Controlled non-uniform random generation of decomposable structures

Abstract

Consider a class of decomposable combinatorial structures, using different types of atoms Z = { Z 1 , … , Z ∣ Z ∣ } . We address the random generation of such structures with respect to a size n and a targeted distribution in k of its distinguished atoms. We consider two variations on this problem. In the first alternative, the targeted distribution is given by k real numbers μ 1 , … , μ k such that 0 < μ i < 1 for all i and μ 1 + ⋯ + μ k ≤ 1 . We aim to generate random structures among the whole set of structures of a given size n , in such a way that the expected frequency of any distinguished atom Z i equals μ i . We address this problem by weighting the atoms with a k -tuple π of real-valued weights, inducing a weighted distribution over the set of structures of size n . We first adapt the classical recursive random generation scheme into an algorithm taking O ( n 1 + o ( 1 ) + m n log n ) arithmetic operations to draw m structures from the π -weighted distribution. Secondly, we address the analytical computation of weights such that the targeted frequencies are achieved asymptotically, i.e. for large values of n . We derive systems of functional equations whose resolution gives an explicit relationship between π and μ 1 , … , μ k . Lastly, we give an algorithm in O ( k n 4 ) for the inverse problem, i.e. computing the frequencies associated with a given k -tuple π of weights, and an optimized version in O ( k n 2 ) in the case of context-free languages. This allows for a heuristic resolution of the weights/frequencies relationship suitable for complex specifications. In the second alternative, the targeted distribution is given by k natural numbers n 1 , … , n k such that n 1 + ⋯ + n k + r = n where r ≥ 0 is the number of undistinguished atoms. The structures must be generated uniformly among the set of structures of size n that contain exactly n i atoms Z i ( 1 ≤ i ≤ k ). We give a O ( r 2 ∏ i = 1 k n i 2 + m n k log n ) algorithm for generating m structures, which simplifies into a O ( r ∏ i = 1 k n i + m n ) for regular specifications.