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, 2, 3, 3, 3, 2, 2, 3, 5, 8, 6, 5, 3, 4, 7, 10, 21, 21

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

Table of n, a(n) for n=0..20.

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

Adjacent sequences: A366541 A366542 A366543 * A366545 A366546 A366547

Keyword

nonn,hard,more

Author

Denis Ivanov, Oct 12 2023

Status

approved