Skip to main content
Log in

O(log logn)-time integer geometry on the CRCW PRAM

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We study problems in computational geometry on PRAMs under the assumption that input objects are specified by points withO(logn)-bit coordinates, or, equivalently, with polynomially bounded integer coordinates. We show that in this setting many geometric problems can be solved in time O(log logn). The following five specific problems are investigated:closest pair of points, intersection of convex polygons, intersection of manhattan line segments, dominating set, andlargest empty square. Algorithms solving them are developed which operate in time O(log logn) on the arbitrary CRCW PRAM. The number of processors used is eitherO(n) orO(n logn).

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. A. Aggarwal, B. Chazelle, L. Guibas, C. O'Dunlaing, and C. K. Yap, Parallel computational geometry,Algorithmica,3 (1988), 293–326.

    Google Scholar 

  2. M. J. Atallah and M. Goodrich, Efficient parallel solutions to some geometric problems,J. Parallel Distributed Comput.,3 (1986), 492–507.

    Google Scholar 

  3. P. Beame and J. Hastad, Optimal bounds for decision problems on the CRCW PRAM,Proceedings of the 19th ACM Symposium on Theory of Computing, 1987, pp. 83–93.

  4. O. Berkman, D. Breslauer, Z. Galil, B. Schieber, and U. Vishkin, Highly-parallelizable problems,Proceedings of the 21st Annual ACM Symposium on Theory of Computing, 1989, pp. 309–319.

  5. O. Berkman, J. JáJá, S. Krishnamurthy, R. Thurimella, and U. Vishkin, Some triply-logarithmic parallel algorithms,Proceedings of the 31st Annual Symposium on Foundations of Computer Science, 1980, pp. 871–881.

  6. P. C. P. Bhatt, K. Diks, T, Hagerup, V. C. Prasad, T. Radzik, and S. Saxena, Improved deterministic parallel integer sorting,Inform. and Comput.,94 (1991), 29–47.

    Google Scholar 

  7. B. S. Chlebus, K. Diks, T. Hagerup, and T. Radzik, Efficient simulations between concurrent-read concurrent-write PRAM models,Proceedings of the 13th Symposium on Mathematical Foundations of Computer Science, 1988, Lecture Notes in Computer Science, Vol. 324, Springer-Verlag, Berlin, pp. 231–239.

    Google Scholar 

  8. R. Cole and M. T. Goodrich, Optimal parallel algorithms for polygon and pointset problems,Proceedings of the 4th ACM Symposium on Computational Geometry, 1988, pp. 205–214.

  9. T. Hagerup, The log-star revolution,Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science, 1992, Lecture Notes in Computer Science, Vol. 577, Springer-Verlag, Berlin, pp. 259–278.

    Google Scholar 

  10. P. D. MacKenzie and Q. F. Stout, Ultra-fast expected time parallel algorithms,Proceedings of the 2nd Annual ACM-SIAM Symposium on Discrete Algorithms, 1991, pp. 414–423.

  11. Y. Shiloach and U. Vishkin, Finding the maximum, merging, and sorting in a parallel computation model,J. Algorithms,3 (1981), 57–67.

    Google Scholar 

  12. Q. F. Stout, Constant-time geometry on PRAMs,Proceedings of the International Conference on Parallel Processing, 1988, pp. 104–107.

  13. U. Vishkin, Structural parallel algorithmics, inLectures on Parallel Computation, edited by A. Gibbons and P. Spirakis, Cambridge University Press, Cambridge, 1993, pp. 1–18.

    Google Scholar 

  14. H. Wagener, Optimal parallel hull construction for simple polygons in O(log logn) time,Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science, 1992, pp. 593–599.

  15. D. E. Willard and Y. C. Wee, Quasi-valid range querying and its implications for nearest neighbor problems,Proceedings of the 4th ACM Symposium on Computational Geometry, 1988, pp. 34–43.

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by K. Melhorn.

This research was supported in part by Grants KBN 2-2044-92-03, KBN 2-2043-92-03, and KBN 2-1190-91-01.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chlebus, B.S., Diks, K. & Kowaluk, M. O(log logn)-time integer geometry on the CRCW PRAM. Algorithmica 14, 52–69 (1995). https://doi.org/10.1007/BF01300373

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01300373

Key words

Navigation