, Volume 15, Issue 2, pp 104–125 | Cite as

Parallel algorithms for arrangements

  • R. Anderson
  • P. Beanie
  • E. Brisson


We give the first efficient parallel algorithms for solving the arrangement problem. We give a deterministic algorithm for the CREW PRAM which runs in nearly optimal bounds ofO (logn log*n) time andn2/logn processors. We generalize this to obtain anO (logn log*n)-time algorithm usingn d /logn processors for solving the problem ind dimensions. We also give a randomized algorithm for the EREW PRAM that constructs an arrangement ofn lines on-line, in which each insertion is done in optimalO (logn) time usingn/logn processors. Our algorithms develop new parallel data structures and new methods for traversing an arrangement.

Key words

Parallel algorithms Computational geometry Arrangement problem Incremental algorithms 


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  1. [1]
    A. Aggarwal, B. Chazelle, L. Guibas, C. O'Dunlaing, and C. K. Yap. Parallel computational geometry.Algorithmica, 3:293–326, 1988.Google Scholar
  2. [2]
    A. Aggarwal and J. Wein. Computational geometry: lecture notes for 18.409, spring 1988. Technical Report MIT/LCS/RSS 3, MIT Laboratory for Computer Science, 1988.Google Scholar
  3. [3]
    R. J. Anderson, P. Beame, and E. Brisson. Parallel algorithms for arrangements.Proceedings of the Second Annual Symposium on Parallel Algorithms and Architectures, 1990, pp. 298–306.Google Scholar
  4. [4]
    D. Arnon, G. Collins, and S. McCallum. Cylindrical algebraic decomposition,I andII. SIAM Journal on Computing, 13(4):865–889, 1984.Google Scholar
  5. [5]
    M. J. Atallah, R. Cole, and M. T. Goodrich. Cascading divide-and-conquer: a technique for designing parallel algorithms.SIAM Journal on Computing, 18:499–532, 1989.Google Scholar
  6. [6]
    J. Canny. A new algebraic method for robot motion planning and real geometry.Proceedings of the 28th Symposium on Foundations of Computer Science, 1987, pp. 29–38.Google Scholar
  7. [7]
    B. Chazelle. Intersecting is easier than sorting.Proceedings of the 16th ACM Symposium on Theory of Computation, 1984, pp. 125–134.Google Scholar
  8. [8]
    B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality.BIT, 25:76–90, 1985.Google Scholar
  9. [9]
    R. Cole and U. Vishkin. Approximate parallel scheduling. Part I: The basic technique with applications to optimal parallel list ranking in logarithmic time.SIAM Journal on Computing, 17:128–142, 1988.Google Scholar
  10. [10]
    J. Driscoll, N. Sarnak, D. Sleator, and R. Tarjan. Making data structures persistent.Journal of Computer and System Sciences, 38:86–124, 1989.Google Scholar
  11. [11]
    H. Edelsbrunner.Algorithms in Combinatorial Geometry. Springer-Verlag, New York, 1987.Google Scholar
  12. [12]
    H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications.SIAM Journal on Computing, 15(2):341–363, 1986.Google Scholar
  13. [13]
    M. T. Goodrich. Intersecting line segments in parallel with an output-sensitive number of processors.Proceedings of the First Annual Symposium on Parallel Algorithms and Architectures, 1989, pp. 127–136.Google Scholar
  14. [14]
    M. T. Goodrich. Constructing arrangements optimal in parallel.Proceedings of the Third Annual Symposium on Parallel Algorithms and Architectures, 1991, pp. 169–179. Also inDiscrete & Computational Geometry, 9:371–385, 1993.Google Scholar
  15. [15]
    T. Hagerup, H. Jung, and E. Welzl. Efficient parallel computation of arrangements of hyperplanes ind dimensions.Proceedings of the Second Annual Symposium on Parallel Algorithms and Architectures, 1990, pp. 290–297.Google Scholar
  16. [16]
    J. JáJá.An Introduction to Parallel Algorithms. Addison-Wesley, Reading, MA, 1992.Google Scholar
  17. [17]
    M. McKenna. Worst-case optimal hidden-surface removal.ACM Transactions on Graphics, 6(1): 19–28, 1987.Google Scholar
  18. [18]
    J. H. Reif and S. Sen. Polling: a new randomized sampling technique for computational geometry.Proceedings of the 21st ACM Symposium on Theory of Computation, 1989, pp. 394–404.Google Scholar
  19. [19]
    A. Tarski.A Decision Method for Elementary Algebra and Geometry. University of California Press, Berkeley, CA, 1951.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1996

Authors and Affiliations

  • R. Anderson
    • 1
  • P. Beanie
    • 1
  • E. Brisson
    • 1
  1. 1.Department of Computer Science and Engineering, FR-35University of WashingtonSeattleUSA

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