The set of binary decision diagrams , an efficient data structure representing Boolean functions, is extensively used in many distinct contexts like model verification, machine learning, cryptography, and resolution of combinatorial problems. The most famous variant, called reduced ordered binary decision diagram ( robdd ), can be viewed as the result of a specific compaction of a complete decision tree. A great property is that, once an order over the Boolean variables is fixed, each Boolean function is represented by exactly one robdd . In this article, we aim at computing the exact distribution of the Boolean functions in \( k \) variables according to the robdd size . Recall the number of Boolean functions with \( k \) variables is equal to \(2^{2^{k}}\!,\) which is of double exponential growth with respect to the number of variables. The maximal size of an robdd with \( k \) variables is \(M_{k}\approx 2^{k}/k\) . In this article, we develop the first polynomial algorithm to derive the distribution of Boolean functions over \( k \) variables with respect to robdd size denoted by \( n \) . It performs \(O(k\;n^{3}\log n)\) arithmetic operations on integers and necessitates to store \(O(n^{2})\) integers in memory storage; note that the maximal size of integers involved in the computations is \(O(k\;2^{k})\) bits. Our new approach relies on a decomposition of robdd s layer by layer and on an enumerative inclusion-exclusion argument.

The Twelvefold Way represents Rota’s classification, addressing the most fundamental enumeration problems and their associated combinatorial counting formulas. These distinct problems are connected to enumerating functions defined from a set of elements denoted by 𝒩 into another one 𝒦. The counting solutions for the twelve problems are well known. We are interested in unranking algorithms. Such an algorithm is based on an underlying total order on the set of structures we aim at constructing. By taking the rank of an object, i.e. its number according to the total order, the algorithm outputs the structure itself after having built it. One famous total order is the lexicographic order: it is probably the one that is the most used by people when one wants to order things. While the counting solutions for Rota’s classification have been known for years it is interesting to note that three among the problems have yet no lexicographic unranking algorithm. In this paper we aim at providing algorithms for the last three cases that remain without such algorithms. After presenting in detail the solution for set partitions associated with the famous Stirling numbers of the second kind, we explicitly explain how to adapt the algorithm for the two remaining cases. Additionally, we propose a detailed and fine-grained complexity analysis based on the number of bitwise arithmetic operations.

For three decades binary decision diagrams, a data structure efficiently representing Boolean functions, have been widely used in many distinct contexts like model verification, machine learning, cryptography and also resolution of combinatorial problems. The most famous variant, called reduced ordered binary decision diagram (robdd for short), can be viewed as the result of a compaction procedure on the full decision tree. A useful property is that once an order over the Boolean variables is fixed, each Boolean function is represented by exactly one robdd. In this paper we aim at computing the {exact distribution of the Boolean functions in k variables according to the robdd size}, where the robdd size is equal to the number of decision nodes of the underlying directed acyclic graph (dag) structure. Recall the number of Boolean functions with k variables is equal to 2^{2^k}, which is of double exponential growth with respect to the number of variables. The maximal size of a robdd with k variables is M_k ≈ 2^k / k. Apart from the natural combinatorial explosion observed, another difficulty for computing the distribution according to size is to take into account dependencies within the dag structure of robdds. In this paper, we develop the first polynomial algorithm to derive the distribution of Boolean functions over k variables with respect to robdd size denoted by n. The algorithm computes the (enumerative) generating function of robdds with k variables up to size n. It performs O(k n⁴) arithmetical operations on integers and necessitates storing O((k+n) n²) integers with bit length O(nlog n). Our new approach relies on a decomposition of robdds layer by layer and on an inclusion-exclusion argument.

Directed Acyclic Graphs (DAGs) are directed graphs in which there is no path from a vertex to itself. DAGs are an omnipresent data structure in computer science and the problem of counting the DAGs of a given number of vertices has been solved in the 70’s by Robinson. In many applications one needs to construct connected DAGs and to control their number of edges, but the adaptation of Robinson’s enumeration to take this into account led to counting formulas based on the inclusion-exclusion principle, inducing a high computational cost for the uniform random sampling of DAGs based on this formula. In the present paper we propose two contributions. First we enumerate a new class of DAGs, enriched with an independent ordering of the children of each vertex, according to their numbers of vertices and edges. We obtain a constructive recursive counting formula for them (i.e. without using the inclusion-exclusion principle) using a new decomposition scheme. Then we show the applicability of our method by proposing a constructive enumeration of Robinson’s labelled DAGs, by vertices and edges, based on the same decomposition. As a consequence we are able to derive efficient uniform random samplers for both models.

In this paper we focus on concurrent processes built on synchronization by means of futures. This concept is an abstraction for processes based on a main execution thread but allowing to delay some computations. The structure of a general concurrent process is a directed acyclic graph (DAG). Since the quantitative study of increasingly labeled DAG (directly related to processes) seems out of reach (this is a #P-complete problem), we restrict ourselves to the study of arch processes, a simplistic model of processes with futures. They are based on two parameters related to their sizes and their numbers of arches. The increasingly labeled structures seems not to be specifiable in the classical sense of Analytic Combinatorics, but we manage to derive a recurrence equation for the enumeration. For this model we first exhibit an exact and an asymptotic formula for the number of runs of a given process. The second main contribution is composed of a uniform random sampler algorithm and an unranking one that allow efficient generation and exhaustive enumeration of the runs of a given arch process.

A deep connection exists between the interleaving semantics of concurrent processes and increasingly labelled combinatorial structures. In this paper we further explore this connection by studying the rich combinatorics of partially increasing structures underlying the operator of non-deterministic choice. Following the symbolic method of analytic combinatorics, we study the size of the computation trees induced by typical non-deterministic processes, providing a precise quantitative measure of the so-called "combinatorial explosion" phenomenon. Alternatively, we can see non-deterministic choice as encoding a family of tree-like partial orders. Measuring the (rather large) size of this family on average offers a key witness to the expressiveness of the choice operator. As a practical outcome of our quantitative study, we describe an efficient algorithm for generating computation paths uniformly at random.