings of the 25th FOCS, 387–391, October, 1984.Google Scholar
  6. 6.
    H. Edelsbrunner and E. Welzl, On the number of line separations of a finite set in the plane,J. Combin. Theory Ser. A 38 (1985), 15–29.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    H. Edelsbrunner, H. A. Maurer, F. P. Preparata, A. L. Rosenberg, E. Welzl, and D. Wood, Stabbing line segments,BIT 22 (1982), 274–281.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    P. Erdös, L. Lovász, A. Simmons, and E. G. Straus, Dissection graphs of planar point sets, inA Survey of Combinatorial Theory, (J. N. Srivastavaet al. ed.), 139–149, Amsterdam, North-Holland, 1973.Google Scholar
  9. 9.
    R. J. Fowler, M. S. Paterson, and S. L. Tanimoto, Optimal packing and covering in the plane are NP-complete,Inform. Process. Lett. 12 (1981), 133–137.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    M. R. Garey and D. S. Johnson,Computers and Intractability, Freeman, New York, 1979.MATHGoogle Scholar
  11. 11.
    D. S. Johnson, NP-completeness column,J. Algorithms 3 (1982), 182–195.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    L. Lovász, On the number of halving lines,Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 14 (1971), 107–108.Google Scholar
  13. 13.
    S. Masuyama, T. Ibaraki, and T. Hasegawa, The computational complexity of them-centers problem in the plane,Trans. IECE Japan, (1981), 57–64.Google Scholar
  14. 14.
    M. I. Shamos,Computational Geometry, University Microfilms, Ann Arbor, MI, 1978.Google Scholar
  15. 15.
    K. Supowit, Topics in Computational Geometry, Report No. UIUCDCS-R-81-1062, Department of Computer Science, University of Illinois, Urbana, IL, 1981.Google Scholar
  16. 16.
    D. E. Willard, Polygon retrieval,SIAM J. Comput. 11 (1982), 149–165.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    F. Yao, 3-Space partition and its applications,Proceedings of the 15th STOC, Boston, MA, 1983.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • David Avis
    • 1
  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada

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