Skip to main content
Log in

The power of geometric duality

  • Part I Computer Science
  • Ordinary Papers
  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

This paper uses a new formulation of the notion of duality that allows the unified treatment of a number of geometric problems. In particular, we are able to apply our approach to solve two long-standing problems of computational geometry: one is to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane; the other is to produce an optimal algorithm for the half-plane range query problem. This problem is to preprocessn points in the plane, so that given a test half-plane, one can efficiently determine all points lying in the half-plane. We describe an optimalO(k + logn) time algorithm for answering such queries, wherek is the number of points to be reported. The algorithm requiresO(n) space andO(n logn) preprocessing time. Both of these results represent significant improvements over the best methods previously known. In addition, we give a number of new combinatorial results related to the computation of line arrangements.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. K. Q. Brown,Geometric transforms for fast geometric algorithms, PhD thesis, Carnegie-Mellon Univ., 1979.

  2. B. Chazelle,Optimal algorithms for computing depths and layers, Brown University, Technical Report, CS-83-13, March 1983.

  3. B. Chazelle,Filtering search: A new approach to query-answering, Proc. 24th Annual FOCS Symp., pp. 122–132, November 1983.

  4. B. Chazelle and D. Dobkin,Detection is easier than computation, Proc. 12th Annual SIGACT Symp., Los Angeles, California, pp. 146–153, May 1980.

  5. H. Edelsbrunner,Private Communication, June 1983.

  6. H. Edelsbrunner, D. G. Kirkpatrick and H. A. Maurer,Polygonal intersection searching, Inf. Proc. Lett., 14, pp. 74–79, 1982.

    Article  Google Scholar 

  7. H. Edelsbrunner, J. O'Rourke and R. Seidel,Constructing arrangements of lines and hyperplanes with applications, Proc. 24th Annual FOCS Symp., pp. 83–91, November 1983.

  8. H. Edelsbrunner and E. Welzl,Halfplanar range estimation, Tech. Univ. of Graz, Tech Rep. F98, 1982.

  9. H. Edelsbrunner and E. Welzl,Halfplanar range search in linear space and O(n 0.695 ) query time, Tech. Univ. of Graz, Tech Rep. F111, 1983.

  10. M. R. Garey, D. S. Johnson, F. P. Preparata and R. E. Tarjan,Triangulating a simple polygon, Inf. Proc. Lett. 7(4), pp. 175–179, 1978.

    Article  Google Scholar 

  11. L. J. Guibas and J. Stolfi,Primitives for the manipulation of general subdivisions and the computations of Voronoi diagrams, Proc. 15th Annual SIGACT Symp., pp. 221–234, April 1983.

  12. L. Guibas, L. Ramshaw and J. Stolfi,A kinetic framework for computational geometry, Proc. 24th Annual FOCS Symp., pp. 100–111, November 1983.

  13. D. G. Kirkpatrick,Optimal search in planar subdivisions, SIAM J. on Comp., Vol. 12, No. 1, pp. 28–35, February 1983.

    Article  Google Scholar 

  14. R. J. Lipton and R. E. Tarjan,Applications of a planar separator theorem, SIAM J. Comp., 9(3), pp. 615–627, 1980.

    Article  Google Scholar 

  15. D. E. Muller and F. P. Preparata,Finding the intersection of two convex polyhedra, Theoret. Comput. Sci. 7 (1978), pp. 217–236.

    Article  Google Scholar 

  16. M. H. Overmars and J. van Leeuwen,Maintenance of configurations in the plane, Journal of Computer and System Sciences, 23, p. 166–204, 1981.

    Article  Google Scholar 

  17. D. E. Willard,Polygon retrieval, SIAM J. Comp., 11, pp. 149–165, 1982.

    Article  Google Scholar 

  18. F. F. Yao,A 3-space partition and its applications, Proc. 15th Annual SIGACT Symp., pp. 258–263, April 1983.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chazelle, B., Guibas, L.J. & Lee, D.T. The power of geometric duality. BIT 25, 76–90 (1985). https://doi.org/10.1007/BF01934990

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01934990

Keywords

Navigation