BIT Numerical Mathematics

, Volume 28, Issue 2, pp 227–241 | Cite as

Scanline algorithms on a grid

  • Rolf G. Karlsson
  • Mark H. Overmars
Part I Computer Science


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.


E.2 F.2.2 

Keywords and phrases

scanline techniques grid geometry maximal elements rectangle problems connected components line segment intersection 


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  1. 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. 2.
    B. Chazelle,Intersecting is easier than sorting, Proc. 16th Annual ACM Symposium on Theory of Computing (1984), 125–234.Google Scholar
  3. 3.
    H. Edelsbrunner,A new approach to rectangle intersections, Part II, Int. J. Comput. Math. 13 (1983), 221–229.Google Scholar
  4. 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. 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. 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. 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.Google Scholar
  8. 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. 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. 10.
    J. M. Keil and D. G. Kirkpatrick,Computational geometry on integer grids, Proc. 19th Annual Allerton Conference (1981), 41–50.Google Scholar
  11. 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. 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. 13.
    E. M. McCreight,Efficient algorithms for enumerating intersecting intervals and rectangles, Xerox Alto Res. Center, Report PARC CSL-80-9, 1980.Google Scholar
  14. 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. 15.
    H. Müller,Rastered point location, Proc. Workshop on Graphtheoretic Concepts in Computer Science (WG85), Trauner Verlag, 1985, 281–293.Google Scholar
  16. 16.
    M. H. Overmars,Efficient data structures for range searching on a grid, to appear in J. of Algorithms.Google Scholar
  17. 17.
    F. P. Preparata and M. I. Shamos,Computational Geometry, An Introduction, Springer-Verlag, 1985.Google Scholar
  18. 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.Google Scholar
  19. 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. 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. 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. 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.Google Scholar

Copyright information

© BIT Foundations 1988

Authors and Affiliations

  • Rolf G. Karlsson
    • 1
    • 2
  • Mark H. Overmars
    • 1
    • 2
  1. 1.Dept. of Computer ScienceLinköping UniversityLinköpingSweden
  2. 2.Dept. of Computer ScienceUniversity of UtrechtTA UtrechtThe Netherlands

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