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A366544
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a(n) is a lower bound for the number of distinct stable centroidal Voronoi tessellations (CVTs) of a square with n generators (seeds).
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1
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1, 1, 1, 1, 2, 3, 3, 3, 2, 2, 3, 5, 8, 6, 5, 3, 4, 7, 10, 21, 21
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OFFSET
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0,5
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COMMENTS
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Stable CVTs are local minimizers of the CVT function (see first paper).
There are other CVTs which are saddle points.
Lloyd's process converges only to stable CVTs.
An efficient two-step semi-manual algorithm is used to recognize identical patterns and a fast code for the Lloyd's process.
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REFERENCES
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Lin Lu, F. Sun, and H. Pan, Global optimization Centroidal Voronoi Tessellation with Monte Carlo Approach, 2012 IEEECS Log Number TVCG-2011-03-0067.
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LINKS
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Yang Liu, Wenping Wang, Bruno Lévy, Feng Sun, Dong-Ming Yan, Lin Lu, and Chenglei Yang, On centroidal Voronoi tessellation—Energy smoothness and fast computation, ACM Transactions on Graphics, Volume 28, Issue 4, Article No. 101, pp. 1-17, 2009.
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EXAMPLE
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As initialization, clustering centers for a large number of points in the square are used. For every set of centers, Lloyd's algorithm is iterated and all variants symmetric with respect to rotations and reflections are removed.
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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