Let $X_{n}$ be an ensemble of combinatorial structures of size $N$, equipped with a measure. Consider the algorithmic problem of exactly sampling from this measure. When this ensemble has a ‘combinatorial specification, the celebrated Boltzmann sampling algorithm allows to solve this problem with a complexity which is, typically, of order $N(3/2)$. Here, a factor $N$ is inherent to the problem, and implied by the Shannon bound on the average number of required random bits, while the extra factor $N$.