Analysis of Top To Random Shuffles

Diaconis, Persi; Fill, James Allen; Pitman, Jim

doi:10.1017/s0963548300000158

Cited by 67 publications

(97 citation statements)

References 12 publications

(26 reference statements)

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“…A thorough treatment of the uniform-weights case is provided by Diaconis, Fill, and Pitman [3]. We establish the remaining results [as we could also have established (b) (Zipf's law weights.)…”

Section: Table 1 Rates Of Convergence For ᏸ(T Cftp ) and ᏸ(T Fmmr )supporting

confidence: 65%

Speeding up the FMMR perfect sampling algorithm: A case study revisited

Dobrow

Fill

2003

Random Struct Algorithms

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ABSTRACT:In a previous paper by the second author, two Markov chain Monte Carlo perfect sampling algorithms-one called coupling from the past (CFTP) and the other (FMMR) based on rejection sampling-are compared using as a case study the move-to-front (MTF) self-organizing list chain. Here we revisit that case study and, in particular, exploit the dependence of FMMR on the user-chosen initial state. We give a stochastic monotonicity result for the running time of FMMR applied to MTF and thus identify the initial state that gives the stochastically smallest running time; by contrast, the initial state used in the previous study gives the stochastically largest running time. By changing from worst choice to best choice of initial state we achieve remarkable speedup of FMMR for MTF; for example, we reduce the running time (as measured in Markov chain steps) from exponential in the length n of the list nearly down to n when the items in the list are requested according to a geometric distribution. For this same example, the running time for CFTP grows exponentially in n.

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Section: Table 1 Rates Of Convergence For ᏸ(T Cftp ) and ᏸ(T Fmmr )supporting

confidence: 65%

Speeding up the FMMR perfect sampling algorithm: A case study revisited

Dobrow

Fill

2003

Random Struct Algorithms

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“…Remarkably, as we shall see that this further consequence of Theorem 0.1 yields an entirely new proof of a result of Phatarford in [15] and Diaconis et al in [5] to the effect that the multiplicity of the eigenvalue i in the action of τ n on Q(S n ) is given by the number of permutations with i fixed points. In fact, as we shall see, this proof yields also further explicit results concerning all the eigenspaces of the operator τ n .…”

Section: Introductionmentioning

confidence: 57%

r-Qsym is free over Sym

Garsia¹,

Wallach²

2007

Journal of Combinatorial Theory, Series A

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Our main result is a proof of the Florent Hivert conjecture [F. Hivert, Local action of the symmetric group and generalizations of quasi-symmetric functions, in preparation] that the algebras of r-Quasi-Symmetric polynomials in x 1 , x 2 , . . . , x n are free modules over the ring of Symmetric polynomials. The proof rests on a theorem that reduces a wide variety of freeness results to the establishment of a single dimension bound. We are thus able to derive the Etingof-Ginzburg [P. Etingof, V. Ginzburg, On m-quasi-invariants of a Coxeter group, Mosc. Math. J. 2 (2002) 555-566] Theorem on m-Quasi-Invariants and our r-QuasiSymmetric result as special cases of a single general principle. Another byproduct of the present treatment is a remarkably simple new proof of the freeness theorem for 1-Quasi-Symmetric polynomials given in [A.M. Garsia, N. Wallach, Qsym over Sym is free, J.We begin by fixing notation and recall some basic facts. We will throughout set X n = {x 1 , x 2 , . . . , x n } and denote by F[X n ] the algebra of polynomials in x 1 , x 2 , . . . , x n with coefficients in a field F of characteristic zero. If V is a graded vector space we denote by H m (V) the subspace of homogeneous elements of degree m in V. We then have the direct sum decomposition

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“…Hence top in at random presents a cutoff for all p ∈ (1, ∞). The cutoff time is not known although one might be able to find it using the results in [10]. Note that Theorem 3.3 does not treat the case p = ∞.…”

Section: Some Examples Of Cutoffsmentioning

confidence: 99%

The Cutoff Phenomenon for Ergodic Markov Processes

Chen

Saloff‐Coste

2008

Electron. J. Probab.

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We consider the cutoff phenomenon in the context of families of ergodic Markov transition functions. This includes classical examples such as families of ergodic finite Markov chains and Brownian motion on families of compact Riemannian manifolds. We give criteria for the existence of a cutoff when convergence is measured in L p -norm, 1 < p < ∞. This allows us to prove the existence of a cutoff in cases where the cutoff time is not explicitly known. In the reversible case, for 1 < p ≤ ∞, we show that a necessary and sufficient condition for the existence of a max-L p cutoff is that the product of the spectral gap by the max-L p mixing time tends to infinity. This type of condition was suggested by Yuval Peres. Illustrative examples are discussed.

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Analysis of Top To Random Shuffles

Cited by 67 publications

References 12 publications

Speeding up the FMMR perfect sampling algorithm: A case study revisited

Speeding up the FMMR perfect sampling algorithm: A case study revisited

r-Qsym is free over Sym

The Cutoff Phenomenon for Ergodic Markov Processes

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