Peres-Style Recursive Algorithms

TL;DR

This paper introduces a binarization tree framework that clarifies the workings of Peres-style recursive algorithms, enabling new algorithm discoveries and demonstrating their near-optimal efficiency in unbiased random bit generation from biased sources.

Contribution

The paper presents a novel binarization tree approach that enhances understanding and development of Peres-style algorithms for unbiased random bit extraction, including generalizations to multi-valued sources.

Findings

Binarization trees clarify the Peres algorithm's mechanism.

New Peres-style algorithms are discovered using the binarization tree.

Algorithms approach the information-theoretic upper bound.

Abstract

Peres algorithm applies the famous von Neumann trick recursively to produce unbiased random bits from biased coin tosses. Its recursive nature makes the algorithm simple and elegant, and yet its output rate approaches the information-theoretic upper bound. However, it is relatively hard to explain why it works, and it appears partly due to this difficulty that its generalization to many-valued source was discovered only recently. Binarization tree provides a new conceptual tool to understand the innerworkings of the original Peres algorithm and the recently-found generalizations in both aspects of the uniform random number generation and asymptotic optimality. Furthermore, it facilitates finding many new Peres-style recursive algorithms that have been arguably very hard to come by without this new tool.