Boys-and-girls Birthdays and Hadamard Products

Abstract

Boltzmann models from statistical physics, combined with methods from analytic combinatorics, give rise to efficient and easy-to-write algorithms for the random generation of combinatorial objects. This paper proposes to extend Boltzmann generators to a new field of applications by uniformly sampling a Hadamard product. Under an abstract real-arithmetic computation model, our algorithm achieves approximate-size sampling in expected time ${\cal O}$(n${\sqrt n}$) or ${\cal O}$(nσ) depending on the objects considered, with σ the standard deviation of smallest order for the component object sizes. This makes it possible to generate random objects of large size on a standard computer. The analysis heavily relies on a variant of the so-called birthday paradox, which can be modelled as an occupancy urn problem.