Advertisement

C-sensitive triangulations approximate the minmax length triangulation

  • Christos Levcopoulos
  • Andrzej Lingas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 652)

Abstract

We introduce the notion of a c-sensitive triangulation based on the local notion of a c-sensitive triangulation edge. We show that any c-sensitive triangulation of a planar point set approximates the minmax length triangulation of the set within the factor 2(c+1). On the other hand we prove that the greedy triangulation and the Dclaunay triangulation of a planar straight-line graph are respectively 4-sensitive and 1-sensitive. We also generalize the relationship between c-sensitive triangulations and the minmax length triangulation to include appropriately augmented planar straight-line graphs. This enables us to obtain a O(n) log n)-time heuristic for the minmax length triangulation of an arbitrary planar straight-line graph with the approximation factor bounded by 3. A modification of the above heuristic for simple polygons runs in linear time.

Keywords

Linear Time Voronoi Diagram Delaunay Triangulation Convex Polygon Simple Polygon 
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.

References

  1. [1]
    J. Bramble and M. Zlamal. Triangular elements in the finite element method. Math. Computation 24 (1970), pp. 809–820.Google Scholar
  2. [2]
    B. Chazelle. Triangulating a Simple Polygon in Linear Time. Proc. 31st IEEE FOCS Symposium, 1990.Google Scholar
  3. [3]
    B. Delaunay. Sur la sphere vide. Izv. Akad. Nauk SSSR, Otdelenic Mathematicheskii i Estestvennyka Nauk 7 (1934), pp. 793–800.Google Scholar
  4. [4]
    G. Das and D. Joseph. Which Triangulations Approximate the Complete Graph? In Proc. Int. Symp. on Optimal Algorithms, LNCS 401, pp. 168–192, Springer Verlag.Google Scholar
  5. [5]
    H. Edelsbrunner. Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science 10, 1987, Springer Verlag.Google Scholar
  6. [6]
    H. Edelsbrunner and T. S. Tan. A Quadratic Time Algorithm for the MinMax Length Triangulation. In Proc. 32nd Ann. IEEE Sympos. Found. Comput. Sci., 1991, pp. 414–423.Google Scholar
  7. [7]
    H. Edelsbrunner, T. S. Tan and R. Waupotisch. An O(n2 log n) time algorithm for the minmax angle triangulation. In Proc. 6th Ann. Sympos. Comput. Geom., 1990, pp. 44–52.Google Scholar
  8. [8]
    D.G. Kirkpatrick, A Note on Delaunay and Optimal Triangulations. IPL, Vol. 10, No. 3, pp. 127–131.Google Scholar
  9. [9]
    R. Klein and A. Lingas On Computing Voronoi Diagrams for Simple Polygons. To appear in Proc. 8th ACM Symposium on Computational Geometry, Berlin, 1992.Google Scholar
  10. [10]
    G. T. Klincsek. Minimal triangulations of polygonal domains. Annals Discrete Math. 9 (1980), pp. 121–123.Google Scholar
  11. [11]
    D.T. Lee. Two-Dimensional Voronoi Diagrams in the L p-metric. JACM, 27(4), 1980, pp. 604–618.CrossRefGoogle Scholar
  12. [12]
    D.T. Lee and A. Lin. Generalized Delaunay Triangulations for Planar Graphs. Discrete and Computational Geometry 1, 1986, Springer Verlag, pp. 201–217.Google Scholar
  13. [13]
    C. Levcopoulos and A. Lingas. On approximation behavior of the greedy triangulation for convex polygons. Algorithmica 2, 1987, pp. 175–193.CrossRefGoogle Scholar
  14. [14]
    C. Levcopoulos and A. Lingas. Fast Algorithms for Greedy Triangulation. Proc. SWAT'90, Lecture Notes in Computer Science 447, Springer Verlag, pp. 238–250.Google Scholar
  15. [15]
    A. Lingas. A new heuristic for minimum weight triangulation. SIAM J. Algebraic Discrete Methods 8 (1987), pp. 646–658.Google Scholar
  16. [16]
    F.P. Preparata and M.I. Shamos. Computational Geometry: An Introduction. Texts and Monographs in Theoretical Computer Science, Springer Verlag, New York, 1985.Google Scholar
  17. [17]
    V. T. Rajan. Optimality of the Delaunay Triangulation in R d. In Proc. 7th Ann. Sympos. Comput. Geom., 1991, pp. 357–363.Google Scholar
  18. [18]
    R. Sibson. Locally equiangular triangulations. Comput. J. 21 (1978), pp. 243–245.CrossRefGoogle Scholar
  19. [19]
    F. W. Wilson, R. K. Goodrich and W. Spratte. Lawson's triangulation is nearly optimal for controlling error bounds. SIAM J. Numer. Anal. 27 (1990), pp. 190–197.CrossRefGoogle Scholar
  20. [20]
    C. Wang and L. Schubert. An Optimal Algorithm for Constructing the Delaunay Triangulation of a Set of Line Segments. Proc. 3rd ACM Symposium on Computational Geometry, Waterloo, pp. 223–232, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Christos Levcopoulos
    • 1
  • Andrzej Lingas
    • 1
  1. 1.Department of Computer ScienceLund UniversityLundSweden

Personalised recommendations