32,704 research outputs found
Expander -Decoding
We introduce two new algorithms, Serial- and Parallel- for
solving a large underdetermined linear system of equations when it is known that has at most
nonzero entries and that is the adjacency matrix of an unbalanced left
-regular expander graph. The matrices in this class are sparse and allow a
highly efficient implementation. A number of algorithms have been designed to
work exclusively under this setting, composing the branch of combinatorial
compressed-sensing (CCS).
Serial- and Parallel- iteratively minimise by successfully combining two desirable features of previous CCS
algorithms: the information-preserving strategy of ER, and the parallel
updating mechanism of SMP. We are able to link these elements and guarantee
convergence in operations by assuming that the signal
is dissociated, meaning that all of the subset sums of the support of
are pairwise different. However, we observe empirically that the signal need
not be exactly dissociated in practice. Moreover, we observe Serial-
and Parallel- to be able to solve large scale problems with a larger
fraction of nonzeros than other algorithms when the number of measurements is
substantially less than the signal length; in particular, they are able to
reliably solve for a -sparse vector from expander
measurements with and up to four times greater than what is
achievable by -regularization from dense Gaussian measurements.
Additionally, Serial- and Parallel- are observed to be able to
solve large problems sizes in substantially less time than other algorithms for
compressed sensing. In particular, Parallel- is structured to take
advantage of massively parallel architectures.Comment: 14 pages, 10 figure
Algorithms for flows and disjoint paths in planar graphs
In this dissertation we describe several algorithms for computing flows, connectivity, and disjoint paths in planar graphs. In all cases, the algorithms are either the first polynomial-time algorithms or are faster than all previously-known algorithms.
First, we describe algorithms for the maximum flow problem in directed planar graphs with integer capacities on both vertices and arcs and with multiple sources and sinks. The algorithms are the first to solve the problem in near-linear time when the number of terminals is fixed and the capacities are polynomially bounded. As a byproduct, we get the first algorithm to solve the vertex-disjoint S-T paths problem in near-linear time when the number of terminals is fixed but greater than 2. We also modify our algorithms to handle real capacities in near-linear time when they are three terminals.
Second, we describe algorithms to compute element-connectivity and a related structure called the reduced graph. We show that global element-connectivity in planar graphs can be found in linear time if the terminals can be covered by O(1) faces. We also show that the reduced graph can be computed in subquadratic time in planar graphs if the number of terminals is fixed.
Third, we describe algorithms for solving or approximately solving the vertex-disjoint paths problem when we want to minimize the total length of the paths. For planar graphs, we describe: (1) an exact algorithm for the case of four pairs of terminals on a single face; and (2) a k-approximation algorithm for the case of k pairs of terminals on a single face.
Fourth, we describe algorithms and a hardness result for the ideal orientation problem. We show that the problem is NP-hard in planar graphs. On the other hand, we show that the problem is polynomial-time solvable in planar graphs when the number of terminals is fixed, the terminals are all on the same face, and no two of the terminal pairs cross. We also describe an algorithm for serial instances of a generalization of the ideal orientation problem called the k-min-sum orientation problem
A Parallel Algorithm for Exact Bayesian Structure Discovery in Bayesian Networks
Exact Bayesian structure discovery in Bayesian networks requires exponential
time and space. Using dynamic programming (DP), the fastest known sequential
algorithm computes the exact posterior probabilities of structural features in
time and space, if the number of nodes (variables) in the
Bayesian network is and the in-degree (the number of parents) per node is
bounded by a constant . Here we present a parallel algorithm capable of
computing the exact posterior probabilities for all edges with optimal
parallel space efficiency and nearly optimal parallel time efficiency. That is,
if processors are used, the run-time reduces to
and the space usage becomes per
processor. Our algorithm is based the observation that the subproblems in the
sequential DP algorithm constitute a - hypercube. We take a delicate way
to coordinate the computation of correlated DP procedures such that large
amount of data exchange is suppressed. Further, we develop parallel techniques
for two variants of the well-known \emph{zeta transform}, which have
applications outside the context of Bayesian networks. We demonstrate the
capability of our algorithm on datasets with up to 33 variables and its
scalability on up to 2048 processors. We apply our algorithm to a biological
data set for discovering the yeast pheromone response pathways.Comment: 32 pages, 12 figure
A Parallel Solver for Graph Laplacians
Problems from graph drawing, spectral clustering, network flow and graph
partitioning can all be expressed in terms of graph Laplacian matrices. There
are a variety of practical approaches to solving these problems in serial.
However, as problem sizes increase and single core speeds stagnate, parallelism
is essential to solve such problems quickly. We present an unsmoothed
aggregation multigrid method for solving graph Laplacians in a distributed
memory setting. We introduce new parallel aggregation and low degree
elimination algorithms targeted specifically at irregular degree graphs. These
algorithms are expressed in terms of sparse matrix-vector products using
generalized sum and product operations. This formulation is amenable to linear
algebra using arbitrary distributions and allows us to operate on a 2D sparse
matrix distribution, which is necessary for parallel scalability. Our solver
outperforms the natural parallel extension of the current state of the art in
an algorithmic comparison. We demonstrate scalability to 576 processes and
graphs with up to 1.7 billion edges.Comment: PASC '18, Code: https://github.com/ligmg/ligm
Forbidden Subgraphs in Connected Graphs
Given a set of connected non acyclic graphs, a
-free graph is one which does not contain any member of as copy.
Define the excess of a graph as the difference between its number of edges and
its number of vertices. Let {\gr{W}}_{k,\xi} be theexponential generating
function (EGF for brief) of connected -free graphs of excess equal to
(). For each fixed , a fundamental differential recurrence
satisfied by the EGFs {\gr{W}}_{k,\xi} is derived. We give methods on how to
solve this nonlinear recurrence for the first few values of by means of
graph surgery. We also show that for any finite collection of non-acyclic
graphs, the EGFs {\gr{W}}_{k,\xi} are always rational functions of the
generating function, , of Cayley's rooted (non-planar) labelled trees. From
this, we prove that almost all connected graphs with nodes and edges
are -free, whenever and by means of
Wright's inequalities and saddle point method. Limiting distributions are
derived for sparse connected -free components that are present when a
random graph on nodes has approximately edges. In particular,
the probability distribution that it consists of trees, unicyclic components,
, -cyclic components all -free is derived. Similar results are
also obtained for multigraphs, which are graphs where self-loops and
multiple-edges are allowed
Parallel Peeling Algorithms
The analysis of several algorithms and data structures can be framed as a
peeling process on a random hypergraph: vertices with degree less than k are
removed until there are no vertices of degree less than k left. The remaining
hypergraph is known as the k-core. In this paper, we analyze parallel peeling
processes, where in each round, all vertices of degree less than k are removed.
It is known that, below a specific edge density threshold, the k-core is empty
with high probability. We show that, with high probability, below this
threshold, only (log log n)/log(k-1)(r-1) + O(1) rounds of peeling are needed
to obtain the empty k-core for r-uniform hypergraphs. Interestingly, we show
that above this threshold, Omega(log n) rounds of peeling are required to find
the non-empty k-core. Since most algorithms and data structures aim to peel to
an empty k-core, this asymmetry appears fortunate. We verify the theoretical
results both with simulation and with a parallel implementation using graphics
processing units (GPUs). Our implementation provides insights into how to
structure parallel peeling algorithms for efficiency in practice.Comment: Appears in SPAA 2014. Minor typo corrections relative to previous
versio
Efficient Parallel Translating Embedding For Knowledge Graphs
Knowledge graph embedding aims to embed entities and relations of knowledge
graphs into low-dimensional vector spaces. Translating embedding methods regard
relations as the translation from head entities to tail entities, which achieve
the state-of-the-art results among knowledge graph embedding methods. However,
a major limitation of these methods is the time consuming training process,
which may take several days or even weeks for large knowledge graphs, and
result in great difficulty in practical applications. In this paper, we propose
an efficient parallel framework for translating embedding methods, called
ParTrans-X, which enables the methods to be paralleled without locks by
utilizing the distinguished structures of knowledge graphs. Experiments on two
datasets with three typical translating embedding methods, i.e., TransE [3],
TransH [17], and a more efficient variant TransE- AdaGrad [10] validate that
ParTrans-X can speed up the training process by more than an order of
magnitude.Comment: WI 2017: 460-46
- …