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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)

Abstract

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.

Keywords

Short Path Voronoi Diagram Delaunay Triangulation Close Pair Voronoi Region 
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.

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References

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    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
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    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
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    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
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    Preparata, F. and M. Shamos, Computational Geometry: an Introduction, Springer-Verlag, 1985.Google Scholar
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    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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