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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 (list; graph; refs; listen; history; text; internal format)
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 two-step semi-manual 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 10.1007/10915-014-9894-1
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, 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 TVCG-2011-03-0067.
LINKS
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 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
Cf. A366544 (square).
Sequence in context: A070548 A209628 A132011 * A054893 A090617 A053693
KEYWORD
nonn,hard,more
AUTHOR
Denis Ivanov, Oct 18 2023
STATUS
approved

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Last modified July 23 19:18 EDT 2024. Contains 374553 sequences. (Running on oeis4.)