An $m$ -extracting procedure produces unbiased random bits from a loaded dice with $m$ faces, and its output rate, the average number of output per input is bounded by the Shannon entropy of the source. This information-theoretic upper bound can be achieved only asymptotically as the input size increases, by certain extracting procedures that we call asymptotically optimal. Although a computationally efficient asymptotically optimal 2-extracting procedure has been known for a while, its counterparts for $m$ -ary input, $m>2$ , was found only recently, and they are still relatively complicated to describe. A binarization takes inputs from an $m$ -faced dice and produce bit sequences to be fed into a binary extracting procedure to obtain random bits. Thus, binary extracting procedures give rise to an $m$ -extracting procedure via a binarization. A binarization is to be called complete, if it preserves the asymptotic optimality, and such a procedure has been proposed by Zhou and Bruck. We show that a complete binarization naturally arises from a binary tree with $m$ leaves. Therefore, there exist complete binarizations in abundance and Zhou-Bruck scheme is an instance of them. We now have a relatively simple way to obtain an asymptotically optimal and computationally efficient $m$ -extracting procedure, from a binary one, because these binarizations are both conceptually and computationally simple. The well-known leaf entropy theorem and a closely related structure lemma play important roles in the arguments.

中文翻译：

---

二值化树和随机数生成

一个 百万美元 -提取程序 从加载的骰子产生无偏随机位 百万美元 面及其输出率，每个输入的平均输出数受源的香农熵的限制。这个信息论上界只能随着输入大小的增加而逐渐实现，通过我们称之为渐近最优的某些提取过程。尽管计算上有效的渐近最优 2-提取过程已经为人所知一段时间了，但它的对应物 百万美元 -ary 输入， $m>2$ ，是最近才发现的，描述起来还是比较复杂的。一种二值化 从一个输入 百万美元 -faced 骰子并产生位序列，将其送入二进制提取程序以获得随机位。因此，二进制提取过程产生了一个 百万美元 - 通过二值化提取过程。一个二值化被称为完全的，如果它保持渐近最优性，并且这样的过程已经由 Zhou 和 Bruck 提出。我们证明了一个完整的二值化自然产生于一棵二叉树 百万美元 树叶。因此，存在大量完整的二值化，Zhou-Bruck 方案就是其中的一个例子。我们现在有一个相对简单的方法来获得渐近最优和计算效率 百万美元 - 从二进制中提取程序，因为这些二进制化在概念上和计算上都很简单。众所周知的叶熵定理 和一个密切相关的结构引理在论证中起着重要的作用。