

A363822


a(n) is the conjectured number of stable distinct centroidal Voronoi tessellations (CVTs) of a unit disk with n generators (seeds).


1



1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 6, 6, 5, 5, 5, 6, 9, 10
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OFFSET

0,5


COMMENTS

Stable CVTs are local minimizers of the CVT function (see Hateley, Wei,and Chen article).
There are other CVTs which are saddle points.
Lloyd's process converges only to stable CVTs from which different with respect to rotation symmetry are selected.
An efficient twostep semimanual algorithm is used to recognize identical patterns and a fast code for the Lloyd's process.
Code in Mathematica and details published on Github.


REFERENCES

J. C. Hateley, H. Wei, and L. Chen, Fast Methods for Computing Centroidal Voronoi Tessellations, 2014 J Sci Comput DOI 10.1007/1091501498941
Yang Liu, Wenping Wang, Bruno Lévy, Feng Sun, DongMing 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. 117, 2009, DOI 10.1145/1559755.1559758
Lin Lu, F. Sun, and H. Pan, Global optimization Centroidal Voronoi Tessellation with Monte Carlo Approach, 2012 IEEECS Log Number TVCG2011030067.


LINKS



EXAMPLE

As initialization, clustering centers for a large number of points in the unit disk are used. For every set of centers, Lloyd's algorithm is iterated and all variants symmetric with respect to rotations are removed.


CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



STATUS

approved



