Four results on randomized incremental constructions

  • Kenneth L. Clarkson
  • Kurt Mehlhorn
  • Raimund Seidel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 577)


We prove four results on randomized incremental constructions (RICs):
  • an analysis of the expected behavior under insertion and deletions,

  • a fully dynamic data structure for convex hull maintenance in arbitrary dimensions,

  • a tail estimate for the space complexity of RICs,

  • a lower bound on the complexity of a game related to RICs.


Convex Hull Voronoi Diagram Computational Geometry Base Facet Neighborhood Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BDS+]
    J.D. Boissonnat, O. Devillers, R. Schott, M. Teillaud, and M. Yvinec. Applications of random sampling to on-line algorithms in computational geometry. Discrete and Computational Geometry. To be published. Available as Technical Report INRIA 1285. Abstract published in IMACS 91 in Dublin.Google Scholar
  2. [CEG+91]
    B. Chazelle, H. Edelsbrunner, L.J. Guibas, M. Sharir, and J. Snoeyink. Computing a face in an arrangement of line segments. 2nd Ann. AGM-SIAM Symp. on Discrete Algorithms, pages 441–448, 1991.Google Scholar
  3. [CS89]
    K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II. Journal of Discrete and Computational Geometry, pages 387–421, 1989.Google Scholar
  4. [Dev]
    O. Devillers. Randomization yields simple o(n log* n) algorithms for difficult Ω(n) problems. International Journal on Computational Geometry and Aplications. To be published. Full paper available as Technical Report INRIA 1412. Abstract published in the Third Canadian Conference on Computational Geometry 1991 in Vancouver.Google Scholar
  5. [DMT91]
    O. Devillers, S. Meiser, and M. Teillaud. Fully dynamic Delaunay triangulation in logarithmic expected time per operation. In WADS 91, volume LNCS 519. Springer Verlag, 1991. Full version available as Technical Report INRIA 1349.Google Scholar
  6. [Ede87]
    H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer Berlin-Heidelberg, 1987.Google Scholar
  7. [GKS90]
    L.J. Guibas, D.E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Proc. of ICALP, pages 414–431, 1990. also to appear in Algorithmica.Google Scholar
  8. [MM091]
    K. Mehlhorn, St. Meiser, and C. Ò'Dunlaing. On the construction of abstract voronoi diagrams. Discrete Comput. Geom., 6:211–224, 1991.Google Scholar
  9. [MN90]
    K. Mehlhorn and St. Näher. Bounded ordered dictionaries in O(loglog n) time and O(n) space. Information Processing Letters, 35:183–189, 1990.Google Scholar
  10. [Mul88]
    K. Mulmuley. A fast planar partition algorithm, I. Proc. of the 29th FOCS, pages 580–589, 1988.Google Scholar
  11. [Mul91a]
    K. Mulmuley. Randomized multidimensional search trees: dynamic sampling. Proc. ACM Symposium on Computational Geometry, 1991.Google Scholar
  12. [Mul91b]
    K. Mulmuley. Randomized multidimensional search trees: further results in dynamic sampling. 32nd IEEE FOCS, pages 216–227, 1991.Google Scholar
  13. [Mul91c]
    K. Mulmuley. Randomized multidimensional search trees: Lazy balancing and dynamic shuffling. 32nd IEEE FOCS, pages 180–194, 1991.Google Scholar
  14. [Sch91]
    O. Schwarzkopf. Dynamic maintenance of geometric structures made easy. 32nd IEEE FOCS, pages 197–206, 1991.Google Scholar
  15. [Sei90]
    R. Seidel. Linear programming and convex hulls made easy. Proc. 6th Ann. ACM Symp. Computational Geometry, pages 211–215, 1990.Google Scholar
  16. [Sei91]
    R. Seidel. A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. Computational Geometry: Theory and Applications, 1:51–64, 1991.Google Scholar
  17. [Sto87]
    A. J. Stolfi. Oriented projective geometry (extended abstract). Proceedings of the 3rd Annual ACM Symp. on Computational Geometry, pages 76–85, 1987.Google Scholar
  18. [vKZ77]
    P. van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Math. Systems Theory, 10:99–127, 1977.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Kenneth L. Clarkson
    • 1
  • Kurt Mehlhorn
    • 2
  • Raimund Seidel
    • 3
  1. 1.AT&T Bell LaboratoriesUSA
  2. 2.Max Planck Institut für Informatik and Universität des SaarlandesGermany
  3. 3.Computer Science DivisionUniversity of California at BerkeleyUSA

Personalised recommendations