自组织列表的混合时间与有偏置换
Mixing Times of Self-Organizing Lists and Biased Permutations
摘要：
S_n的抽样排列是概率论中的一个基本问题。已知最近邻换位链\cal{M} }_{nn}在时间Θ上收敛（n^3 \log n）和恒定偏置情况下的时间Θ（n^2），其中我们以概率p \neq 1/2将相邻元素有序放置，以概率1-p将相邻元素无序放置。这里我们考虑可变偏置情况，其中我们以概率p{x，y}并且以概率1- p_{x,y}无序。M_{nn}的混合率边界问题是由Fill提出的，并由Move-Ahead-One自组织列表更新算法驱动。证明了如果对于所有x < y，1/2 \leq p_{x,y} \leq 1，则链将总是快速混合，但这仅在恒定偏差的情况下或当p_{x,y}等于1/2或1时是已知的，这种情况对应于对偏序的线性扩展进行采样。我们证明了两个类的链是快速混合的：“选择你的武器”，其中我们给定r_1，...，具有r_i \geq 1/2和p_{x,y} =r_x的r_{n-1}（因此主导玩家选择游戏，从而固定他或她的获胜概率），以及“联盟层级”，其中存在两个联盟并且来自A联盟的玩家具有击败来自B联盟的玩家的固定概率，每个联盟内的玩家类似地被划分为具有可能不同的固定概率的子联盟，以此类推。这两类都包括具有恒定偏差的置换作为特殊情况。此外，我们还证明了最一般的猜想是假的，通过构造一个反例，其中1/2 \leq p_{x,y} \leq 1对所有x< y，但最近邻换位链需要指数时间收敛。
英文摘要：
Sampling permutations from S_n is a fundamental problem from probability theory. The nearest neighbor transposition chain \cal{M}}_{nn} is known to converge in time Θ(n^3 \log n) in the uniform case and time Θ(n^2) in the constant bias case, in which we put adjacent elements in order with probability p \neq 1/2 and out of order with probability 1-p. Here we consider the variable bias case where we put adjacent elements x<y in order with probability p{x,y} and out of order with probability 1-p_{x,y}. The problem of bounding the mixing rate of M_{nn} was posed by Fill 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 \leq p_{x,y} \leq 1 for all x < y, but this was only known in the case of constant bias or when p_{x,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 r_1,..., r_{n-1} with r_i \geq 1/2 and p_{x,y}=r_x 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 by constructing a counterexample where 1/2 \leq p_{x,y} \leq 1 for all x< y, but for which the nearest neighbor transposition chain requires exponential time to converge.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
