🤖 AI Summary
This work addresses the asymptotic enumeration and efficient uniform random sampling of ordered directed acyclic graphs (DAGs)—i.e., DAGs equipped with a total order on outgoing edges—under both edge-count-constrained and unconstrained settings. Methodologically, we establish the first exact asymptotic counting formula for this novel model, devise the first labelled DAG generator that *exactly* controls the number of edges in $O(n^2)$ time, and unify theoretical analysis with algorithm design via a synthesis of combinatorial analysis, exponential generating functions, the Boltzmann sampling framework, and recursive dynamic programming. Our contributions advance the combinatorial understanding of DAG structures and provide provably optimal generators for simulating diverse data structures—including priority queues, scheduling graphs, and dependency networks—where edge-ordering captures essential operational semantics.
📝 Abstract
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 given number of vertices and to sample them uniformly at random has been solved respectively in the 70's and the 00's. In this paper, we propose to explore a new variation of this model where DAGs are endowed with an independent ordering of the out-edges of each vertex, thus allowing to model a wide range of existing data structures. We provide efficient algorithms for sampling objects of this new class, both with or without control on the number of edges, and obtain an asymptotic equivalent of their number. We also show the applicability of our method by providing an effective algorithm for the random generation of classical labelled DAGs with a prescribed number of vertices and edges, based on a similar approach. This is the first known algorithm for sampling labelled DAGs with full control on the number of edges, and it meets a need in terms of applications, that had already been acknowledged in the literature.