The Delaunay triangulation closely approximates the complete Euclidean graph

  • J. Mark Keil
  • Carl A. Gutwin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 382)


Let p and q be a pair of points in a set S of N points in the plane. Let d(p,q) be the Euclidean distance between p and q and let DT(p,q) be the length of the shortest path from p to q in the Delaunay triangulation of S. We show that that the ratio
$$\frac{{DT(p,q)}}{{d(p,q)}} \leqslant \frac{{2\pi }}{{3\cos (\frac{\pi }{6})}} \approx 2.42$$
independent of S and N.


Short Path Voronoi Diagram Delaunay Triangulation Close Pair Voronoi Region 
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  1. [1]
    Chew, P., There is a planar graph almost as good as the complete graph, Proceedings of the 2nd Symposium on Computational Geometry, Yorktown Heights NY, 1986, 169–177.Google Scholar
  2. [2]
    Dobkin, D., S. Friedman and K. Supowit, Delaunay Graphs are Almost as Good as Complete Graphs, Proceedings of the 28th Annual Symposium on Foundations of Computing, Los Angeles Ca., 1987, 20–26.Google Scholar
  3. [3]
    Keil. J.M., Approximating the Complete Euclidean Graph, Proceedings of the First Scandinavian Workshop on Algorithm Theory, Halmstad, Sweden, July 1988, Springer-Verlag Lecture Notes in Computer Science No. 318, 208–213.Google Scholar
  4. [4]
    Preparata, F. and M. Shamos, Computational Geometry: an Introduction, Springer-Verlag, 1985.Google Scholar
  5. [5]
    Sedgewick, R. and J. Vitter, Shortest paths in Euclidean graphs, Algorithmica, 1,1(1986), 31–48.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • J. Mark Keil
    • 1
  • Carl A. Gutwin
    • 1
  1. 1.Department of Computational ScienceUniversity of SaskatchewanSaskatoonCanada

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