Mixing Times of Markov Chains for Self-Organizing Lists and Biased Permutations

Bhakta, Prateek; Miracle, Sarah; Randall, Dana; Streib, Amanda Pascoe

doi:10.1137/1.9781611973105.1

Cited by 12 publications

(55 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The chain M nn connects the state space Ω and has the stationary distribution (see e.g., [3]) π(σ) = i<j p σ(i),σ(j) Z −1 , where Z is the normalizing constant σ∈Ω i<j p σ(i),σ(j) . For our main result we prove that if a set of probabilities P are weakly monotonic and form a bounded k-class then the Markov chain M nn is rapidly mixing.…”

Section: The Nearest Neighbor Markov Chain M Nnmentioning

confidence: 99%

“…For our main result we prove that if a set of probabilities P are weakly monotonic and form a bounded k-class then the Markov chain M nn is rapidly mixing. We will require the weakly monotonic condition defined in [3] rather than the stronger monotonic condition defined in [7,8].…”

Section: The Nearest Neighbor Markov Chain M Nnmentioning

confidence: 99%

“…Let λ * = max i<j p i,j /p j,i then π * = min x∈Ω π(x) ≥ (λ ( n 2 ) * n!) −1 (see [3] for more details), so log(1/ǫπ * ) = O(n 2 ln ǫ −1 ) since λ is bounded from above by a positive constant. Applying Theorem 5(b) and (1), we have τ k+1 (ǫ) ≤ O(n 2(k−1) n 2 ln ǫ −1 ).…”

Section: K-particle Processes MIX Rapidlymentioning

confidence: 99%

“…By Lemma 3, M k+1 has mixing time O(n 2k ln ǫ −1 ). To bound the mixing time of M k , we will use the following theorem due to Bhakta et al [3], which bounds the mixing time of a product of independent Markov chains. 2 If the input probabilities P satisfy the weak monotonicity property 3 instead then we would need to modify M tk to instead allow swaps between elements in different particle classes across elements whose particle class is larger (instead of smaller).…”

Section: From K Particle Process To K-classmentioning

confidence: 99%

“…In both cases, we extend their results to allow γ < 2. In addition, we extend the work of Bhakta et al [3] on distributions based on binary trees to include trees with maximum degree at most k.…”

Section: Introductionmentioning

confidence: 97%

See 4 more Smart Citations

Rapid Mixing of k-Class Biased Permutations

Miracle

Streib

2018

LATIN 2018: Theoretical Informatics

Self Cite

View full text Add to dashboard Cite

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

show abstract

Section: The Nearest Neighbor Markov Chain M Nnmentioning

confidence: 99%

Section: The Nearest Neighbor Markov Chain M Nnmentioning

confidence: 99%

Section: K-particle Processes MIX Rapidlymentioning

confidence: 99%

Section: From K Particle Process To K-classmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

See 3 more Smart Citations

Rapid Mixing of k-Class Biased Permutations

Miracle

Streib

2018

LATIN 2018: Theoretical Informatics

Self Cite

View full text Add to dashboard Cite

show abstract