How to generate an object under an ordinary Boltzmann distribution via an exponential Boltzmann sampler

This short note presents an efficient way to derive from an exponential Boltzmann sampler a ordinary Boltzmann sampler

💡 Research Summary

The paper presents a simple and efficient method to convert an exponential Boltzmann sampler into an ordinary Boltzmann sampler for any combinatorial class C. Boltzmann samplers, introduced by Duchon, Flajolet, Louchard and Schaeffer, are a cornerstone of random generation of combinatorial structures. Two variants exist: the exponential sampler (\hat\Gamma_x C) draws an object (\omega) of size n with probability (x^n/(n!,\hat C(x))), where (\hat C(x)) is the exponential generating function (EGF) of C; the ordinary sampler (\Gamma_x C) draws with probability (x^n/C(x)), where (C(x)) is the ordinary generating function (OGF). The main difficulty is the factor (n!) present in the exponential case but absent in the ordinary case.

The key idea is to randomise the Boltzmann parameter itself. Define a continuous probability density on (u\ge0) by
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