Boltzmann Samplers, P\'olya Theory, and Cycle Pointing
Abstract
We introduce a general method to count unlabeled combinatorial structures and to efficiently generate them at random. The approach is based on pointing unlabeled structures in an "unbiased" way that a structure of size n gives rise to n pointed structures. We extend Polya theory to the corresponding pointing operator, and present a random sampling framework based on both the principles of Boltzmann sampling and on P\'olya operators. All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results. Our method is illustrated on several examples: in each case, we provide enumerative results and efficient random samplers. The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families of planar maps.
Cite
@article{arxiv.1003.4546,
title = {Boltzmann Samplers, P\'olya Theory, and Cycle Pointing},
author = {Manuel Bodirsky and Éric Fusy and Mihyun Kang and Stefan Vigerske},
journal= {arXiv preprint arXiv:1003.4546},
year = {2011}
}
Comments
41 pages. This is an extended and revised journal version of a conference paper with the title "An unbiased pointing operator for unlabeled structures, with applications to counting and sampling", Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA'07), 2007, New Orleans. The second Arxiv version incorporates many improvements suggested by anonymous referees of the journal version