Improved bounds for the mixing time of the random-to-random insertion shuffle

TL;DR

This paper establishes a tighter upper bound on the mixing time of the random-to-random insertion shuffle, reducing the previous estimate from 2 n log n to approximately 1.5324 n log n using a non-Markovian coupling analysis.

Contribution

It introduces a novel analysis technique using non-Markovian coupling to improve the upper bound on the shuffle's mixing time.

Findings

Upper bound of 1.5324 n log n for mixing time

Improved from previous 2 n log n bound

Analysis based on non-Markovian coupling

Abstract

We prove an upper bound of $1.5324 n \log n$ for the mixing time of the random-to-random insertion shuffle, improving on the best known upper bound of $2 n \log n$. Our proof is based on the analysis of a non-Markovian coupling.