Skip to main content

Greedy triangulation can be efficiently implemented in the average case

detailed abstract

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 344))

Abstract

Let S be a set of n points uniformly distributed in a unit square. We show that the greedy triangulation of S can be computed in O(nlog1.5 n) expected time (without bucketing). The best previously known upper-bound on the expected-time performance of an algorithm for the greedy triangulation was O(n 2).

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Angluin and L.G. Valiant, Fast probabilistic algorithms for Hamiltonian circuits and matchings, Proc. 9th Ann. ACM Symp. on Theory of Computing, New York.

    Google Scholar 

  2. J.L. Bentley, B.W. Weide, A.C. Yao, Optimal expected-time algorithms for closest point problems, ACM Transactions on Mathematical Software 6, pp. 563–580.

    Google Scholar 

  3. R.C. Chang and R.C.T Lee, On the average length of Delaunay triangulations, BIT 24, pp. 269–273.

    Google Scholar 

  4. A. Fournier and D.Y. Montuno, Triangulating Simple Polygons and Equivalent Problems, ACM Transactions on Graphics 3(2), pp. 153174.

    Google Scholar 

  5. P.D. Gilbert, New Results in Planar Triangulations, M.S. Thesis, Coordinated Science Laboratory, University of Illinois, Urbana, Illinois.

    Google Scholar 

  6. D.G. Kirkpatrick, A Note on Delaunay and Optimal Triangulations, IPL, Vol. 10, No. 3, pp. 127–131.

    Google Scholar 

  7. A. Lingas, The greedy and Delaunay triangulations are not bad in the average case, IPL 22, pp. 25–31.

    Google Scholar 

  8. A. Lingas, The shortest diagonal problem, submitted to IPL.

    Google Scholar 

  9. A. Lingas, A space efficient algorithm for the greedy triangulation, presented at the 13th IFIP Conference on System Modelling and Optimization, Tokyo, Japan 1987.

    Google Scholar 

  10. C. Levcopoulos and A. Lingas, On Approximation Behavior and Implementation of the Greedy Triangulation for Convex Polygons, Algorithmica 2, pp. 175193.

    Google Scholar 

  11. E.L. Lloyd, On Triangulations of a Set of Points in the Plane, Proc. of the 18th Annual IEEE Conference on the Foundations of Computer Science, Providence.

    Google Scholar 

  12. G.K. Manacher, and A.L. Zorbrist, Neither the greedy nor the Delaunay triangulation of a planar point set approximates the optimal triangulation, Information Processing Letters, Vol.9, No. 1, pp. 31–34.

    Google Scholar 

  13. G.K. Manacher, and A.L. Zorbrist, The use of probabilistic methods and of heaps for fast-average-case, space-optimal greedy algorithms, manuscript, 1982.

    Google Scholar 

  14. K. Mehlhorn, Data Structures and Algorithms, EATS Monographs on Theoretical Computer Science, Springer Verlag, New York.

    Google Scholar 

  15. J. O'Rourke, The Computational Geometry Column, ACM SIGACT News 18(1).

    Google Scholar 

  16. F.P. Preparata and M.I. Shamos, Computational Geometry, An Introduction, Texts and Monographs in Computer Science, Springer Verlag, New York.

    Google Scholar 

  17. C. Wang and L. Shubert, An optimal algorithm for constructing the Delaunay triangulation of a set of line segments, in the proceedings of the 3rd ACM Symposium on Computational Geometry, Waterloo, pp. 223–232, 1987.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

J. van Leeuwen

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lingas, A. (1989). Greedy triangulation can be efficiently implemented in the average case. In: van Leeuwen, J. (eds) Graph-Theoretic Concepts in Computer Science. WG 1988. Lecture Notes in Computer Science, vol 344. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-50728-0_48

Download citation

  • DOI: https://doi.org/10.1007/3-540-50728-0_48

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50728-4

  • Online ISBN: 978-3-540-46076-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics