494 research outputs found

### Exactly solvable model with two conductor-insulator transitions driven by impurities

We present an exact analysis of two conductor-insulator transitions in the
random graph model. The average connectivity is related to the concentration of
impurities. The adjacency matrix of a large random graph is used as a hopping
Hamiltonian. Its spectrum has a delta peak at zero energy. Our analysis is
based on an explicit expression for the height of this peak, and a detailed
description of the localized eigenvectors and of their contribution to the
peak. Starting from the low connectivity (high impurity density) regime, one
encounters an insulator-conductor transition for average connectivity
1.421529... and a conductor-insulator transition for average connectivity
3.154985.... We explain the spectral singularity at average connectivity
e=2.718281... and relate it to another enumerative problem in random graph
theory, the minimal vertex cover problem.Comment: 4 pages revtex, 2 fig.eps [v2: new title, changed intro, reorganized
text

### Discordant voting processes on finite graphs

We consider an asynchronous voting process on graphs which we call discordant voting, and which can be described as follows. Initially each vertex holds one of two opinions, red or blue say. Neighbouring vertices with different opinions interact pairwise. After an interaction both vertices have the same colour. The quantity of interest is T, the time to reach consensus, i.e. the number of interactions needed for all vertices have the same colour. An edge whose endpoint colours differ (i.e. one vertex is coloured red and the other one blue) is said to be discordant. A vertex is discordant if its is incident with a discordant edge. In discordant voting, all interactions are based on discordant edges. Because the voting process is asynchronous there are several ways to update the colours of the interacting vertices. Push: Pick a random discordant vertex and push its colour to a random discordant neighbour. Pull: Pick a random discordant vertex and pull the colour of a random discordant neighbour. Oblivious: Pick a random endpoint of a random discordant edge and push the colour to the other end point. We show that ET, the expected time to reach consensus, depends strongly on the underlying graph and the update rule. For connected graphs on n vertices, and an initial half red, half blue colouring the following hold. For oblivious voting, ET = n2/4 independent of the underlying graph. For the complete graph Kn, the push protocol has ET = =(n log n), whereas the pull protocol has ET = =(2n). For the cycle Cn all three protocols have ET = =(n2). For the star graph however, the pull protocol has ET = O(n2), whereas the push protocol is slower with ET = =(n2 log n). The wide variation in ET for the pull protocol is to be contrasted with the well known model of synchronous pull voting, for which ET = O(n) on many classes of expanders

### The decimation process in random k-SAT

Let F be a uniformly distributed random k-SAT formula with n variables and m
clauses. Non-rigorous statistical mechanics ideas have inspired a message
passing algorithm called Belief Propagation Guided Decimation for finding
satisfying assignments of F. This algorithm can be viewed as an attempt at
implementing a certain thought experiment that we call the Decimation Process.
In this paper we identify a variety of phase transitions in the decimation
process and link these phase transitions to the performance of the algorithm

### Wear Minimization for Cuckoo Hashing: How Not to Throw a Lot of Eggs into One Basket

We study wear-leveling techniques for cuckoo hashing, showing that it is
possible to achieve a memory wear bound of $\log\log n+O(1)$ after the
insertion of $n$ items into a table of size $Cn$ for a suitable constant $C$
using cuckoo hashing. Moreover, we study our cuckoo hashing method empirically,
showing that it significantly improves on the memory wear performance for
classic cuckoo hashing and linear probing in practice.Comment: 13 pages, 1 table, 7 figures; to appear at the 13th Symposium on
Experimental Algorithms (SEA 2014

### Spatial Mixing of Coloring Random Graphs

We study the strong spatial mixing (decay of correlation) property of proper
$q$-colorings of random graph $G(n, d/n)$ with a fixed $d$. The strong spatial
mixing of coloring and related models have been extensively studied on graphs
with bounded maximum degree. However, for typical classes of graphs with
bounded average degree, such as $G(n, d/n)$, an easy counterexample shows that
colorings do not exhibit strong spatial mixing with high probability.
Nevertheless, we show that for $q\ge\alpha d+\beta$ with $\alpha>2$ and
sufficiently large $\beta=O(1)$, with high probability proper $q$-colorings of
random graph $G(n, d/n)$ exhibit strong spatial mixing with respect to an
arbitrarily fixed vertex. This is the first strong spatial mixing result for
colorings of graphs with unbounded maximum degree. Our analysis of strong
spatial mixing establishes a block-wise correlation decay instead of the
standard point-wise decay, which may be of interest by itself, especially for
graphs with unbounded degree

### Trajectories in phase diagrams, growth processes and computational complexity: how search algorithms solve the 3-Satisfiability problem

Most decision and optimization problems encountered in practice fall into one
of two categories with respect to any particular solving method or algorithm:
either the problem is solved quickly (easy) or else demands an impractically
long computational effort (hard). Recent investigations on model classes of
problems have shown that some global parameters, such as the ratio between the
constraints to be satisfied and the adjustable variables, are good predictors
of problem hardness and, moreover, have an effect analogous to thermodynamical
parameters, e.g. temperature, in predicting phases in condensed matter physics
[Monasson et al., Nature 400 (1999) 133-137]. Here we show that changes in the
values of such parameters can be tracked during a run of the algorithm defining
a trajectory through the parameter space. Focusing on 3-Satisfiability, a
recognized representative of hard problems, we analyze trajectories generated
by search algorithms using growth processes statistical physics. These
trajectories can cross well defined phases, corresponding to domains of easy or
hard instances, and allow to successfully predict the times of resolution.Comment: Revtex file + 4 eps figure

### Simulating Auxiliary Inputs, Revisited

For any pair $(X,Z)$ of correlated random variables we can think of $Z$ as a
randomized function of $X$. Provided that $Z$ is short, one can make this
function computationally efficient by allowing it to be only approximately
correct. In folklore this problem is known as \emph{simulating auxiliary
inputs}. This idea of simulating auxiliary information turns out to be a
powerful tool in computer science, finding applications in complexity theory,
cryptography, pseudorandomness and zero-knowledge. In this paper we revisit
this problem, achieving the following results:
\begin{enumerate}[(a)] We discuss and compare efficiency of known results,
finding the flaw in the best known bound claimed in the TCC'14 paper "How to
Fake Auxiliary Inputs". We present a novel boosting algorithm for constructing
the simulator. Our technique essentially fixes the flaw. This boosting proof is
of independent interest, as it shows how to handle "negative mass" issues when
constructing probability measures in descent algorithms. Our bounds are much
better than bounds known so far. To make the simulator
$(s,\epsilon)$-indistinguishable we need the complexity $O\left(s\cdot
2^{5\ell}\epsilon^{-2}\right)$ in time/circuit size, which is better by a
factor $\epsilon^{-2}$ compared to previous bounds. In particular, with our
technique we (finally) get meaningful provable security for the EUROCRYPT'09
leakage-resilient stream cipher instantiated with a standard 256-bit block
cipher, like $\mathsf{AES256}$.Comment: Some typos present in the previous version have been correcte

### Fast Scalable Construction of (Minimal Perfect Hash) Functions

Recent advances in random linear systems on finite fields have paved the way
for the construction of constant-time data structures representing static
functions and minimal perfect hash functions using less space with respect to
existing techniques. The main obstruction for any practical application of
these results is the cubic-time Gaussian elimination required to solve these
linear systems: despite they can be made very small, the computation is still
too slow to be feasible.
In this paper we describe in detail a number of heuristics and programming
techniques to speed up the resolution of these systems by several orders of
magnitude, making the overall construction competitive with the standard and
widely used MWHC technique, which is based on hypergraph peeling. In
particular, we introduce broadword programming techniques for fast equation
manipulation and a lazy Gaussian elimination algorithm. We also describe a
number of technical improvements to the data structure which further reduce
space usage and improve lookup speed.
Our implementation of these techniques yields a minimal perfect hash function
data structure occupying 2.24 bits per element, compared to 2.68 for MWHC-based
ones, and a static function data structure which reduces the multiplicative
overhead from 1.23 to 1.03

### Parameter estimators of random intersection graphs with thinned communities

This paper studies a statistical network model generated by a large number of
randomly sized overlapping communities, where any pair of nodes sharing a
community is linked with probability $q$ via the community. In the special case
with $q=1$ the model reduces to a random intersection graph which is known to
generate high levels of transitivity also in the sparse context. The parameter
$q$ adds a degree of freedom and leads to a parsimonious and analytically
tractable network model with tunable density, transitivity, and degree
fluctuations. We prove that the parameters of this model can be consistently
estimated in the large and sparse limiting regime using moment estimators based
on partially observed densities of links, 2-stars, and triangles.Comment: 15 page

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