🤖 AI Summary
This study addresses the problem of efficiently generating uniformly random directed acyclic graphs (DAGs) of a fixed size. The authors propose two novel algorithms, one of which constitutes the first asymptotically optimal exact-size sampler for DAGs, achieving an expected time complexity of $\frac{n^2}{2} + o(n^2)$. This method extends the Boltzmann sampling framework by integrating structural decompositions of DAGs with generating function techniques and an optimized strategy for random number usage. Compared to the current state-of-the-art, the proposed approach yields significant improvements both in theoretical time complexity and practical runtime performance, thereby enabling, for the first time, uniform and efficient sampling of DAGs at a prescribed scale.
📝 Abstract
We propose two efficient algorithms for generating uniform random directed acyclic graphs, including an asymptotically optimal exact-size sampler that performs $\frac{n^2}{2} + o(n^2)$ operations and requests to a random generator. This was achieved by extending the Boltzmann model for graphical generating functions and by using various decompositions of directed acyclic graphs. The presented samplers improve upon the state-of-the-art algorithms in terms of theoretical complexity and offer a significant speed-up in practice.