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A366544 a(n) is a lower bound for the number of distinct stable centroidal Voronoi tessellations (CVTs) of a square with n generators (seeds). 1
1, 1, 1, 1, 2, 3, 3, 3, 2, 2, 3, 5, 8, 6, 5, 3, 4, 7, 10, 21, 21 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
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.
REFERENCES
Lin Lu, F. Sun, and H. Pan, Global optimization Centroidal Voronoi Tessellation with Monte Carlo Approach, 2012 IEEECS Log Number TVCG-2011-03-0067.
LINKS
Denis Ivanov, Code, explanations and results (github).
J. C. Hateley, H. Wei, and L. Chen, Fast Methods for Computing Centroidal Voronoi Tessellations, J. Sci. Comput., 63, pp. 185-212, 2015.
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.
Wikipedia, Centroidal Voronoi tessellation (unfortunately, article is a stub and contains inaccuracies).
Wikipedia, Lloyd's algorithm.
EXAMPLE
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.
CROSSREFS
Cf. A363822 (disk).
Sequence in context: A185437 A335660 A210681 * A096520 A236552 A352628
KEYWORD
nonn,hard,more
AUTHOR
Denis Ivanov, Oct 12 2023
STATUS
approved

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Last modified May 28 01:34 EDT 2024. Contains 372900 sequences. (Running on oeis4.)