Sampling elements of a finite group: efficiency of the product replacement algorithm with an accumulator

TL;DR

This paper analyzes a refined product replacement algorithm that efficiently samples individual elements of a finite group, achieving near-uniform distribution after a polynomial number of steps, with proof leveraging spectral gap estimates and computational methods.

Contribution

It introduces a refined algorithm for sampling elements of a finite group with improved mixing time bounds, supported by spectral analysis and computational verification.

Findings

Achieves near-uniform sampling in O(k^2 log|G|) steps

Provides spectral gap estimates for the algorithm

Utilizes computer-assisted calculations for proof

Abstract

Let $G$ be a finite group generated by $k$ elements. The well-known product replacement algorithm provides an effective method for sampling generating sets of $G$. We study a refinement of this algorithm that is designed to output individual elements of $G$. We show that after $O(k^2\log|G|)$ steps, the distribution of the output is close to uniform on $G$, which improves upon the best results known to date. The proof proceeds via spectral gap estimates and uses computer assisted calculations.