Years and Authors of Summarized Original Work
1998; Levcopoulos, Krznaric
Problem Definition
Given a set S of n points in the Euclidean plane, a triangulation T of S is a maximal set of nonintersecting 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 is NP-hard [12]. Their proof of NP-completeness is not given explicitly; it relies on extensive calculations which they performed with a computer. Remy and Steger have shown a quasi-polynomial time approximation scheme for MWT [13]. These results are stated in the following theorem:
Theorem 1
The problem of computing the MWT (minimum weight...
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Beirouti R, Snoeyink J (1998) Implementations of the LMT heuristic for minimum weight triangulation. In: Symposium on computational geometry, Minneapolis, 7–10 June 1998, pp 96-105
Borgelt C, Grantson M, Levcopoulos C (2008) Fixed parameter algorithms for the minimum weight triangulation problem. Int J Comput Geom Appl 18(3):185–220
de Berg M, van Kreveld M, Overmars M, Schwarzkopf O (2000) Computational geometry – algorithms and applications, 2nd edn. Springer, Heidelberg
Grantson M, Borgelt C, Levcopoulos C (2005) Minimum weight triangulation by cutting out triangles. In: Proceedings of the 16th annual international symposium on algorithms and computation (ISAAC 2005), Sanya. Lecture notes in computer science, vol 3827. Springer, Heidelberg, pp 984–994
Gudmundsson J, Levcopoulos C (2000) A parallel approximation algorithm for minimum weight triangulation. Nord J Comput 7(1):32–57
Hjelle Ø, Dæhlen M (2006) Triangulations and applications. In: Mathematics and visualization, vol IX. Springer, Heidelberg. ISBN:978-3-540-33260-2
Levcopoulos C, Krznaric D (1998) Quasi-greedy triangulations approximating the minimum weight triangulation. J Algorithms 27(2):303–338
Levcopoulos C, Krznaric D (1999) The greedy triangulation can be computed from the Delaunay triangulation in linear time. Comput Geom 14(4):197–220
Levcopoulos C, Lingas A (1987) On approximation behavior of the greedy triangulation for convex polygons. Algorithmica 2:15–193
Levcopoulos C, Lingas A (2014) A note on a QPTAS for maximum weight triangulation of planar point sets. Inf Process Lett 114:414–416
Lingas A (1998) Subexponential-time algorithms for minimum weight triangulations and related problems. In: Proceedings 10th Canadian conference on computational geometry (CCCG), McGill University, Montreal, 10–12 Aug 1998
Mulzer W, Rote G (2006) Minimum-weight triangulation is NP-hard. In: Proceedings of the 22nd annual ACM symposium on computational geometry (SoCG’06), Sedona. ACM, New York. The journal version in J ACM 55(2):Article No. 11 (2008)
Remy J, Steger A (2006) A quasi-polynomial time approximation scheme for minimum weight triangulation. In: Proceedings of the 38th ACM symposium on theory of computing (STOC’06), Seattle. ACM, New York. The journal version in J ACM 56(3):Article No. 15 (2009)
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Levcopoulos, C. (2016). Minimum Weight Triangulation. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_241
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DOI: https://doi.org/10.1007/978-1-4939-2864-4_241
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