The overhand shuffle mixes in Θ(n^2 n) steps

Electronic Journal of Probability, 2011

Consider a deck of n cards. Let p 1 , p 2 ,. .. , p n be a probability vector and consider the mixing time of the card shuffle which at each step of time picks a position according to the p i 's and move the card in that position to the top. This setup was introduced in [5], where a few special cases were studied. In particular the case p n−k = p n = 1/2, k = Θ(n), turned out to be challenging and only a few lower bounds were produced. These were improved in [1] where it was shown that the relaxation time for the motion of a single card is Θ(n 2) when k/n approaches a rational number. In this paper we give the first upper bounds. We focus on the case m := n − k = n/2. It is shown that for the additive symmetrization as well as the lazy version of the shuffle, the mixing time is O(n 3 log n). We then consider two other modifications of the shuffle. The first one is the case p n−k = p n−k+1 = 1/4 and p n = 1/2. Using the entropy technique developed by Morris [7], we show that mixing time is O(n 2 log 3 n) for the shuffle itself as well as for the symmetrization. The second modification is a variant of the first, where the moves are made in pairs so that if the first move involves position n, then the second move must be taken from positions m or m + 1 and vice versa. Interestingly, this shuffle is much slower; the mixing time is at least of order n 3 log n and at most of order n 3 log 3 n. It is also observed that results of [1] can be modified to improve lower bounds for some k = o(n) .

arXiv preprint arXiv:1207.3406, 2012

Abstract: The Card-Cyclic-to-Random shuffle on $ n $ cards is defined as follows: at time $ t $ remove the card with label $ t $ mod $ n $ and randomly reinsert it back into the deck. Pinsky introduced this shuffle and asked how many steps are needed to mix the deck. He showed $ n $ steps do not suffice. Here we show that the mixing time is on the order of $\ Theta (n\ log n) $.

A Markov chain is a random process {Xt}, where, in discrete time, t ∈ Z+ and in continuous time, t ∈ R+ and the Xt’s take values in some state space S, such that given the process up to time t, the distribution of the future of the process only depends on Xt. When S is the symmetric group, i.e. the set of permutations of a number of elements, we think of the Markov chain as a card shuffling chain. In this course, all Markov chains will live on a finite state space S. By enumerating the states 1, 2, . . . , |S|, we may identify S with [|S|] = {1, 2, . . . , |S|}. A Markov chain in discrete time, {X0, X1, X2, . . .} is governed by the starting distribution P(X0 ∈ ·) and the transition matrix P = [pij]i,j∈S where the pij’s are the transition probabilities

Electronic Communications in Probability, 2015

A famous result of Bayer and Diaconis is that the Gilbert-Shannon-Reeds (GSR) model for the riffle shuffle of n cards mixes in 3 2 log 2 n steps and that for 52 cards about 7 shuffles suffices to mix the deck. In this paper, we study variants of the GSR shuffle that have been proposed to model more realistically how people actually shuffle a deck of cards. The clumpy riffle shuffle and dealer riffle shuffle differ from the GSR model in that when a card is dropped from one hand, the conditional probability that the next card is dropped from the same hand is higher/lower than for the GSR model. Until now, no nontrivial rigorous results have been known for the clumpy shuffle or dealer shuffle. In this paper we show that the mixing time is O(log 4 n).

2005

In this essay we shall discuss mathematical models of card-shuffling. The basic question to answer is” how many times do you need to shuffle a deck of cards for it to become sufficiently randomized?”. This obviously depends on what we mean by shuffling and sufficiently randomized so we shall dwell quite a bit on these points too.

2005

Recently Wilson [12] introduced an important new technique for lower bounding the mixing time of a Markov chain. In this paper we extend Wilson's technique to find lower bounds of the correct order for card shuffling Markov chains where at each time step a random card is picked and put at the top of the deck. Two classes of such shuffles are addressed, one where the probability that a given card is picked at a given time step depends on its identity, the so called move-to-front scheme, and one where it depends on its position. For the move-to-front scheme a test function that is a combination of several different eigenvectors of the transition matrix is used. A general method for finding and using such a test function, under a natural negative dependence condition, is introduced. It is shown that the correct order of the mixing time is given by the biased coupon collector's problem corresponding to the move-to-front scheme at hand. For the second class, a version of Wilson's technique for complex-valued eigenvalues/eigenvectors is used. Such variants were presented in [11] and [10]. Here we present another such variant which seems to be the most natural one for this particular class of problems. To find the eigenvalues for the general case of the second class of problems is difficult, so we restrict attention to two special cases. In the first case the card that is moved to the top is picked uniformly at random from the bottom k = k(n) = o(n) cards, and we find the lower bound (n 3 /(4π 2 k(k − 1))) log n. Via a coupling an upper bound exceeding this by only a factor 4 is found. This generalizes Wilson's [11] result on the Rudvalis shuffle and Goel's [4] result on topto-bottom shuffles. In the second case the card moved to the top is with probability 1/2 the bottom card and with probability 1/2 the card at position n − k. Here the lower bound is again of order (n 3 /k 2) log n, but in this case this does not seem to be tight unless k = O(1). What the correct order of mixing is in this case is an open question. We show that when k = n/2 it is at least Θ(n 2).

2020

We examine the most common model of card shuffling, the GSRshuffle. We prove and analyze the meaning of the famous result that “seven riffle shuffles are sufficient to randomize a deck.” Then, this proof is extended to a deck with repeated cards.

The Annals of Probability, 2006

By a well-known result of Bayer and Diaconis, the maximum entropy model of the common riffle shuffle implies that the number of riffle shuffles necessary to mix a standard deck of 52 cards is either 7 or 11-with the former number applying when the metric used to define mixing is the total variation distance and the latter when it is the separation distance. This and other related results assume all 52 cards in the deck to be distinct and require all 52! permutations of the deck to be almost equally likely for the deck to be considered well mixed. In many instances, not all cards in the deck are distinct and only the sets of cards dealt out to players, and not the order in which they are dealt out to each player, needs to be random. We derive transition probabilities under riffle shuffles between decks with repeated cards to cover some instances of the type just described. We focus on decks with cards all of which are labeled either 1 or 2 and describe the consequences of having a symmetric starting deck of the form 1, . . . , 1, 2, . . . , 2 or 1, 2, . . . , 1, 2. Finally, we consider mixing times for common card games.

Stochastic Processes and their Applications, 2007

In this paper we study random orderings of the integers with a certain invariance property. We describe all such orders in a simple way. We define and represent random shuffles of a countable set of labels and then give an interpretation of these orders in terms of a class of generalized riffle shuffles.

Theory of Cryptography, 2020

The shuffle model of differential privacy [Bittau et al. SOSP 2017; Erlingsson et al. SODA 2019; Cheu et al. EUROCRYPT 2019] was proposed as a viable model for performing distributed differentially private computations. Informally, the model consists of an untrusted analyzer that receives messages sent by participating parties via a shuffle functionality, the latter potentially disassociates messages from their senders. Prior work focused on one-round differentially private shuffle model protocols, demonstrating that functionalities such as addition and histograms can be performed in this model with accuracy levels similar to that of the curator model of differential privacy, where the computation is performed by a fully trusted party. A model closely related to the shuffle model was presented in the seminal work of Ishai et al. on establishing cryptography from anonymous communication [FOCS 2006]. Focusing on the round complexity of the shuffle model, we ask in this work what can be computed in the shuffle model of differential privacy with two rounds. Ishai et al. showed how to use one round of the shuffle to establish secret keys between every two parties. Using this primitive to simulate a general secure multi-party protocol increases its round complexity by one. We show how two parties can use one round of the shuffle to send secret messages without having to first establish a secret key, hence retaining round complexity. Combining this primitive with the two-round semi-honest protocol of Applebaun, Brakerski, and Tsabary [TCC 2018], we obtain that every randomized functionality can be computed in the shuffle model with an honest majority, in merely two rounds. This includes any differentially private computation. We hence move to examine differentially private computations in the shuffle model that (i) do not require the assumption of an honest majority, or (ii) do not admit one-round protocols, even with an honest majority. For that, we introduce two computational tasks: common element, and nested common element with parameter α. For the common element problem we show that for large enough input domains, no one-round differentially private shuffle protocol exists with constant message complexity and negligible δ, whereas a two-round protocol exists where every party sends a single message in every round. For the nested common element we show that no one-round differentially private protocol exists for this problem with adversarial coalition size αn. However, we show that it can be privately computed in two rounds against coalitions of size cn for every c < 1. This yields a separation between one-round and two-round protocols. We further show a one-round protocol for the nested common element problem that is differentially private with coalitions of size smaller than cn for all 0 < c < α < 1/2.