8,441 research outputs found
On the Computational Power of Radio Channels
Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity.
Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes\u27 inputs. Our techniques are general, and apply to many types of radio channels studied in the literature
DNF Sparsification and a Faster Deterministic Counting Algorithm
Given a DNF formula on n variables, the two natural size measures are the
number of terms or size s(f), and the maximum width of a term w(f). It is
folklore that short DNF formulas can be made narrow. We prove a converse,
showing that narrow formulas can be sparsified. More precisely, any width w DNF
irrespective of its size can be -approximated by a width DNF with
at most terms.
We combine our sparsification result with the work of Luby and Velikovic to
give a faster deterministic algorithm for approximately counting the number of
satisfying solutions to a DNF. Given a formula on n variables with poly(n)
terms, we give a deterministic time algorithm
that computes an additive approximation to the fraction of
satisfying assignments of f for \epsilon = 1/\poly(\log n). The previous best
result due to Luby and Velickovic from nearly two decades ago had a run-time of
.Comment: To appear in the IEEE Conference on Computational Complexity, 201
Deterministic Approximate Counting for Juntas of Degree-2 Polynomial Threshold Functions
Let be any Boolean function and
be any degree-2 polynomials over We give a \emph{deterministic}
algorithm which, given as input explicit descriptions of and
an accuracy parameter \eps>0, approximates \Pr_{x \sim
\{-1,1\}^n}[g(\sign(q_1(x)),\dots,\sign(q_k(x)))=1] to within an additive
\pm \eps. For any constant \eps > 0 and the running time of our
algorithm is a fixed polynomial in . This is the first fixed polynomial-time
algorithm that can deterministically approximately count satisfying assignments
of a natural class of depth-3 Boolean circuits.
Our algorithm extends a recent result \cite{DDS13:deg2count} which gave a
deterministic approximate counting algorithm for a single degree-2 polynomial
threshold function \sign(q(x)), corresponding to the case of our
result.
Our algorithm and analysis requires several novel technical ingredients that
go significantly beyond the tools required to handle the case in
\cite{DDS13:deg2count}. One of these is a new multidimensional central limit
theorem for degree-2 polynomials in Gaussian random variables which builds on
recent Malliavin-calculus-based results from probability theory. We use this
CLT as the basis of a new decomposition technique for -tuples of degree-2
Gaussian polynomials and thus obtain an efficient deterministic approximate
counting algorithm for the Gaussian distribution. Finally, a third new
ingredient is a "regularity lemma" for \emph{-tuples} of degree-
polynomial threshold functions. This generalizes both the regularity lemmas of
\cite{DSTW:10,HKM:09} and the regularity lemma of Gopalan et al \cite{GOWZ10}.
Our new regularity lemma lets us extend our deterministic approximate counting
results from the Gaussian to the Boolean domain
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
We give the best known pseudorandom generators for two touchstone classes in
unconditional derandomization: an -PRG for the class of size-
depth- circuits with seed length , and an -PRG for the class of -sparse
polynomials with seed length . These results bring the state of the art for
unconditional derandomization of these classes into sharp alignment with the
state of the art for computational hardness for all parameter settings:
improving on the seed lengths of either PRG would require breakthrough progress
on longstanding and notorious circuit lower bounds.
The key enabling ingredient in our approach is a new \emph{pseudorandom
multi-switching lemma}. We derandomize recently-developed
\emph{multi}-switching lemmas, which are powerful generalizations of
H{\aa}stad's switching lemma that deal with \emph{families} of depth-two
circuits. Our pseudorandom multi-switching lemma---a randomness-efficient
algorithm for sampling restrictions that simultaneously simplify all circuits
in a family---achieves the parameters obtained by the (full randomness)
multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and
H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into
the optimality (given current circuit lower bounds) of our PRGs for
and sparse polynomials
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