Mixing of Permutations by Biased Transpositions

Haddadan, Shahrzad; Winkler, Peter

doi:10.1007/s00224-018-9899-5

Cited by 7 publications

(11 citation statements)

References 17 publications

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“…This improves the mixing time bound given in [11] for k = 3. We also analyze M pp over k-particle systems, and find the mixing time is O(n 2k+4 ), matching the bounds from [11] for k = 3. In both cases, we extend their results to allow γ < 2.…”

Section: Introductionsupporting

confidence: 59%

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Rapid Mixing of k-Class Biased Permutations

Miracle

Streib

2018

LATIN 2018: Theoretical Informatics

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In this paper, we study a biased version of the nearest-neighbor transposition Markov chain on the set of permutations where neighboring elements i and j are placed in order (i, j) with probability p i,j . Our goal is to identify the class of parameter sets P = {p i,j } for which this Markov chain is rapidly mixing. Specifically, we consider the open conjecture of Jim Fill that all monotone, positively biased distributions are rapidly mixing.We resolve Fill's conjecture in the affirmative for distributions arising from k-class particle processes, where the elements are divided into k classes and the probability of exchanging neighboring elements depends on the particular classes the elements are in. We further require that k is a constant, and all probabilities between elements in different classes are bounded away from 1/2. These particle processes arise in the context of self-organizing lists and our result also applies beyond permutations to the setting where all particles in a class are indistinguishable. Additionally we show that a broader class of distributions based on trees is also rapidly mixing, which generalizes a class analyzed by Bhakta et. al. (SODA '13). Our work generalizes recent work by Haddadan and Winkler (STACS '17) studying 3-class particle processes.Our proof involves analyzing a generalized biased exclusion process, which is a nearestneighbor transposition chain applied to a 2-particle system. Biased exclusion processes are of independent interest, with applications in self-assembly. We generalize the results of Greenberg et al. (SODA '09) and Benjamini et. al (Trans. AMS '05) on biased exclusion processes to allow the probability of swapping neighboring elements to depend on the entire system, as long as the minimum bias is bounded away from 1.The fundamental problem of generating a random permutation has a long history in computer science, beginning as early as 1969 [13]. One way to generate a random permutation is to use the nearest-neighbor Markov chain M nn which repeatedly swaps the elements in a random pair of adjacent positions. The chain M nn was among the first considered in the study of the computational efficiency of Markov chains for sampling [1,6,4] and has subsequently been studied extensively. After a series of papers, the mixing time of M nn (Θ(n 3 log n) [25]) is now well-understood when sampling from the uniform distribution on the permutation group S n .The nearest-neighbor chain M nn can also be used to sample from more general probability distributions by allowing non-uniform swap probabilities. Suppose we have a set of parameters P = {p i,j } and that M nn puts neighboring elements i and j in order (i, j) with probability p i,j , where p j,i = 1 − p i,j . Despite the simplicity of this natural extension, much less is known about the mixing time of M nn in the non-uniform case. In this paper we look at the question for which parameter sets P is M nn rapidly (polynomially) mixing? We say P is positively

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Section: Introductionsupporting

confidence: 59%

“…and Winkler [11] recently studied 3-value w-distributions. They showed if w 2 /w 3 , w 1 /w 2 ≥ 2, then M nn has mixing time O(n 18 ).…”

Section: Introductionmentioning

confidence: 99%

Rapid Mixing of k-Class Biased Permutations

Miracle

Streib

2018

LATIN 2018: Theoretical Informatics

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“…Bhakta et al [2] identified certain classes of P for which M n is actually a product of independent Markov chains. Subsequently, two papers analyzed biased k-classes [14,22], where there are k classes of particles and particles from class i and class j interact with the same probability p i,j . When k = n, this is the same as the original permutation problem.…”

Section: Application: Biased Permutationsmentioning

confidence: 99%

“…They considered bounded k-classes, where p i,j /p j,i ≤ q for all i < j for some constant q < 1. In [14], Haddadan and Winkler showed a polynomial time bound on the mixing time when k = 3 and Miracle and Streib [22] generalized this to all constant k. They proved a bound of Ω(n −2(k−1) ) on the spectral gap of M p , which after applying the comparison technique [24] and relating the gap to the mixing time, leads to a bound of O(n 2k+6 ln k) on the mixing time of the permutation process.…”

Section: Application: Biased Permutationsmentioning

confidence: 99%

“…The decomposition method for Markov chains allows one to bound the spectral gap of a Markov chain in terms of the spectral gaps of constituent (hopefully simpler) Markov chains. The method was first introduced by Madras and Randall [17], and has been subsequently used and modified to produce the first polynomial time bounds on the spectral gaps of many interesting Markov chains [3,4,8,10,13,14,15,18,19,21,23,25]. In this paper, we consider a disjoint decomposition in the style of [19].…”

Section: Introductionmentioning

confidence: 99%

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Iterated Decomposition of Biased Permutations Via New Bounds on the Spectral Gap of Markov Chains

Miracle¹,

Streib²,

Streib³

2019

Preprint

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The spectral gap of a Markov chain can be bounded by the spectral gaps of constituent "restriction" chains and a "projection" chain, and the strength of such a bound is the content of various decomposition theorems. In this paper, we introduce a new parameter that allows us to improve upon these bounds. We further define a notion of orthogonality between the restriction chains and "complementary" restriction chains. This leads to a new Complementary Decomposition theorem, which does not require analyzing the projection chain. For -orthogonal chains, this theorem may be iterated O(1/ ) times while only giving away a constant multiplicative factor on the overall spectral gap. As an application, we provide a 1/n-orthogonal decomposition of the nearest neighbor Markov chain over k-class biased monotone permutations on [n], as long as the number of particles in each class is at least C log n. This allows us to apply the Complementary Decomposition theorem iteratively n times to prove the first polynomial bound on the spectral gap when k is as large as Θ(n/ log n). The previous best known bound assumed k was at most a constant.

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

Bhakta

Miracle

Randall

et al. 2022

Random Struct Algorithms

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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

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Mixing of Permutations by Biased Transpositions

Cited by 7 publications

References 17 publications

Rapid Mixing of k-Class Biased Permutations

Rapid Mixing of k-Class Biased Permutations

Iterated Decomposition of Biased Permutations Via New Bounds on the Spectral Gap of Markov Chains

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

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