Analytic Samplers and the Combinatorial Rejection Method

Boltzmann samplers, introduced by Duchon et al. in 2001, make it possible to uniformly draw approximate size objects from any class which can be specified through the symbolic method. This, through by evaluating the associated generating functions to obtain the correct branching probabilities. But these samplers require generating functions, in particular in the neighborhood of their sunglarity, which is a complex problem; they also require picking an appropriate tuning value to best control the size of generated objects. Although Pivoteau et al.have brought a sweeping question to the first question, with the introduction of their Newton oracle, questions remain. By adapting the rejection method, a classical tool from the random, we show how to obtain a variant of the Boltzmann sampler framework, which is tolerant of approximation, even large ones. Our goal for this is twofold: this allows for exact sampling with approximate values; but this also allows much more flexibility in tuning samplers. For the class of simple trees, we will show how this could be used to more easily calibrate samplers. Introduction Being able to randomly generate large objects of any given combinatorial class (for instance described by a grammar), is a fundamental problem with countless applications in scientific modeling. Nijenhuis and Wilf introduced the recursive method [16] in the late 70s (later extended by Flajolet et al. [12]), the first automatic random generation method; so termed automatic because it can directly derive random samplers from any combinatorial description—no bijection, no clever algorithm, no complicated equations are needed. The drawback is that this method is costly, notably in preprocessing: to compute the probabilities involved in generating an object of size n, the method requires knowing the complete enu∗LIPN, UMR 7030, Université Paris 13/CNRS, F-93430 Villetaneuse, France. †Department of Computer Science, Princeton University, 35 Olden Street, Princeton, NJ 08540, USA. meration of the combinatorial class up to size n; and predictably when n is large, this enumeration is significant both to calculate and to store. Enter Boltzmann sampling, introduced by Duchon et al. in 2002 [7, 8], of which the key insight was that a class’ enumeration is not required to compute the correct branching probabilities: instead, such probabilities can be obtained by evaluating the counting generating functions—for an unlabelled combinatorial class C, for which there are cn elements of size n, its counting generating function is defined as