Boltzmann Sampling of Unlabelled Structures

Abstract

Boltzmann models from statistical physics combined with methods from analytic combinatorics give rise to efficient algorithms for the random generation of unlabelled objects. The resulting algorithms generate in an unbiased manner discrete configurations that may have nontrivial symmetries, and they do so by means of real-arithmetic computations. We present a collection of construction rules for such samplers, which applies to a wide variety of combinatorial classes, including integer partitions, necklaces, unlabelled functional graphs, dictionaries, series-parallel circuits, term trees and acyclic molecules obeying a variety of constraints, and so on. Under an abstract real-arithmetic computation model, the algorithms are, for many classical structures, of linear complexity provided a small tolerance is allowed on the size of the object drawn. As opposed to many of their discrete competitors, the resulting programs routinely make it possible to generate random objects of sizes in the range 10 4 -10 6 .

On the diversity of pattern distributions in rational language.

“…Contrasting with the so-called recursive method, the key idea here was to draw objects of any size within a Boltzmann-induced distribution of parameter x, and reject those of unsuitable sizes. A careful fine-tuning of the parameter x allowed for Θ(n 2 ) exact-size and Θ(n) approximate-size samplers for a large number of operators [15], later extended by subsequent efforts [16,14,12,13]. In particular, two of the authors generalized this idea to multidimensional objects [11] and proved the effectiveness of the Boltzmann generator for the regular and context-free languages when the limit law of the parameters is a multidimensional Gaussian law.…”

Random Generation of Finitely Generated Subgroups of a Free Group

“…Corollary 4.1 shows that the mean value of the rank of H is E(rank) = (|A| − 1)n − |A| √ n + 1, with variance σ 2 (rank) = o(n). If rank(H) ≤ k, then in particular |rank(H) − E(rank)| ≥ E(rank) − k. It follows, by Chebyshev's inequality (see Equation (8) …”

“…A method to systematically produce Boltzmann samplers was recently introduced by Duchon, Flajolet, Louchard and Schaeffer [7] for labeled structures (Flajolet, Fusy and Pivoteau for unlabeled structures in [8]). The evaluation of x is the only required precomputation and the complexity of generation itself is linear as long as small variations in size are allowed.…”

Scaling limit of random k-trees

“…They were introduced by Duchon, Flajolet, Louchard and Schaeffer [18] and were further developed by Flajolet, Fusy and Pivoteau [23]. Here we refer the readers to their papers [18,23] for a detailed description of the Boltzmann samplers. We just mention that the Boltzmann sampler ΓM (x) is a random generator which chooses an object c ∈ M with probability…”