Bernard-Letac公平采样构造的算法、复杂性与熵
Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling Construction
- Toronto Metropolitan University（多伦多都会大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究扩展Bernard-Letac公平采样构造的分析，提出5种带正确性保证的算法，分析素数m=p情形，推导期望抽取次数公式，用七状态自动机降低公平分配代价复杂度。
AI中文摘要：
Bernard和Letac于1971年提出了一种方法，用于从未知的有偏独立同分布符号源中对m个结果进行均匀随机采样。该过程在累积符号计数的多项系数模m等于0时终止。本研究通过提出5种具有形式化正确性保证和全面复杂性分析的算法，扩展了对该构造的计算与信息论分析。对于素数m=p，对Bernard-Letac框架进行了更详细的分析。源的Rényi熵给出了期望抽取次数的精确乘积公式。一阶近似始终高估该值，且从未达到熵下界。当p趋近于1时，期望代价收敛于大于1的常数，该常数由整个源分布决定。此外，一个七状态自动机可计算二元游走的模2首达核，将公平分配代价从二次复杂度降至近似线性。
英文摘要：
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 Rényi 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 $p$, treated as a continuous parameter, approaches 1, the expected cost converges to a constant greater than 1, determined by the entire source distribution. Furthermore, for every prime modulus $p$ and every finite alphabet $I$, an explicit automaton with $p + |I| + 3$ states computes the mod-$p$ first-passage kernel of the walk from the base-$p$ digits of its arguments, reducing the fair assignment cost from quadratic to nearly linear.
