6,461 research outputs found
Percolation games, probabilistic cellular automata, and the hard-core model
Let each site of the square lattice be independently assigned
one of three states: a \textit{trap} with probability , a \textit{target}
with probability , and \textit{open} with probability , where
. Consider the following game: a token starts at the origin, and two
players take turns to move, where a move consists of moving the token from its
current site to either or . A player who moves the token
to a trap loses the game immediately, while a player who moves the token to a
target wins the game immediately. Is there positive probability that the game
is \emph{drawn} with best play -- i.e.\ that neither player can force a win?
This is equivalent to the question of ergodicity of a certain family of
elementary one-dimensional probabilistic cellular automata (PCA). These
automata have been studied in the contexts of enumeration of directed lattice
animals, the golden-mean subshift, and the hard-core model, and their
ergodicity has been noted as an open problem by several authors. We prove that
these PCA are ergodic, and correspondingly that the game on has
no draws.
On the other hand, we prove that certain analogous games \emph{do} exhibit
draws for suitable parameter values on various directed graphs in higher
dimensions, including an oriented version of the even sublattice of
in all . This is proved via a dimension reduction to a
hard-core lattice gas in dimension . We show that draws occur whenever the
corresponding hard-core model has multiple Gibbs distributions. We conjecture
that draws occur also on the standard oriented lattice for
, but here our method encounters a fundamental obstacle.Comment: 35 page
Sparse approaches for the exact distribution of patterns in long state sequences generated by a Markov source
We present two novel approaches for the computation of the exact distribution
of a pattern in a long sequence. Both approaches take into account the sparse
structure of the problem and are two-part algorithms. The first approach relies
on a partial recursion after a fast computation of the second largest
eigenvalue of the transition matrix of a Markov chain embedding. The second
approach uses fast Taylor expansions of an exact bivariate rational
reconstruction of the distribution. We illustrate the interest of both
approaches on a simple toy-example and two biological applications: the
transcription factors of the Human Chromosome 5 and the PROSITE signatures of
functional motifs in proteins. On these example our methods demonstrate their
complementarity and their hability to extend the domain of feasibility for
exact computations in pattern problems to a new level
Deterministic parallel algorithms for bilinear objective functions
Many randomized algorithms can be derandomized efficiently using either the
method of conditional expectations or probability spaces with low independence.
A series of papers, beginning with work by Luby (1988), showed that in many
cases these techniques can be combined to give deterministic parallel (NC)
algorithms for a variety of combinatorial optimization problems, with low time-
and processor-complexity.
We extend and generalize a technique of Luby for efficiently handling
bilinear objective functions. One noteworthy application is an NC algorithm for
maximal independent set. On a graph with edges and vertices, this
takes time and processors, nearly
matching the best randomized parallel algorithms. Other applications include
reduced processor counts for algorithms of Berger (1997) for maximum acyclic
subgraph and Gale-Berlekamp switching games.
This bilinear factorization also gives better algorithms for problems
involving discrepancy. An important application of this is to automata-fooling
probability spaces, which are the basis of a notable derandomization technique
of Sivakumar (2002). Our method leads to large reduction in processor
complexity for a number of derandomization algorithms based on
automata-fooling, including set discrepancy and the Johnson-Lindenstrauss
Lemma
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