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arXiv · 2608.20234

Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling Construction

Claude Gravel
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Abstract

Bernard and Letac (1971) introduced a method for uniform random sampling among m outcomes from an unknown biased source of independent and identically distributed symbols. The process terminates when the multinomial coefficient of the cumulative symbol counts equals zero modulo m. This study extends the computational and information-theoretic analysis of their construction by presenting five algorithms with formal correctness guarantees and comprehensive complexity analyses. For prime m = p, the Bernard-Letac framework is analyzed in greater detail. The R\'enyi entropies of the source yield an exact product formula for the expected number of draws. A first-order approximation consistently overestimates this value, and the entropy lower bound is never attained. As p approaches 1, the expected cost converges to a constant greater than 1, determined by the entire source distribution. Furthermore, a seven-state automaton computes the mod-2 first-passage kernel of the binary walk, reducing the fair assignment cost from quadratic to nearly linear.

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Claude Gravel. 2026-08-20. Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling Construction. https://arxiv.org/abs/2608.20234

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