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Maximum Weight Triangulation and Its Application on Graph Drawing

  • Cao An Wang
  • Francis Y. Chin
  • Bo Ting Yang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1449)

Abstract

In this paper, we investigate the maximum weight triangulation of a polygon inscribed in a circle (simply inscribed polygon). A complete characterization of maximum weight triangulation of such polygons has been obtained. As a consequence of this characterization, an O(n 2) algorithm for finding the maximum weight triangulation of an inscribed n-gon is designed. In case of a regular polygon, the complexity of this algorithm can be reduced to O(n). We also show that a tree admits a maximum weight drawing if its internal node connects at most 2 non-leaf nodes. The drawing can be done in O(n) time. Furthermore, we prove a property of maximum planar graphs which do not admit a maximum weight drawing on any set of convex points.

Keywords

Span Tree Maximum Weight Minimum Weight Boundary Edge Regular 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.

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References

  1. [BeEp92]
    Bern M. and Eppstein D., Mesh generation and optimal triangulation, Technical Report, Xerox Palo Alto Research Center, 1992.Google Scholar
  2. [DETT94]
    Di Battista G., Eades P., Tamassia R., and Tollos I., Algorithms for automatic graph drawing: A annotated bibliography, Computational Geometry: Theory and Applications, 4(1994), pp. 235–282.zbMATHMathSciNetGoogle Scholar
  3. [DM96]
    Dickerson M., Montague M., The exact minimum weight triangulation, Proc. 12th Ann. Symp. Computational Geometry, Philadelphia, Association for Computing Machinery, 1996, pp..Google Scholar
  4. [EW94]
    Eades P. and Whitesides S., The realization problem for Euclidean minimum spanning tree is NP-hard, Proceedings of 10th ACM Symposium on Computational Geometry, Stony Brook, NY (1994), pp. 49–56.Google Scholar
  5. [Gilb79]
    Gilbert P., New results on planar triangulations, Tech. Rep. ACT-15 (1979), Coord. Sci. Lab., University of Illinois at Urbana.Google Scholar
  6. [Keil94]
    Keil M., Computing a subgraph of the minimum weight triangulation, Computational Geometry: Theory and Applications, 4(1994), pp.13–26.zbMATHMathSciNetGoogle Scholar
  7. [Kirk80]
    Kirkpatrick D., A note on Delaunay and optimal triangulations, Information Processing Letters 10(3) (1980), pp. 127–128.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Kl80]
    Klincsek G., Minimal triangulations of polygonal domains, Annual Discrete Mathematics 9(1980) pp. 121–123.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [LeLi87]
    Levcopoulos C. and Lingas A., On approximation behavior of the greedy triangulation for convex polygon, Algorithmica 2 (1987), pp. 175–193.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [LeLi90]
    Lingas A., A new heuristic for minimum weight triangulation, SIAM J. Algebraic Discrete Methods, (1987), pp. 646–658.Google Scholar
  11. [LeLi96]
    Lenhart W. and Liotta G., Drawing outerplanar minimum weight triangulations, Information Processing Letters, 57 (1996) pp.253–260.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [MR92]
    Meijeri H. and Rappaport D., Computing the minimum weight triangulation of a set of linearly ordered points, Information Processing Letters, 42 (1992), pp. 35–38.CrossRefMathSciNetGoogle Scholar
  13. [MS91]
    Monna C. and Suri S., Transitions in geometric minimum spanning trees, Proceedings of 7th ACM Symposium on Computational Geometry, North Conway, NH (1991), pp.239–249.Google Scholar
  14. [PrSh85]
    Preparata F. and Shamos M., Computational Geometry (1985), Springer-Verlag.Google Scholar
  15. [WCX97]
    Wang C., Chin F., and Xu Y., A new subgraph of minimum weight triangulations, Journal of Combinatorial Optimization, Volume 1, No. 2 (1997), pp. 115–127.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Cao An Wang
    • 1
  • Francis Y. Chin
    • 2
  • Bo Ting Yang
    • 2
  1. 1.Department of Computer ScienceMemorial University of NewfoundlandSt. John’sCanada
  2. 2.Department of Computer ScienceUniversity of Hong KongHong Kong

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