135 research outputs found
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
On the Distributed Complexity of Large-Scale Graph Computations
Motivated by the increasing need to understand the distributed algorithmic
foundations of large-scale graph computations, we study some fundamental graph
problems in a message-passing model for distributed computing where
machines jointly perform computations on graphs with nodes (typically, ). The input graph is assumed to be initially randomly partitioned among
the machines, a common implementation in many real-world systems.
Communication is point-to-point, and the goal is to minimize the number of
communication {\em rounds} of the computation.
Our main contribution is the {\em General Lower Bound Theorem}, a theorem
that can be used to show non-trivial lower bounds on the round complexity of
distributed large-scale data computations. The General Lower Bound Theorem is
established via an information-theoretic approach that relates the round
complexity to the minimal amount of information required by machines to solve
the problem. Our approach is generic and this theorem can be used in a
"cookbook" fashion to show distributed lower bounds in the context of several
problems, including non-graph problems. We present two applications by showing
(almost) tight lower bounds for the round complexity of two fundamental graph
problems, namely {\em PageRank computation} and {\em triangle enumeration}. Our
approach, as demonstrated in the case of PageRank, can yield tight lower bounds
for problems (including, and especially, under a stochastic partition of the
input) where communication complexity techniques are not obvious.
Our approach, as demonstrated in the case of triangle enumeration, can yield
stronger round lower bounds as well as message-round tradeoffs compared to
approaches that use communication complexity techniques
Near-Optimal Distributed Approximation of Minimum-Weight Connected Dominating Set
This paper presents a near-optimal distributed approximation algorithm for
the minimum-weight connected dominating set (MCDS) problem. The presented
algorithm finds an approximation in rounds,
where is the network diameter and is the number of nodes.
MCDS is a classical NP-hard problem and the achieved approximation factor
is known to be optimal up to a constant factor, unless P=NP.
Furthermore, the round complexity is known to be
optimal modulo logarithmic factors (for any approximation), following [Das
Sarma et al.---STOC'11].Comment: An extended abstract version of this result appears in the
proceedings of 41st International Colloquium on Automata, Languages, and
Programming (ICALP 2014
Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model
We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question.
Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires a near-quadratic number of rounds in the CONGEST model, as well as any algorithm for computing the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and near-quadratic number of rounds.
Finally, we address the problem of computing an exact solution to weighted all-pairs-shortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple linear lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a sub-linear complexity. We also formally prove that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question
A time- and message-optimal distributed algorithm for minimum spanning trees
This paper presents a randomized Las Vegas distributed algorithm that
constructs a minimum spanning tree (MST) in weighted networks with optimal (up
to polylogarithmic factors) time and message complexity. This algorithm runs in
time and exchanges messages (both with
high probability), where is the number of nodes of the network, is the
diameter, and is the number of edges. This is the first distributed MST
algorithm that matches \emph{simultaneously} the time lower bound of
[Elkin, SIAM J. Comput. 2006] and the message
lower bound of [Kutten et al., J.ACM 2015] (which both apply to
randomized algorithms).
The prior time and message lower bounds are derived using two completely
different graph constructions; the existing lower bound construction that shows
one lower bound {\em does not} work for the other. To complement our algorithm,
we present a new lower bound graph construction for which any distributed MST
algorithm requires \emph{both} rounds and
messages
Towards a Stronger Theory for Permutation-based Evolutionary Algorithms
While the theoretical analysis of evolutionary algorithms (EAs) has made
significant progress for pseudo-Boolean optimization problems in the last 25
years, only sporadic theoretical results exist on how EAs solve
permutation-based problems.
To overcome the lack of permutation-based benchmark problems, we propose a
general way to transfer the classic pseudo-Boolean benchmarks into benchmarks
defined on sets of permutations. We then conduct a rigorous runtime analysis of
the permutation-based EA proposed by Scharnow, Tinnefeld, and Wegener
(2004) on the analogues of the \textsc{LeadingOnes} and \textsc{Jump}
benchmarks. The latter shows that, different from bit-strings, it is not only
the Hamming distance that determines how difficult it is to mutate a
permutation into another one , but also the precise cycle
structure of . For this reason, we also regard the more
symmetric scramble mutation operator. We observe that it not only leads to
simpler proofs, but also reduces the runtime on jump functions with odd jump
size by a factor of . Finally, we show that a heavy-tailed version
of the scramble operator, as in the bit-string case, leads to a speed-up of
order on jump functions with jump size~.%Comment: To appear in the proceedings of GECCO 2022. This version contains the
proofs omitted in the proceedings version for reasons of spac
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