Abstract
We describe a fixed parameter algorithm for computing the minimum weight triangulation (MWT) of a simple polygon with (n–k) 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.
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References
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)
Downey, R., Fellows, M.: Parameterized Complexity. Springer, New York (1999)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to Theory of NP-Completeness. Freeman, New York (1979)
Gilbert, P.D.: New Results in Planar Triangulations. Report R-850. University of Illinois, Coordinated Science Lab. (1979)
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)
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)
Klincsek, G.T.: Minimal Triangulations of Polygonal Domains. In: Annals of Discrete Mathematics, pp. 121–123. ACM Press, New York (1980)
Lodi, E., Luccio, F., Mugnai, C., Pagli, L.: On Two-Dimensional Data Organization, Part I. Fundaments Informaticae 2, 211–226 (1979)
Lubiw, A.: The Boolean Basis Problem and How to Cover Some Polygons by Rectangles. SIAM Journal on Discrete Mathematics 3, 98–115 (1990)
Moitra, D.: Finding a Minimum Cover for Binary Images: An Optimal Parallel Algorithm. Algorithmica 6, 624–657 (1991)
Plaisted, D., Hong, J.: A Heuristic Triangulation Algorithm. Journal of Algorithms 8, 405–437 (1987)
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)
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Grantson, M., Borgelt, C., Levcopoulos, C. (2005). Minimum Weight Triangulation by Cutting Out Triangles. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_98
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DOI: https://doi.org/10.1007/11602613_98
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30935-2
Online ISBN: 978-3-540-32426-3
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