Discrete & Computational Geometry

, Volume 11, Issue 2, pp 163–191

Approximating the minimum weight steiner triangulation

  • David Eppstein
Article

Abstract

We show that the length of the minimum weight Steiner triangulation (MWST) of a point set can be approximated within a constant factor by a triangulation algorithm based on quadtrees. InO(n logn) time we can compute a triangulation withO(n) new points, and no obtuse triangles, that approximates the MWST. We can also approximate the MWST with triangulations having no sharp angles. We generalize some of our results to higher-dimensional triangulation problems. No previous polynomial-time triangulation algorithm was known to approximate the MWST within a factor better thanO(logn).

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References

  1. 1.
    M. Bern, D. Eppstein, and J. Gilbert. Provably good mesh generation.Proc. 31st IEEE Symp. on Foundations of Computer Science, 1990, pp. 231–241. Some cited results appear only in the full version,J. Comput. System. Sci., to appear.Google Scholar
  2. 2.
    K. Clarkson. Approximation algorithms for planar traveling salesman tours and minimum-length triangulations.Proc. 2nd ACM-SIAM Symp. on Discrete Algorithms, 1991, pp. 17–23.Google Scholar
  3. 3.
    E. F. D'Azevedo and R. B. Simpson. On optimal interpolation triangle incidences.SIAM J. Sci. Statist. Comput. 10 (1989), 1063–1075.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    D. Z. Du and F. K. Hwang. An approach to proving lower bounds: solution of Gilbert-Pollack's conjecture on Steiner ratio.Proc. 31st IEEE Symp. Foundations of Computer Science, 1990, pp. 76–85.Google Scholar
  5. 5.
    H. Edelsbrunner and T. S. Tan. A quadratic time algorithm for the minmax length triangulation.Proc. 32nd IEEE Symp. on Foundations of Computer Science, 1991, pp. 414–423.Google Scholar
  6. 6.
    H. Edelsbrunner, T. S. Tan, and R. Waupotitsch. A polynomial time algorithm for the minmax angle triangulation.Proc. 6th ACM Symp. on Computational Geometry, 1990, pp. 44–52.Google Scholar
  7. 7.
    D. Eppstein. The farthest point Delaunay triangulation minimizes angles.Comput. Geom. Theory Appl. 1 (1992), 143–148.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Eppstein. Approximating the minimum weight triangulation.Proc. 2nd ACM-SIAM Symp. on Discrete Algorithms, 1992, pp. 48–57.Google Scholar
  9. 9.
    M. R. Garey and D. S. Johnson,Computers and intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, CA, 1979.MATHGoogle Scholar
  10. 10.
    P. D. Gilbert. New results in planar triangulations. Report R-850, Coordinated Science Laboratory, University of Illionis (1979).Google Scholar
  11. 11.
    D. G. Kirkpatrick. A note on Delaunay and optimal triangulations,Inform. Process. Lett. 10 (1980), 127–128.MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    G. T. Klincsek. Minimal triangulations of polygonal domains.Ann. Discrete Math. 9 (1980), 121–123.MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    C. Levcopoulos, An\(\Omega (\sqrt n )\) lower bound for non-optimality of the greedy triangulation.Inform. Process. Lett. 25 (1987), 247–251.MathSciNetCrossRefGoogle Scholar
  14. 14.
    C. Levcopoulos and A. Lingas. On approximation behavior of the greedy triangulation for convex polygons,Algorithmica 2 (1987), 175–193.MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    C. Levcopoulos and A. Lingas. Fast algorithms for greedy triangulation.Proc. 2nd Scadinavian Workshop on Algorithm Theory, LNCS, vol. 447, Springer-Verlag, Berlin, 1990, pp. 238–250.Google Scholar
  16. 16.
    A. Lingas. Advances in minimum weight triangulation. Ph.D. thesis, Linköping University (1983).Google Scholar
  17. 17.
    E. L. Lloyd. On triangulations of a set of points in the plane.Proc. 18th IEEE Symp on Foundations of Computer Science, 1977, pp. 228–240.Google Scholar
  18. 18.
    R. Löhner. Some useful data structures for the generation of unstructured grids.Comm. Appl. Numerical Methods 4 (1988), 123–135.MATHCrossRefGoogle Scholar
  19. 19.
    G. K. Manacher and A. L. Zobrist. Neither the greedy nor the Delaunay triangulation approximates the optimum.Inform. Process. Lett. 9 (1979), 31–34.MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    D. M. Mount and A. Saalfeld. Globally-equiangular triangulations of co-circular points inO(n logn) time.Proc. 4th ACM Symp. on Computational Geometry 1988, pp. 143–152.Google Scholar
  21. 21.
    D. A. Plaisted and J. Hong. A heuristic triangulation algorithm.J. Algorithms 8 (1987), 405–437.MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    W. C. Rheinboldt and C. K. Mesztenyi, On a data structure for adaptive finite element mesh refinements.ACM Trans. Math. Software 6 (1980), 166–187.MATHCrossRefGoogle Scholar
  23. 23.
    H. Samet. The quadtree and related hierarchical data structures.Comput. Surveys 16 (1984), 188–260.MathSciNetCrossRefGoogle Scholar
  24. 24.
    R. Sibson. Locally equiangular triangulations.Comput. J. 21 (1978), 243–245.MathSciNetCrossRefGoogle Scholar
  25. 25.
    W. D. Smith. Implementing the Plaisted-Hong min-length plane triangulation heuristic. Manuscript cited by [2] (1989).Google Scholar
  26. 26.
    M. A. Yerry and M. S. Shephard. A modified quadtree approach to finite element mesh generation.IEEE Comput. Graphics and Applications 3 (1983), 39–46.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • David Eppstein
    • 1
  1. 1.Department of Information and Computer ScienceUniversity of CaliforniaIrvineUSA

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