Advertisement

On intersection searching problems involving curved objects

  • Prosenjit Gupta
  • Ravi Janardan
  • Michiel Smid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 824)

Abstract

Efficient solutions are given for the problem of preprocessing a set of linear or curved geometric objects (e.g. lines, line segments, circles, circular arcs, d-balls, d-spheres) such that the ones that are intersected by a curved query object can be reported (or counted) quickly. The problem is considered both in the standard setting (where one is interested in all the objects intersected) and in a generalized setting (where the input objects come aggregated in disjoint groups and one is interested in the disjoint groups that are intersected). The solutions are based on geometric transformations, simplex compositions, persistence, and, for the generalized problem, on a method to progressively eliminate groups that cannot possibly be intersected.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AHL90]
    A. Aggarwal, M. Hansen, and T. Leighton. Solving query-retrieval problems by compacting Voronoi diagrams. In Proc. 18th Annual ACM Symposium on Theory of Computing, pages 331–340, 1990.Google Scholar
  2. [AM92]
    P. K. Agarwal and J. Matoušek. On range searching with semialgebraic sets. In Proc. 17th Internat. Sympos. Math. Found. Comput. Sci., LNCS 629, pages 1–13. Springer-Verlag, 1992.Google Scholar
  3. [AS91]
    P.K. Agarwal and M. Sharir. Counting circular arc intersections. In Proc. 7th Annual Symposium on Computational Geometry, pages 10–20, 1991.Google Scholar
  4. [AvK93]
    P.K. Agarwal and Marc van Kreveld. Connected component and simple polygon intersection searching. In Proc. 3rd WADS, LNCS 709, pages 36–47. Springer-Verlag, 1993.Google Scholar
  5. [AvKO93]
    P.K. Agarwal, Marc van Kreveld, and Marc Overmars. Intersection queries for curved objects. Journal of Algorithms, 15:229–266, 1993.Google Scholar
  6. [CW89]
    B. Chazelle and E. Welzl. Quasi-optimal range searching in spaces with finite VC-dimension. Discrete and Computational Geometry, 4:467–489, 1989.Google Scholar
  7. [DK83]
    D.P. Dobkin and D.G. Kirkpatrick. Fast detection of polyhedral intersection. Theoretical Computer Science, 27:241–253, 1983.Google Scholar
  8. [Ede87]
    H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, 1987.Google Scholar
  9. [GJS94]
    P. Gupta, R. Janardan, and M. Smid. Efficient algorithms for generalized intersection searching on non-iso-oriented objects. To appear in Proc. 10th Annual ACM Symposium on Computational Geometry, 1994.Google Scholar
  10. [GJS93b]
    P. Gupta, R. Janardan, and M. Smid. Further results on generalized intersection searching problems: counting, reporting and dynamization. In Proc. WADS 1993, LNCS 709, pages 361–372. Springer Verlag, 1993. (To appear in Journal of Algorithms.)Google Scholar
  11. [JL93]
    R. Janardan and M. Lopez. Generalized intersection searching problems. International Journal on Computational Geometry & Applications, 3(1):39–69, 1993.Google Scholar
  12. [KOA90]
    M. Van Kreveld, M. Overmars, and P. K. Agarwal. Intersection queries in sets of disks. In Proc., SWAT 1990, LNCS 447, pages 393–403. Springer Verlag, 1990.Google Scholar
  13. [Pel92]
    M. Pellegrini. A new algorithm for counting circular arc intersections. TR-92-010, International Computer Science Institute, Berkeley, 1992.Google Scholar
  14. [Sha91]
    M. Sharir. The k-set problem for arrangement of curves and surfaces. Discrete and Computational Geometry, 6:593–613, 1991.Google Scholar
  15. [vK92]
    M. van Kreveld. New results on data structures in computational geometry. PhD thesis, Department of Computer Science, University of Utrecht, Utrecht, the Netherlands, 1992.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Prosenjit Gupta
    • 1
  • Ravi Janardan
    • 1
  • Michiel Smid
    • 2
  1. 1.Department of Computer ScienceUniversity of MinnesotaMinneapolisUSA
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations