30,241 research outputs found
Best of Two Local Models: Local Centralized and Local Distributed Algorithms
We consider two models of computation: centralized local algorithms and local
distributed algorithms. Algorithms in one model are adapted to the other model
to obtain improved algorithms.
Distributed vertex coloring is employed to design improved centralized local
algorithms for: maximal independent set, maximal matching, and an approximation
scheme for maximum (weighted) matching over bounded degree graphs. The
improvement is threefold: the algorithms are deterministic, stateless, and the
number of probes grows polynomially in , where is the number of
vertices of the input graph.
The recursive centralized local improvement technique by Nguyen and
Onak~\cite{onak2008} is employed to obtain an improved distributed
approximation scheme for maximum (weighted) matching. The improvement is
twofold: we reduce the number of rounds from to for a
wide range of instances and, our algorithms are deterministic rather than
randomized
Distributed Approximation of Maximum Independent Set and Maximum Matching
We present a simple distributed -approximation algorithm for maximum
weight independent set (MaxIS) in the model which completes
in rounds, where is the maximum
degree, is the number of rounds needed to compute a maximal
independent set (MIS) on , and is the maximum weight of a node. %Whether
our algorithm is randomized or deterministic depends on the \texttt{MIS}
algorithm used as a black-box.
Plugging in the best known algorithm for MIS gives a randomized solution in
rounds, where is the number of nodes.
We also present a deterministic -round algorithm based
on coloring.
We then show how to use our MaxIS approximation algorithms to compute a
-approximation for maximum weight matching without incurring any additional
round penalty in the model. We use a known reduction for
simulating algorithms on the line graph while incurring congestion, but we show
our algorithm is part of a broad family of \emph{local aggregation algorithms}
for which we describe a mechanism that allows the simulation to run in the
model without an additional overhead.
Next, we show that for maximum weight matching, relaxing the approximation
factor to () allows us to devise a distributed algorithm
requiring rounds for any constant
. For the unweighted case, we can even obtain a
-approximation in this number of rounds. These algorithms are
the first to achieve the provably optimal round complexity with respect to
dependency on
New Insights into History Matching via Sequential Monte Carlo
The aim of the history matching method is to locate non-implausible regions
of the parameter space of complex deterministic or stochastic models by
matching model outputs with data. It does this via a series of waves where at
each wave an emulator is fitted to a small number of training samples. An
implausibility measure is defined which takes into account the closeness of
simulated and observed outputs as well as emulator uncertainty. As the waves
progress, the emulator becomes more accurate so that training samples are more
concentrated on promising regions of the space and poorer parts of the space
are rejected with more confidence. Whilst history matching has proved to be
useful, existing implementations are not fully automated and some ad-hoc
choices are made during the process, which involves user intervention and is
time consuming. This occurs especially when the non-implausible region becomes
small and it is difficult to sample this space uniformly to generate new
training points. In this article we develop a sequential Monte Carlo (SMC)
algorithm for implementation which is semi-automated. Our novel SMC approach
reveals that the history matching method yields a non-implausible distribution
that can be multi-modal, highly irregular and very difficult to sample
uniformly. Our SMC approach offers a much more reliable sampling of the
non-implausible space, which requires additional computation compared to other
approaches used in the literature
Distributed Maximum Matching in Bounded Degree Graphs
We present deterministic distributed algorithms for computing approximate
maximum cardinality matchings and approximate maximum weight matchings. Our
algorithm for the unweighted case computes a matching whose size is at least
(1-\eps) times the optimal in \Delta^{O(1/\eps)} +
O\left(\frac{1}{\eps^2}\right) \cdot\log^*(n) rounds where is the number
of vertices in the graph and is the maximum degree. Our algorithm for
the edge-weighted case computes a matching whose weight is at least (1-\eps)
times the optimal in
\log(\min\{1/\wmin,n/\eps\})^{O(1/\eps)}\cdot(\Delta^{O(1/\eps)}+\log^*(n))
rounds for edge-weights in [\wmin,1].
The best previous algorithms for both the unweighted case and the weighted
case are by Lotker, Patt-Shamir, and Pettie~(SPAA 2008). For the unweighted
case they give a randomized (1-\eps)-approximation algorithm that runs in
O((\log(n)) /\eps^3) rounds. For the weighted case they give a randomized
(1/2-\eps)-approximation algorithm that runs in O(\log(\eps^{-1}) \cdot
\log(n)) rounds. Hence, our results improve on the previous ones when the
parameters , \eps and \wmin are constants (where we reduce the
number of runs from to ), and more generally when
, 1/\eps and 1/\wmin are sufficiently slowly increasing functions
of . Moreover, our algorithms are deterministic rather than randomized.Comment: arXiv admin note: substantial text overlap with arXiv:1402.379
On the Benefit of Merging Suffix Array Intervals for Parallel Pattern Matching
We present parallel algorithms for exact and approximate pattern matching
with suffix arrays, using a CREW-PRAM with processors. Given a static text
of length , we first show how to compute the suffix array interval of a
given pattern of length in
time for . For approximate pattern matching with differences or
mismatches, we show how to compute all occurrences of a given pattern in
time, where is the size of the alphabet
and . The workhorse of our algorithms is a data structure
for merging suffix array intervals quickly: Given the suffix array intervals
for two patterns and , we present a data structure for computing the
interval of in sequential time, or in
parallel time. All our data structures are of size bits (in addition to
the suffix array)
Dynamic Algorithms for the Massively Parallel Computation Model
The Massive Parallel Computing (MPC) model gained popularity during the last
decade and it is now seen as the standard model for processing large scale
data. One significant shortcoming of the model is that it assumes to work on
static datasets while, in practice, real-world datasets evolve continuously. To
overcome this issue, in this paper we initiate the study of dynamic algorithms
in the MPC model.
We first discuss the main requirements for a dynamic parallel model and we
show how to adapt the classic MPC model to capture them. Then we analyze the
connection between classic dynamic algorithms and dynamic algorithms in the MPC
model. Finally, we provide new efficient dynamic MPC algorithms for a variety
of fundamental graph problems, including connectivity, minimum spanning tree
and matching.Comment: Accepted to the 31st ACM Symposium on Parallelism in Algorithms and
Architectures (SPAA 2019
Optimal Dynamic Distributed MIS
Finding a maximal independent set (MIS) in a graph is a cornerstone task in
distributed computing. The local nature of an MIS allows for fast solutions in
a static distributed setting, which are logarithmic in the number of nodes or
in their degrees. The result trivially applies for the dynamic distributed
model, in which edges or nodes may be inserted or deleted. In this paper, we
take a different approach which exploits locality to the extreme, and show how
to update an MIS in a dynamic distributed setting, either \emph{synchronous} or
\emph{asynchronous}, with only \emph{a single adjustment} and in a single
round, in expectation. These strong guarantees hold for the \emph{complete
fully dynamic} setting: Insertions and deletions, of edges as well as nodes,
gracefully and abruptly. This strongly separates the static and dynamic
distributed models, as super-constant lower bounds exist for computing an MIS
in the former.
Our results are obtained by a novel analysis of the surprisingly simple
solution of carefully simulating the greedy \emph{sequential} MIS algorithm
with a random ordering of the nodes. As such, our algorithm has a direct
application as a -approximation algorithm for correlation clustering. This
adds to the important toolbox of distributed graph decompositions, which are
widely used as crucial building blocks in distributed computing.
Finally, our algorithm enjoys a useful \emph{history-independence} property,
meaning the output is independent of the history of topology changes that
constructed that graph. This means the output cannot be chosen, or even biased,
by the adversary in case its goal is to prevent us from optimizing some
objective function.Comment: 19 pages including appendix and reference
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