Advertisement

Which triangulations approximate the complete graph?

Conference abstract
  • Gautam Das
  • Deborah Joseph
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 401)

Abstract

Chew and Dobkin et. al. have shown that the Delaunay triangulation and its variants are sparse approximations of the complete graph, in that the shortest distance between two sites within the triangulation is bounded by a constant multiple of their Euclidean separation. In this paper, we show that other classical triangulation algorithms, such as the greedy triangulation, and more notably, the minimum weight triangulation, also approximate the complete graph in this sense. We also design an algorithm for constructing extremely sparse (nontriangular) planar graphs that approximate the complete graph.

We define a sufficiency condition and show that any Euclidean planar graph constructing algorithm which satisfies this condition always produces good approximations of the complete graph. This condition is quite general because it is satisfied by all the triangulation algorithms mentioned above, and probably by many other graph algorithms as well. We thus partially answer the question posed by the title.

From a theoretical standpoint, our results are interesting because we prove non-trivial properties of minimum weight triangulations, of which little is currently known. From a practical standpoint, the graph algorithms we consider are good alternatives to the Delaunay triangulation, particularly when designing a sparse network under severe constraints on the total edge length. Finally, our general approach may help in identifying or designing other algorithms for constructing sparse networks.

Keywords

Planar Graph Complete Graph Delaunay Triangulation Single Edge Graph Algorithm 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

7. Bibliography

  1. [C]
    Chew: There is a Planar Graph Almost as Good as the Complete Graph: ACM Symposium on Computational Geometry, 1986, 169–177.Google Scholar
  2. [DJ]
    Das, Joseph: Planar Euclidean Graphs and their uses in Network Design: Technical Report, UW-Madison, in preparation.Google Scholar
  3. [DFS]
    Dobkin, Friedman, Supowit: Delaunay Graphs are Almost as Good as Complete Graphs: IEEE Symposium on Foundations of Computer Science, 1987, 20–26.Google Scholar
  4. [K]
    Klincsek: Minimal Triangulations of Polygonal Domains: Annals of Discrete Mathematics 9, 1980, 121–123.Google Scholar
  5. [L]
    Lingas: On Approximation Behavior and Implementation of the Greedy Triangulation for Convex Planar Point Sets: ACM Symposium on Computational Geometry, 1986, 72–79.Google Scholar
  6. [MZ]
    Manacher, Zobrist: Neither the Greedy nor the Delaunay Triangulation of a Planar Point Set Approximates the Optimal Triangulation: Information Processing Letters, July 1979, 31–34.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Gautam Das
    • 1
  • Deborah Joseph
    • 2
  1. 1.University of WisconsinUSA
  2. 2.University of WisconsinUSA

Personalised recommendations