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Consider a class of decomposable combinatorial structures, using differenttypes of atoms $\Atoms = \{\At_1,\ldots ,\At_{|{\Atoms}|}\}$. We address therandom generation of such structures with respect to a size $n$ and a targeteddistribution in $k$ of its \emph{distinguished} atoms. We consider twovariations on this problem. In the first alternative, the targeted distributionis given by $k$ real numbers $\TargFreq_1, \ldots, \TargFreq_k$ such that $0 <\TargFreq_i < 1$ for all $i$ and $\TargFreq_1+\cdots+\TargFreq_k \leq 1$. Weaim to generate random structures among the whole set of structures of a givensize $n$, in such a way that the {\em expected} frequency of any distinguishedatom $\At_i$ equals $\TargFreq_i$. We address this problem by weighting theatoms with a $k$-tuple $\Weights$ of real-valued weights, inducing a weighteddistribution over the set of structures of size $n$. We first adapt theclassical recursive random generation scheme into an algorithm taking$\bigO{n^{1+o(1)}+mn\log{n}}$ arithmetic operations to draw $m$ structures fromthe $\Weights$-weighted distribution. Secondly, we address the analyticalcomputation of weights such that the targeted frequencies are achievedasymptotically, i. e. for large values of $n$. We derive systems of functionalequations whose resolution gives an explicit relationship between $\Weights$and $\TargFreq_1, \ldots, \TargFreq_k$. Lastly, we give an algorithm in$\bigO{k n^4}$ for the inverse problem, {\it i.e.} computing the frequenciesassociated with a given $k$-tuple $\Weights$ of weights, and an optimizedversion in $\bigO{k n^2}$ in the case of context-free languages. This allowsfor a heuristic resolution of the weights/frequencies relationship suitable forcomplex specifications. In the second alternative, the targeted distribution isgiven by a $k$ natural numbers $n_1, \ldots, n_k$ such that$n_1+\cdots+n_k+r=n$ where $r \geq 0$ is the number of undistinguished atoms.The structures must be generated uniformly among the set of structures of size$n$ that contain {\em exactly} $n_i$ atoms $\At_i$ ($1 \leq i \leq k$). We givea $\bigO{r^2\prod_{i=1}^k n_i^2 +m n k \log n}$ algorithm for generating $m$structures, which simplifies into a $\bigO{r\prod_{i=1}^k n_i +m n}$ forregular specifications.

Controlled non uniform random generation of decomposable structures