Boltzmann Samplers for the Random Generation of Combinatorial Structures

Abstract

This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class-an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on real-arithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.

An Experimental Study on Generating Planar Graphs

“…Fusy. The planar graph generator developed by Fusy [13] is based on the principles of a Boltzmann Sampler [11]. Labeled graphs of size n are drawn uniformly at random.…”

Boltzmann sampling of ordered structures

“…Boltzmann model is a very useful tool to efficiently generate combinatorial structures [4,5]. In particular, it is possible to automatically build a sampler according to the specification of a combinatorial class, following recursively the rules described in figure 1 or in [4].…”

“…In the original paper [4], it is proved that, under some conditions on the analytical nature of the generating function, we only need a constant number of trials for approximate-size sampling. Here, these conditions are often true, and the generation stays linear.…”

“…In 2004, Duchon, Flajolet, Louchard and Schaeffer [4] proposed a new model, called Boltzmann model, which leads to systematically construct samplers for random generation of objects in combinatorial classes described by specification systems. This framework has two main features: uniformity (two objects of the same size will have equal chances of being drawn) and quasi-linear time complexity.…”

Random Generation of Deterministic Acyclic Automata Using the Recursive Method

“…For instance, recursive methods [8] or Boltzmann samplers [7], which have been used for deterministic automata [6,2,9], rely on a good recursive description of the input, which is not known for acyclic automata. To our knowledge, the only combinatorial result on acyclic automata is due to Liskovets [11], who gave a close formula for the number of acyclic automata, but which cannot be directly translate into a good recursive description.…”