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Optimal rolling of fair dice using fair coins

Mark Huber, Danny Vargas

TL;DR

The paper tackles efficient sampling from a finite uniform distribution using fair coin flips within the Knuth–Yao framework. It presents a randomness-recycling implementation that tracks a state $(X,m)$ with $[X|m] \sim \mathrm{d}m$ and doubles $m$ until at least $n$, then either accepts when $X \le n$ or reduces to a smaller subproblem via $(X-n,m-n)$. The authors prove correctness and optimality of the resulting DDG, derive the bound $\mathbb{E}[N_n] \le \lceil \log_2(n) \rceil + 1$, and extend the method to general discrete distributions using a bit-decomposition of probabilities. The approach yields a simple, memory-efficient, near-optimal sampler for fair dice and general discrete distributions, with potential applicability to loaded dice when combined with binary-probability representations.

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.

Optimal rolling of fair dice using fair coins

TL;DR

The paper tackles efficient sampling from a finite uniform distribution using fair coin flips within the Knuth–Yao framework. It presents a randomness-recycling implementation that tracks a state with and doubles until at least , then either accepts when or reduces to a smaller subproblem via . The authors prove correctness and optimality of the resulting DDG, derive the bound , and extend the method to general discrete distributions using a bit-decomposition of probabilities. The approach yields a simple, memory-efficient, near-optimal sampler for fair dice and general discrete distributions, with potential applicability to loaded dice when combined with binary-probability representations.

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.
Paper Structure (6 sections, 7 theorems, 14 equations, 2 figures)

This paper contains 6 sections, 7 theorems, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of correctness
  3. The optimal DDG
  4. Upper bounding the expected number of flips
  5. Extending to discrete distributions
  6. Conclusion

Key Result

Theorem 1

The output of $\textsf{optimal\_uniform}(n)$ is uniform over $\{1, \ldots, n \}$.

Figures (2)

  • Figure 1: From each state, each child is equally likely to be chosen. When a state of the form $(i, 5)$ is reached, output $i$. When state $(1, 1)$ is reached, begin again at the root.
  • Figure 2: Two DDGs for $(3 / 8, 1 / 2, 1 / 8)$. The DDG on the left will always use 3 flips to determine the outcome, while the one on the right uses 1 flip with probability $1 / 2$, 2 flips with probability $1 / 4$, or 3 flips with probability $1 / 4$.

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['thm:one']}
  • Theorem 2
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 3 more