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A Fixed Parameter Algorithm for the Minimum Number Convex Partition Problem

  • Magdalene Grantson
  • Christos Levcopoulos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3742)

Abstract

Given an input consisting of an n-vertex convex polygon with k hole vertices or an n-vertex planar straight line graph (PSLG) with k holes and/or reflex vertices inside the convex hull, the parameterized minimum number convex partition (MNCP) problem asks for a partition into a minimum number of convex pieces. We give a fixed-parameter tractable algorithm for this problem that runs in the following time complexities:

– linear time if k is constant,

– time polynomial in n if \(k=O(\frac{{\rm log}n}{{\rm log log}n})\),

or, to be exact, in O(n Open image in new window k \(^{\rm 6{\it k}-5}\) Open image in new window 216k ) time.

Keywords

Convex Polygon Dynamic Programming Algorithm Clockwise Order Solution Table Current Labelling 
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. 1.
    Downey, R., Fellows, M.: Parameterized Complexity. Springer, New York (1999)Google Scholar
  2. 2.
    Garcia, A., Noy, M., Tejel, J.: Lower Bounds on the Number of Crossing-free Subgraphs of K N. In: Computational Geometry, Theory and Applications, vol. 16, pp. 211–221. Elsevier Science, Amsterdam (2000)Google Scholar
  3. 3.
    Grantson, M.: Fixed-Parameter Algorithms and Other Results for Optimal Convex Partitions. LU-CS-TR:2004-231, ISSN 1650-1276 Report 152. Lund University, Sweden (2004)Google Scholar
  4. 4.
    Greene, D.: The Decomposition of Polygons into Convex Parts. In: Preparata, F.P. (ed.) Computational Geometry, Adv. Comput. Res., vol. 1, pp. 235–259. JAI Press, London (1983)Google Scholar
  5. 5.
    Keil, J.: Decomposing a Polygon into Simpler Components. SIAM Journal on Computing. Society of Industrial and Applied Mathematics 14, 799–817 (1985)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Keil, J., Snoeyink, J.: On the Time Bound for Convex Decomposition of Simple Polygons. In: proceedings of the 10th Canadian Conference on Computational Geometry, Montreal, Canada, pp. 54–55 (1998)Google Scholar
  7. 7.
    Lingas, A.: The Power of Non-Rectilinear Holes. LNCS, vol. 140, pp. 369–383. Springer, Heidelberg (1982)Google Scholar
  8. 8.
    Santos, F., Seidel, R.: A Better Upper Bound on the Number of Triangulations of a Planar Point Set. arXiv:math.CO/0204045 v2 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Magdalene Grantson
    • 1
  • Christos Levcopoulos
    • 1
  1. 1.Department of Computer ScienceLund UniversityLundSweden

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