118 research outputs found
Broadcasting on Random Directed Acyclic Graphs
We study a generalization of the well-known model of broadcasting on trees.
Consider a directed acyclic graph (DAG) with a unique source vertex , and
suppose all other vertices have indegree . Let the vertices at
distance from be called layer . At layer , is given a random
bit. At layer , each vertex receives bits from its parents in
layer , which are transmitted along independent binary symmetric channel
edges, and combines them using a -ary Boolean processing function. The goal
is to reconstruct with probability of error bounded away from using
the values of all vertices at an arbitrarily deep layer. This question is
closely related to models of reliable computation and storage, and information
flow in biological networks.
In this paper, we analyze randomly constructed DAGs, for which we show that
broadcasting is only possible if the noise level is below a certain degree and
function dependent critical threshold. For , and random DAGs with
layer sizes and majority processing functions, we identify the
critical threshold. For , we establish a similar result for NAND
processing functions. We also prove a partial converse for odd
illustrating that the identified thresholds are impossible to improve by
selecting different processing functions if the decoder is restricted to using
a single vertex.
Finally, for any noise level, we construct explicit DAGs (using expander
graphs) with bounded degree and layer sizes admitting
reconstruction. In particular, we show that such DAGs can be generated in
deterministic quasi-polynomial time or randomized polylogarithmic time in the
depth. These results portray a doubly-exponential advantage for storing a bit
in DAGs compared to trees, where but layer sizes must grow exponentially
with depth in order to enable broadcasting.Comment: 33 pages, double column format. arXiv admin note: text overlap with
arXiv:1803.0752
On combinatorial structures in linear codes
In this work we show that given a connectivity graph of a
quantum code, there exists , such that , and the 's are
-expander. If the codes are classical we show
instead that the 's are -expander.
We also show converses to these bounds. In particular, we show that the BPT
bound for classical codes is tight in all Euclidean dimensions. Finally, we
prove structural theorems for graphs with no "dense" subgraphs which might be
of independent interest
Finite-dimensional Approximations of Discrete Groups
The main objective of this workshop was to bring together experts from various fields, which are all interested in finite and finite-dimensional approximations of infinite algebraic and analytic objects, such as groups, algebras, dynamical systems, group actions, or even von Neumann algebras
Testing Spreading Behavior in Networks with Arbitrary Topologies
Inspired by the works of Goldreich and Ron (J. ACM, 2017) and Nakar and Ron
(ICALP, 2021), we initiate the study of property testing in dynamic
environments with arbitrary topologies. Our focus is on the simplest
non-trivial rule that can be tested, which corresponds to the 1-BP rule of
bootstrap percolation and models a simple spreading behavior: Every "infected"
node stays infected forever, and each "healthy" node becomes infected if and
only if it has at least one infected neighbor. We show various results for both
the case where we test a single time step of evolution and where the evolution
spans several time steps. In the first, we show that the worst-case query
complexity is or
(whichever is smaller), where and are the maximum degree of a node
and number of vertices, respectively, in the underlying graph, and we also show
lower bounds for both one- and two-sided error testers that match our upper
bounds up to and , respectively. In
the second setting of testing the environment over time steps, we show
upper bounds of and , where is the set of edges of the underlying graph. All of our
algorithms are one-sided error, and all of them are also time-conforming and
non-adaptive, with the single exception of the more complex
-query tester for the case .Comment: 33 pages, 3 figure
Proceedings of AUTOMATA 2010: 16th International workshop on cellular automata and discrete complex systems
International audienceThese local proceedings hold the papers of two catgeories: (a) Short, non-reviewed papers (b) Full paper
Almost uniform sampling via quantum walks
Many classical randomized algorithms (e.g., approximation algorithms for
#P-complete problems) utilize the following random walk algorithm for {\em
almost uniform sampling} from a state space of cardinality : run a
symmetric ergodic Markov chain on for long enough to obtain a random
state from within total variation distance of the uniform
distribution over . The running time of this algorithm, the so-called {\em
mixing time} of , is , where
is the spectral gap of .
We present a natural quantum version of this algorithm based on repeated
measurements of the {\em quantum walk} . We show that it
samples almost uniformly from with logarithmic dependence on
just as the classical walk does; previously, no such
quantum walk algorithm was known. We then outline a framework for analyzing its
running time and formulate two plausible conjectures which together would imply
that it runs in time when is
the standard transition matrix of a constant-degree graph. We prove each
conjecture for a subclass of Cayley graphs.Comment: 13 pages; v2 added NSF grant info; v3 incorporated feedbac
Discrete Quantum Walks on the Symmetric Group
The theory of random walks on finite graphs is well developed with numerous
applications. In quantum walks, the propagation is governed by quantum
mechanical rules; generalizing random walks to the quantum setting. They have
been successfully applied in the development of quantum algorithms. In
particular, to solve problems that can be mapped to searching or property
testing on some specific graph. In this paper we investigate the discrete time
coined quantum walk (DTCQW) model using tools from non-commutative Fourier
analysis. Specifically, we are interested in characterizing the DTCQW on Cayley
graphs generated by the symmetric group (\sym) with appropriate generating
sets. The lack of commutativity makes it challenging to find an analytical
description of the limiting behavior with respect to the spectrum of the
walk-operator. We determine certain characteristics of these walks using a path
integral approach over the characters of \sym
Amenability, locally finite spaces, and bi-lipschitz embeddings
We define the isoperimetric constant for any locally finite metric space and
we study the property of having isoperimetric constant equal to zero. This
property, called Small Neighborhood property, clearly extends amenability to
any locally finite space. Therefore, we start making a comparison between this
property and other notions of amenability for locally finite metric spaces that
have been proposed by Gromov, Lafontaine and Pansu, by Ceccherini-Silberstein,
Grigorchuk and de la Harpe and by Block and Weinberger. We discuss possible
applications of the property SN in the study of embedding a metric space into
another one. In particular, we propose three results: we prove that a certain
class of metric graphs that are isometrically embeddable into Hilbert spaces
must have the property SN. We also show, by a simple example, that this result
is not true replacing property SN with amenability. As a second result, we
prove that \emph{many} spaces with \emph{uniform bounded geometry} having a
bi-lipschitz embedding into Euclidean spaces must have the property SN.
Finally, we prove a Bourgain-like theorem for metric trees: a metric tree with
uniform bounded geometry and without property SN does not have bi-lipschitz
embeddings into finite-dimensional Hilbert spaces.Comment: 15 pages. To appear in Expositiones Mathematica
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