Minimum Weight Triangulation by Cutting Out Triangles

  • Magdalene Grantson
  • Christian Borgelt
  • Christos Levcopoulos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3827)


We describe a fixed parameter algorithm for computing the minimum weight triangulation (MWT) of a simple polygon with (nk) vertices on the perimeter and k hole vertices in the interior, that is, for a total of n vertices. Our algorithm is based on cutting out empty triangles (that is, triangles not containing any holes) from the polygon and processing the parts or the rest of the polygon recursively. We show that with our algorithm a minimum weight triangulation can be found in time at most O(n 3 k ! k), and thus in O(n 3) if k is constant. We also note that k! can actually be replaced by b k for some constant b. We implemented our algorithm in Java and report experiments backing our analysis.


Time Complexity Parameter Algorithm Simple Polygon Screen Shot Index Word 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aichholzer, O., Rote, G., Speckmann, B., Streinu, I.: The Zigzag Path of a Pseudo-Triangulation. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 377–388. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Downey, R., Fellows, M.: Parameterized Complexity. Springer, New York (1999)Google Scholar
  3. 3.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to Theory of NP-Completeness. Freeman, New York (1979)zbMATHGoogle Scholar
  4. 4.
    Gilbert, P.D.: New Results in Planar Triangulations. Report R-850. University of Illinois, Coordinated Science Lab. (1979)Google Scholar
  5. 5.
    Grantson, M., Borgelt, C., Levcopoulos, C.: A Fixed Parameter Algorithm for Minimum Weight Triangulation: Analysis and Experiments. Technical Report LU-CS-TR:2005-234, ISSN 1650-1276 Report 154. Lund University, Sweden (2005)Google Scholar
  6. 6.
    Hoffmann, M., Okamoto, Y.: The Minimum Triangulation Problem with Few Inner Points. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 200–212. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Klincsek, G.T.: Minimal Triangulations of Polygonal Domains. In: Annals of Discrete Mathematics, pp. 121–123. ACM Press, New York (1980)Google Scholar
  8. 8.
    Lodi, E., Luccio, F., Mugnai, C., Pagli, L.: On Two-Dimensional Data Organization, Part I. Fundaments Informaticae 2, 211–226 (1979)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Lubiw, A.: The Boolean Basis Problem and How to Cover Some Polygons by Rectangles. SIAM Journal on Discrete Mathematics 3, 98–115 (1990)Google Scholar
  10. 10.
    Moitra, D.: Finding a Minimum Cover for Binary Images: An Optimal Parallel Algorithm. Algorithmica 6, 624–657 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Plaisted, D., Hong, J.: A Heuristic Triangulation Algorithm. Journal of Algorithms 8, 405–437 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Sharir, M., Welzl, E.: On the Number of Crossing-Free Matchings (Cycles and Partitions). In: Proc. 17th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006 (2006) (to appear)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Magdalene Grantson
    • 1
  • Christian Borgelt
    • 2
  • Christos Levcopoulos
    • 1
  1. 1.Department of Computer ScienceLund UniversityLundSweden
  2. 2.Department of Knowledge Processing and Language EngineeringUniversity of MagdeburgMagdeburgGermany

Personalised recommendations