Mixing times of Markov chains for self‐organizing lists and biased permutations

Abstract

We study the mixing time of a Markov chain on biased permutations, a problem related to self‐organizing lists. We are given probabilities false{pi,jfalse},$$ \left\{{p}_{i,j}\right\}, $$ for all i≠j,$$ i\ne j, $$ such that pi,j=1prefix−pj,i$$ {p}_{i,j}=1-{p}_{j,i} $$. The chain ℳnn$$ {\mathcal{M}}_{nn} $$ iteratively chooses two adjacent elements i$$ i $$ and j$$ j $$, and swaps them with probability pi,j$$ {p}_{i,j} $$. It has been conjectured that ℳnn$$ {\mathcal{M}}_{nn} $$ is rapidly mixing whenever the set of probabilities are “positively biased,” that is, false{pi,j≥1false/2false},$$ \left\{{p}_{i,j}\ge 1/2\right\}, $$ for all i<j$$ i<j $$. We define two general classes and give the first proofs that ℳnn$$ {\mathcal{M}}_{nn} $$ is rapidly mixing for both. We also demonstrate that the chain can have exponential mixing time, disproving the most general version of this conjecture.