Skip to main content
SpringerLink
Log in

A Tight Bound for β-Skeleton of Minimum Weight Triangulations

  • Conference paper
  • First Online:
Algorithms and Data Structures (WADS 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1663))

Included in the following conference series:

Abstract

In this paper, we prove a tight bound for β value \( \left( {\beta = \frac{{\sqrt {2\sqrt 3 + 9} }} {3}} \right) \) ) such that being less than this value,the β-skeleton of a planar point set may not belong to the minimum weight triangulation of this set, while being equal to or greater than this value, the β-skeleton always belongs to the minimum weight triangulation. Thus, we settled the conjecture of the tight bound for β-skeleton of minimum weight triangulation by Mark Keil. We also present a new sufficient condition for identifying a subgraph of minimum weight triangulation of a planar n-point set. The identified subgraph could be different from all the known subgraphs, and the subgraph can be found in O(n 2 log n) time.

This work is partially supported by NSERC grant OPG0041629.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Anagnostou and D. Corneil, Polynomial time instances of the minimum weight triangulation problem, Computational Geometry: Theory and applications, vol. 3, pp. 247–259, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  2. S.-W. Cheng, M. Golin and J. Tsang, Expected case analysis of b-skeletons with applications to the construction of minimum weight triangulations, CCCG Conference Proceedings, P.Q., Canada, pp. 279–284, 1995.

    Google Scholar 

  3. P. Bose, L. Devroye, and W. Evens, Diamonds are not a minimum weight triangulation’s best friend, Proceedings of 8th CCCG, 1996, Ottawa, pp. 68–73.

    Google Scholar 

  4. S.-W. Cheng and Y.-F. Xu, Approaching the largest β-skeleton within the minimum weight triangulation, Proc. 12th Ann. Symp. Computational Geometry, Philadelphia, Association for Computing Machinery, 1996.

    Google Scholar 

  5. Y. Dai and N. Katoh, On computing new classes of optimal triangulations with angular constraints, Proceedings on 4th annual international conference of Computing and Combinatorics, LNCS 1449, pp.15–24.

    Google Scholar 

  6. M. T. Dickerson, M. H. Montague, The exact minimum weight triangulation, Proc. 12th Ann. Symp. Computational Geometry, Philadelphia, Association for Computing Machinery, 1996.

    Google Scholar 

  7. P. D. Gilbert, New results in planar triangulations, TR-850, University of Illinois Coordinated science Lab, 1979.

    Google Scholar 

  8. M. Garey and D. Johnson, Computer and Intractability. A guide to the theory of NP-completeness, Freeman, 1979.

    Google Scholar 

  9. J. M. Keil, Computing a subgraph of the minimum weight triangulation, Computational Geometry: Theory and Applications pp. 13–26, 4 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Kirkpatrick and J. Radke, A framework for computational morphology, in G. Toussaint, ed., Computational Geometry, Elsevier, 1985, pp. 217–248.

    Google Scholar 

  11. G. Klinesek, Minimal triangulations of polygonal domains, Ann. Discrete Math., pp. 121–123, 9 (1980).

    Article  MathSciNet  Google Scholar 

  12. H. Meijer and D. Rappaport, Computing the minimum weight triangulation of a set of linearly ordered points, Information Processing Letters, vol. 42, pp. 35–38, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. O’Rourke, Computational Geometry In C, Cambridge University Press, 1993.

    Google Scholar 

  14. A. Mirzain, C. Wang and Y. Xu, On stable line segments in triangulations, Proceedings of 8th CCCG, Ottawa, 1996, pp.68–73.

    Google Scholar 

  15. C. Wang and Y. Xu, Minimum weight triangulations with convex layers constraint, to appear J. of Global Optimization.

    Google Scholar 

  16. C. Wang, F. Chin, and Y. Xu, A new subgraph of Minimum weight triangulations, J. of Combinational Optimization Vol 1, No. 2, pp. 115–127.

    Google Scholar 

  17. Y. Xu, D. Zhou, Improved heuristics for the minimum weight triangulation, Acta Mathematics Applicatae Sinica, vol. 11, no. 4, pp. 359–368, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  18. B. Yang, A better subgraph of the minimum weight triangulation, The IPL, Vol.56, pp. 255–258.

    Google Scholar 

  19. B. Yang, Y. Xu and Z. You, A chain decomposition algorithm for the proof of a property on minimum weight triangulations, Proc. 5th International Symposium on Algorithms and Computation (ISAAC’94), LNCS 834, Springer-Verlag, pp. 423–427, 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Memorial University of Newfoundland, St.John’s, NFLD, Canada, A1B 3X5

    Cao An Wang & Boting Yang

Authors
  1. Cao An Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boting Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, Canada, K1S 5B6

    Frank Dehne  & Jörg-Rüdiger Sack  & 

  2. School of Computer Science, Simon Fraser University, Burnaby, BC, Canada, V5A 1S6

    Arvind Gupta

  3. Center for Geometric Computing Providence, Brown University, RI, 02912-1910, USA

    Roberto Tamassia

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Wang, C.A., Yang, B. (1999). A Tight Bound for β-Skeleton of Minimum Weight Triangulations. In: Dehne, F., Sack, JR., Gupta, A., Tamassia, R. (eds) Algorithms and Data Structures. WADS 1999. Lecture Notes in Computer Science, vol 1663. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48447-7_27

Download citation

  • DOI: https://doi.org/10.1007/3-540-48447-7_27

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66279-2

  • Online ISBN: 978-3-540-48447-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation