Skip to main content
SpringerLink
Log in

Randomized geometric algorithms and pseudorandom generators

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

It is shown that the bounds on the expected running times of most of the randomized incremental algorithms in computational geometry do not change by more than a constant factor when they are made pseudorandom using a very simple scheme. This reduces the number of random bits used by these algorithms from Ω(nlogn) toO(logn).

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. Alon, O. Goldreich, J. Hastad, R. Perelta, Simple constructions of almostk-wise independent random variables,Proc. of the 31st Annual IEEE Symp. on Foundations of Computer Science, 1990, pp. 544–553.

  2. J. Boissonnat, O. Devillers, R. Schott, M. Teillaud, M. Yvinec, Applications of random sampling to on-line algorithms in computational geometry,Discrete Comput. Geom., 8:51–71, 1992.

    Google Scholar 

  3. B. Chazelle, An optimal convex hull algorithm and new results on cuttings,Proc. of the 32nd Annual IEEE Symp. on Foundations of Compter Science, 1991, pp. 29–38.

  4. B. Chazelle, H. Edelsbrunner, An optimal algorithm for intersecting line segments in the plane,J. Assoc. Comput. Mach., 39:1–54, 1992.

    Google Scholar 

  5. B. Chazelle, J. Friedman, A deterministic view of random sampling and its use in geometry,Combinatorica, 10(3):229–249, 1990.

    Google Scholar 

  6. B. Chazelle, J. Matoušek, On linear-time deterministic algorithms for optimization problems in fixed dimension,Proc. of the 4th ACM Symp. on Discrete Algebra, 1993.

  7. K. Clarkson, P. Shor, Applications of random sampling in computational geometry, II,Discrete Comput. Geom., 4:387–421, 1989.

    Google Scholar 

  8. L. Guibas, D. Knuth, M. Sharir, Randomized incremental construction of Delauney and Voronoi diagrams,Algorithmica, 7:381–413, 1992.

    Google Scholar 

  9. D. Haussler, E. Welzl, Epsilon-nets and simplex range queries,Discrete Comput. Geom., 2:127–151, 1987.

    Google Scholar 

  10. H. Karloff, P. Raghavan, Randomized algorithms and pseudorandom numbers,Proc. of the 20th Annual ACM Symp. on Theory of Computing, 1988, pp. 310–321.

  11. M. Luby, A simple parallel algorithm for the maximal independent set problem,SIAM J. Comput., 15:1036–1053, 1986.

    Google Scholar 

  12. N. Megiddo, Linear programming in linear time when the dimension is fixed,J. Assoc. Comput. Mach., 34:114–127, 1981.

    Google Scholar 

  13. K. Mehlhorn, S. Meiser, C. O'Dunlaing, On the construction of abstract Voronoi diagrams,Discrete Comput. Geom., 6:211–224, 1991.

    Google Scholar 

  14. K. Mulmuley,Computational Geometry: An Introduction Through Randomized Algorithms, Prentice-Hall, Englewood Cliffs, NJ, 1994.

    Google Scholar 

  15. K. Mulmuley, A fast planar partition algorithm, I,Proc. of the 29th Annual IEEE Symp. on Foundations of Computer Science, 1988, pp. 580–589.

  16. K. Mulmuley, An efficient algorithm for hidden surface removal (Proc. of the ACM SIGGKAPH),Comput. Graphics, 23(3):379–388, 1989.

    Google Scholar 

  17. K. Mulmuley, On levels in arrangements and Voronoi diagrams,Discrete Comput. Geom., 6:307–338, 1991.

    Google Scholar 

  18. K. Mulmuley, Randomized multidimensional search trees: lazy balancing and dynamic shuffling,Proc. of the 32nd Annual IEEE Symp. on Foundations of Computer Science, 1991, pp. 180–196.

  19. J. Naor, M. Naor, Small bias probability spaces: efficient constructions and applications,Proc. of the 22nd Annual ACM Symp. on Theory of Computing, 1990, pp. 213–223.

  20. J. Reif, S. Sen, Optimal parallel randomized algorithms for 3-D convex hulls and related problems. Manuscript, preliminary version appears as “Polling: a new randomized sampling technique for computational geometry” inProc. of the 21st Annual ACM Symp. on Theory of Computing, 1989, pp. 394–404.

  21. O. Schwarzkopf, Dynamic maintenance of geometric structures made easy,Proc. of the 32nd Annual IEEE Symp. on Foundations of Computer Science, 1991, pp. 197–206.

  22. R. Seidel, Low dimensional linear programming and convex hulls made easy,Discrete Comput. Geom., 6:423–434, 1991.

    Google Scholar 

  23. R. Seidel, A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons.Comput. Geom. Theory Appl, 1:51–64, 1991.

    Google Scholar 

  24. R. Seidel, Backward analysis of randomized incremental algorithms,New Trends in Discrete and Computational Geometry (J. Pach ed.), Springer-Verlag, New York, 1993, pp. 37–67.

    Google Scholar 

  25. J. Spencer,Ten Lectures on the Probabilistic Method, CBMS-NSF, SIAM, Philadelphia, PA, 1987.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Chicago, 1100 E 58th Street, 60637, Chicago, IL, USA

    K. Mulmuley

Authors
  1. K. Mulmuley
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by M. Luby.

This research was supported by Packard fellowship.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mulmuley, K. Randomized geometric algorithms and pseudorandom generators. Algorithmica 16, 450–463 (1996). https://doi.org/10.1007/BF01940875

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01940875

Key words

Search

Navigation