332 research outputs found
A rule of thumb for riffle shuffling
We study how many riffle shuffles are required to mix n cards if only certain
features of the deck are of interest, e.g. suits disregarded or only the colors
of interest. For these features, the number of shuffles drops from 3/2 log_2(n)
to log_2(n). We derive closed formulae and an asymptotic `rule of thumb'
formula which is remarkably accurate.Comment: 27 pages, 5 table
Biased random-to-top shuffling
Recently Wilson [Ann. Appl. Probab. 14 (2004) 274--325] 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 [Random Walks
and Geometry (2004) 515--532] and [Electron. Comm. Probab. 8 (2003) 77--85].
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 cards, and we find the lower
bound . Via a coupling, an upper bound exceeding
this by only a factor 4 is found. This generalizes Wilson's [Electron. Comm.
Probab. 8 (2003) 77--85] result on the Rudvalis shuffle and Goel's [Ann. Appl.
Probab. 16 (2006) 30--55] result on top-to-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 . Here the lower bound is again of
order , but in this case this does not seem to be tight unless
. What the correct order of mixing is in this case is an open question.
We show that when , it is at least .Comment: Published at http://dx.doi.org/10.1214/10505160600000097 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Stretching and folding versus cutting and shuffling: An illustrated perspective on mixing and deformations of continua
We compare and contrast two types of deformations inspired by mixing
applications -- one from the mixing of fluids (stretching and folding), the
other from the mixing of granular matter (cutting and shuffling). The
connection between mechanics and dynamical systems is discussed in the context
of the kinematics of deformation, emphasizing the equivalence between stretches
and Lyapunov exponents. The stretching and folding motion exemplified by the
baker's map is shown to give rise to a dynamical system with a positive
Lyapunov exponent, the hallmark of chaotic mixing. On the other hand, cutting
and shuffling does not stretch. When an interval exchange transformation is
used as the basis for cutting and shuffling, we establish that all of the map's
Lyapunov exponents are zero. Mixing, as quantified by the interfacial area per
unit volume, is shown to be exponentially fast when there is stretching and
folding, but linear when there is only cutting and shuffling. We also discuss
how a simple computational approach can discern stretching in discrete data.Comment: REVTeX 4.1, 9 pages, 3 figures; v2 corrects some misprints. The
following article appeared in the American Journal of Physics and may be
found at http://ajp.aapt.org/resource/1/ajpias/v79/i4/p359_s1 . Copyright
2011 American Association of Physics Teachers. This article may be downloaded
for personal use only. Any other use requires prior permission of the author
and the AAP
Hopf algebras and Markov chains: Two examples and a theory
The operation of squaring (coproduct followed by product) in a combinatorial
Hopf algebra is shown to induce a Markov chain in natural bases. Chains
constructed in this way include widely studied methods of card shuffling, a
natural "rock-breaking" process, and Markov chains on simplicial complexes.
Many of these chains can be explictly diagonalized using the primitive elements
of the algebra and the combinatorics of the free Lie algebra. For card
shuffling, this gives an explicit description of the eigenvectors. For
rock-breaking, an explicit description of the quasi-stationary distribution and
sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes
will only appear on the version on Amy Pang's website, the arXiv version will
not be updated.
Physical Zero-Knowledge Proofs for Akari, Takuzu, Kakuro and KenKen
Akari, Takuzu, Kakuro and KenKen are logic games similar to Sudoku. In Akari,
a labyrinth on a grid has to be lit by placing lanterns, respecting various
constraints. In Takuzu a grid has to be filled with 0's and 1's, while
respecting certain constraints. In Kakuro a grid has to be filled with numbers
such that the sums per row and column match given values; similarly in KenKen a
grid has to be filled with numbers such that in given areas the product, sum,
difference or quotient equals a given value. We give physical algorithms to
realize zero-knowledge proofs for these games which allow a player to show that
he knows a solution without revealing it. These interactive proofs can be
realized with simple office material as they only rely on cards and envelopes.
Moreover, we formalize our algorithms and prove their security.Comment: FUN with algorithms 2016, Jun 2016, La Maddalena, Ital
Knowing When to Fold\u27em: A Monte Carlo Exploration of Card Shuffling and How Poker Players Can Gain an Advantage
This work demonstrates that the card shuffling procedure commonly performed in casino poker rooms is insufficient for randomizing a deck of cards. We explore this in the context of Texas Holdâem, which has established itself as the most popular form of poker worldwide over the past few decades. We show the degree to which the resulting distribution of the orderings of all 52 cards in the deck after shuffling is not uniform. Rather, any given card may be substantially more (or less) likely to show up as an important card in the subsequent hand. Additionally, we find that the shuffling procedure does not sufficiently separate cards from their starting point; that is, cards are more likely to stay close together after shuffling than they should by chance. Thus, in this work, we demonstrate that Texas Holdâem players can gain an advantage over their opponents by recognizing these deficiencies in the shuffling procedure
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