Mixing Times of Self-Organizing Lists and Biased Permutations

Sampling permutations from Sn is a fundamental problem from probability theory. The nearest neighbor transposition chain Mnn is known to converge in time Θ(n log n) in the uniform case [18] and time Θ(n) in the constant bias case, in which we put adjacent elements in order with probability p 6= 1/2 and out of order with probability 1− p [2]. Here we consider the variable bias case where the probability of putting an adjacent pair of elements in order depends on the two elements, and we put adjacent elements x < y in order with probability px,y and out of order with probability 1 − px,y. The problem of bounding the mixing rate of Mnn was posed by Fill [8, 9] and was motivated by the Move-Ahead-One self-organizing list update algorithm. It was conjectured that the chain would always be rapidly mixing if 1/2 ≤ px,y ≤ 1 for all x < y, but this was only known in the case of constant bias or when px,y is equal to 1/2 or 1, a case that corresponds to sampling linear extensions of a partial order. We prove the chain is rapidly mixing for two classes: “Choose Your Weapon,” where we are given r1, . . . , rn−1 with ri ≥ 1/2 and px,y = rx for all x < y (so the dominant player chooses the game, thus fixing his or her probability of winning), and “League Hierarchies,” where there are two leagues and players from the A-league have a fixed probability of beating players from the B-league, players within each league are similarly divided into sub-leagues with a possibly different fixed probability, and so forth recursively. Both of these classes include permutations with constant bias as a special case. Moreover, we also prove that the most general conjecture is false. We do so by constructing a counterexample where 1/2 ≤ px,y ≤ 1 for all x < y, but for which the nearest neighbor transposition chain requires exponential time to converge. ∗College of Computing, Georgia Institute of Technology, Atlanta, GA 30332-0765. Supported in part by NSF CCF-0830367 and a Georgia Institute of Technology ARC Fellowship. †College of Computing, Georgia Institute of Technology, Atlanta, GA 30332-0765. Supported in part by a DOE Office of Science Graduate Fellowship, NSF CCF-0830367 and a ARCS Scholar Award. ‡College of Computing, Georgia Institute of Technology, Atlanta, GA 30332-0765. Supported in part by NSF CCF-0830367 and CCF-0910584. §School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0280. Supported in part by the National Physical Sciences Consortium Fellowship and NSF CCF-0910584. ar X iv :1 20 4. 32 39 v1 [ cs .D M ] 1 5 A pr 2 01 2