Optimal rolling of fair dice using fair coins

TL;DR

This paper adapts the Knuth-Yao algorithm for fair dice rolling with fair coins, achieving near-optimal efficiency and linear memory usage, improving upon previous bounds for coin flip usage.

Contribution

It presents a simplified, memory-efficient method for simulating fair dice rolls with coins, extending to general discrete distributions with near-optimal flip counts.

Findings

Achieves a bound on coin flips slightly better than Knuth-Yao.

Uses linear memory in the input size.

Extends to general discrete distributions efficiently.

Abstract

In 1976, Knuth and Yao presented an algorithm for sampling from a finite distribution using flips of a fair coin that on average used the optimal number of flips. Here we show how to easily run their algorithm for the special case of rolling a fair die that uses memory linear in the input. Analysis of this algorithm yields a bound on the average number of coin flips needed that is slightly better than the original Knuth-Yao bound. This can then be extended to discrete distributions in a near optimal number of flips again using memory linear in the input.