Skip to main content
SpringerLink
Log in

Scanline algorithms on a grid

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

Abstract

A number of important problems in computational geometry are solved efficiently on 2- or 3-dimensional grids by means of scanline techniques. In the time complexity of solutions to the maximal elements and closure problems, a factor logn is substituted by loglogn, wheren is the number of elements. Next, by using a data structure introduced in the paper, the interval trie, previous solutions to the rectangle intersection and connected component problems are improved upon. Finally, a fast intersection finding algorithm for arbitrarily oriented line segments is presented.

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.

Similar content being viewed by others

References

  1. J. L. Bentley and T. Ottman,Algorithms for reporting and counting geometric intersections, IEEE Trans. Comp. C-28, 9 (1979), 643–647.

    Google Scholar 

  2. B. Chazelle,Intersecting is easier than sorting, Proc. 16th Annual ACM Symposium on Theory of Computing (1984), 125–234.

  3. H. Edelsbrunner,A new approach to rectangle intersections, Part II, Int. J. Comput. Math. 13 (1983), 221–229.

    Google Scholar 

  4. H. Edelsbrunner, J. van Leeuwen, T. Ottmann and D. Wood,Computing the connected components of simple rectilinear geometrical objects in d-space, R.A.I.R.O. Theoretical Informatics 18, 2 (1984), 171–183.

    Google Scholar 

  5. R. A. Jarvis,On the identification of the convex hull of a finite set of points in the plane, Information Processing Lett. 2 (1973), 18–21.

    Google Scholar 

  6. D. B. Johnson,A priority queue in which initialization and queue operations take O(loglogD)time, Math. Systems Theory 15, 4 (1982), 295–310.

    Google Scholar 

  7. R. G. Karlsson,Algorithms in a restricted universe, Ph.D. thesis, University of Waterloo, 1984, Dept of Computer Science Tech. Report CS-84-50.

  8. R. G. Karlsson and J. I. Munro,Proximity on a Grid, Proc. 2nd Symposium on Theoretical Aspects of Computer Science, Springer-Verlag Lecture Notes in Computer Science 182 (1985), 187–196.

    Google Scholar 

  9. R. G. Karlsson and M. H. Overmars,Normalized divide and conquer: A scaling technique for solving multi-dimensional problems, Information Processing Lett. 26 (1988), 307–312.

    Google Scholar 

  10. J. M. Keil and D. G. Kirkpatrick,Computational geometry on integer grids, Proc. 19th Annual Allerton Conference (1981), 41–50.

  11. D. Kirkpatrick and S. Reisch,Upper bounds for sorting integers on random access machines, Theoretical Computer Science 28 (1984), 263–276.

    Google Scholar 

  12. H. T. Kung, F. Luccio and F. P. Preparata,On finding the maxima of a set of vectors, J. ACM 22, 4 (1975), 469–476.

    Google Scholar 

  13. E. M. McCreight,Efficient algorithms for enumerating intersecting intervals and rectangles, Xerox Alto Res. Center, Report PARC CSL-80-9, 1980.

  14. F. W. Myers,An O(E logE +I)expected time algorithm for the planar segment intersection problem, SIAM J. Computing 14 (1985), 625–637.

    Google Scholar 

  15. H. Müller,Rastered point location, Proc. Workshop on Graphtheoretic Concepts in Computer Science (WG85), Trauner Verlag, 1985, 281–293.

  16. M. H. Overmars,Efficient data structures for range searching on a grid, to appear in J. of Algorithms.

  17. F. P. Preparata and M. I. Shamos,Computational Geometry, An Introduction, Springer-Verlag, 1985.

  18. E. Soisalon-Soininen and D. Wood,An optimal algorithm for testing for safety and detecting deadlock in locked transaction systems, Proc. ACM Symposium on Principles of Data Bases (1982), 108–116.

  19. P. van Emde Boas,Preserving order in a forest in less than logarithmic time and linear space, Information Processing Lett. 6, 3 (1977), 80–82.

    Google Scholar 

  20. D. E. Willard,Log-logarithmic worst-case range queries are possible in Space Θ(n), Information Processing Lett. 17, 2 (1983), 81–84.

    Google Scholar 

  21. D. E. Willard,New trie data structures which support very fast search operations, J. Comput. Syst. Sci. 28 (1984), 379–394.

    Google Scholar 

  22. M. Z. Yannakakis, C. H. Papadimitriou and H. T. Kung,Locking policies: safety and freedom for deadlock, Proc. 20th Annual IEEE Symposium on Foundations of Computer Science (1979), 286–297.

Download references

Author information

Authors and Affiliations

  1. Dept. of Computer Science, Linköping University, 581 83, Linköping, Sweden

    Rolf G. Karlsson & Mark H. Overmars

  2. Dept. of Computer Science, University of Utrecht, P.O. Box 80.012, 3508, TA Utrecht, The Netherlands

    Rolf G. Karlsson & Mark H. Overmars

Authors
  1. Rolf G. Karlsson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark H. Overmars
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Karlsson, R.G., Overmars, M.H. Scanline algorithms on a grid. BIT 28, 227–241 (1988). https://doi.org/10.1007/BF01934088

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01934088

CR-categories

Keywords and phrases

Search

Navigation