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

Offset: 0

Author

Denis Ivanov, Oct 18 2023

Keywords

• nonn
• hard
• more

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.

Examples

			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.
		

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

• Denis Ivanov, Github with code, explanations and results.
• Wikipedia, Centroidal Voronoi tessellation (unfortunately, article is a stub and contains inaccuracies).
• Wikipedia, Lloyd's algorithm.

Crossrefs

Cf. A366544 (square).