Birthday paradox, coupon collectors, caching algorithms and self-organizing search
Discrete Applied Mathematics
A calculus for the random generation of labelled combinatorial structures
Theoretical Computer Science
Uniform Random Generation of Balanced Parenthesis Strings
ACM Transactions on Programming Languages and Systems (TOPLAS)
Boltzmann Samplers for the Random Generation of Combinatorial Structures
Combinatorics, Probability and Computing
Analytic Combinatorics
Asymptotic analysis and random sampling of digitally convex polyominoes
DGCI'13 Proceedings of the 17th IAPR international conference on Discrete Geometry for Computer Imagery
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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.