26,337 research outputs found
Probabilistic Computability and Choice
We study the computational power of randomized computations on infinite
objects, such as real numbers. In particular, we introduce the concept of a Las
Vegas computable multi-valued function, which is a function that can be
computed on a probabilistic Turing machine that receives a random binary
sequence as auxiliary input. The machine can take advantage of this random
sequence, but it always has to produce a correct result or to stop the
computation after finite time if the random advice is not successful. With
positive probability the random advice has to be successful. We characterize
the class of Las Vegas computable functions in the Weihrauch lattice with the
help of probabilistic choice principles and Weak Weak K\H{o}nig's Lemma. Among
other things we prove an Independent Choice Theorem that implies that Las Vegas
computable functions are closed under composition. In a case study we show that
Nash equilibria are Las Vegas computable, while zeros of continuous functions
with sign changes cannot be computed on Las Vegas machines. However, we show
that the latter problem admits randomized algorithms with weaker failure
recognition mechanisms. The last mentioned results can be interpreted such that
the Intermediate Value Theorem is reducible to the jump of Weak Weak
K\H{o}nig's Lemma, but not to Weak Weak K\H{o}nig's Lemma itself. These
examples also demonstrate that Las Vegas computable functions form a proper
superclass of the class of computable functions and a proper subclass of the
class of non-deterministically computable functions. We also study the impact
of specific lower bounds on the success probabilities, which leads to a strict
hierarchy of classes. In particular, the classical technique of probability
amplification fails for computations on infinite objects. We also investigate
the dependency on the underlying probability space.Comment: Information and Computation (accepted for publication
Percolation by cumulative merging and phase transition for the contact process on random graphs
Given a weighted graph, we introduce a partition of its vertex set such that
the distance between any two clusters is bounded from below by a power of the
minimum weight of both clusters. This partition is obtained by recursively
merging smaller clusters and cumulating their weights. For several classical
random weighted graphs, we show that there exists a phase transition regarding
the existence of an infinite cluster.
The motivation for introducing this partition arises from a connection with
the contact process as it roughly describes the geometry of the sets where the
process survives for a long time. We give a sufficient condition on a graph to
ensure that the contact process has a non trivial phase transition in terms of
the existence of an infinite cluster. As an application, we prove that the
contact process admits a sub-critical phase on d-dimensional random geometric
graphs and on random Delaunay triangulations. To the best of our knowledge,
these are the first examples of graphs with unbounded degrees where the
critical parameter is shown to be strictly positive.Comment: 50 pages, many figure
The continuum random tree is the scaling limit of unlabelled unrooted trees
We prove that the uniform unlabelled unrooted tree with n vertices and vertex
degrees in a fixed set converges in the Gromov-Hausdorff sense after a suitable
rescaling to the Brownian continuum random tree. This proves a conjecture by
Aldous. Moreover, we establish Benjamini-Schramm convergence of this model of
random trees
Factors and Connected Factors in Tough Graphs with High Isolated Toughness
In this paper, we show that every -tough graph with order and isolated
toughness at least has a factor whose degrees are , except for at most
one vertex with degree . Using this result, we conclude that every
-tough graph with order and isolated toughness at least has a
connected factor whose degrees lie in the set , where .
Also, we show that this factor can be found -tree-connected, when is a
-tough graph with order and isolated toughness at least ,
where and . Next, we prove that
every -tough graph of order at least with high enough
isolated toughness admits an -tree-connected factor with maximum degree at
most . From this result, we derive that every -tough graph
of order at least three with high enough isolated toughness has a spanning
Eulerian subgraph whose degrees lie in the set . In addition, we
provide a family of -tough graphs with high enough isolated toughness
having no connected even factors with bounded maximum degree
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