Keywords and Synonyms
Minimum length triangulation
Problem Definition
Given a set S of n points in the Euclidean plane, a triangulation T of S is a maximal set of non-intersecting straight-line segments whose endpoints are in S. The weight of T is defined as the total Euclidean length of all edges in T. A triangulation that achieves minimum weight is called a minimum weight triangulation, often abbreviated MWT, of S.
Key Results
Since there is a very large number of papers and results dealing with minimum weight triangulation, only relatively very few of them can be mentioned here.
Mulzer and Rote have shown that MWT in NP-hard [11]. Their proof of NP-completeness is not given explicitly; it relies on extensive calculations which they performed with a computer. Also recently, Remy and Steger have shown a quasi-polynomial time approximation scheme for MWT [12]. These results are stated in the following theorem.
Theorem 1
The problem of computing the MWT (minimum weight triangulation) of...
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Recommended Reading
Beirouti, R., Snoeyink, J.: Implementations of the LMT Heuristic for Minimum Weight Triangulation. Symposium on Computational Geometry, pp. 96–105, Minneapolis, Minnesota, June 7–10, 1998
Borgelt, C., Grantson, M., Levcopoulos, C.: Fixed-Parameter Algorithms for the Minimum Weight Triangulation Problem. Technical Report LU-CS-TR:2006-238, ISSN 1650-1276 Report 158. Lund University, Lund (An extended version has been submitted to IJCGA) (2006)
de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry – Algorithms and Applications, 2nd edn. Springer, Heidelberg (2000)
Grantson, M., Borgelt, C., Levcopoulos, C.: Minimum Weight Triangulation by Cutting Out Triangles. In: Proceedings 16th Annual International Symposium on Algorithms and Computation, ISAAC 2005, Sanya, China, pp. 984–994. Lecture Notes in Computer Science, vol. 3827. Springer, Heidelberg (2005)
Gudmundsson, J., Levcopoulos, C.: A Parallel Approximation Algorithm for Minimum Weight Triangulation. Nordic J. Comput. 7(1), 32–57 (2000)
Hjelle, Ø., Dæhlen, M.: Triangulations and Applications. In: Mathematics and Visualization, vol. IX. Springer, Heidelberg (2006). ISBN 978-3-540-33260-2
Levcopoulos, C., Krznaric, D.: Quasi-Greedy Triangulations Approximating the Minimum Weight Triangulation. J. Algorithms 27(2), 303–338 (1998)
Levcopoulos, C., Krznaric, D.: The Greedy Griangulation can be Computed from the Delaunay Triangulation in Linear Time. Comput. Geom. 14(4), 197–220 (1999)
Levcopoulos, C., Lingas, A.: On Approximation Behavior of the Greedy Triangulation for Convex Polygons. Algorithmica 2, 15–193 (1987)
Lingas, A.: Subexponential-time algorithms for minimum weight triangulations and related problems. In: Proceedings 10th Canadian Conference on Computational Geometry (CCCG), McGill University, Montreal, Quebec, 10–12 August 1998
Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. In: Proceedings 22nd Annual ACM Symposium on Computational Geometry, SoCG'06, Sedona, AZ, USA. ACM Press, New York, NY, USA (2006)
Remy, J., Steger, A.: AÂ Quasi-Polynomial Time Approximation Scheme for Minimum Weight Triangulation. In: Proceedings 38th ACM Symposium on Theory of Computing (STOC'06). ACM Press, New York, NY, USA (2006)
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Levcopoulos, C. (2008). Minimum Weight Triangulation. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_241
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DOI: https://doi.org/10.1007/978-0-387-30162-4_241
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