8,868 research outputs found
Faster tuple lattice sieving using spherical locality-sensitive filters
To overcome the large memory requirement of classical lattice sieving
algorithms for solving hard lattice problems, Bai-Laarhoven-Stehl\'{e} [ANTS
2016] studied tuple lattice sieving, where tuples instead of pairs of lattice
vectors are combined to form shorter vectors. Herold-Kirshanova [PKC 2017]
recently improved upon their results for arbitrary tuple sizes, for example
showing that a triple sieve can solve the shortest vector problem (SVP) in
dimension in time , using a technique similar to
locality-sensitive hashing for finding nearest neighbors.
In this work, we generalize the spherical locality-sensitive filters of
Becker-Ducas-Gama-Laarhoven [SODA 2016] to obtain space-time tradeoffs for near
neighbor searching on dense data sets, and we apply these techniques to tuple
lattice sieving to obtain even better time complexities. For instance, our
triple sieve heuristically solves SVP in time . For
practical sieves based on Micciancio-Voulgaris' GaussSieve [SODA 2010], this
shows that a triple sieve uses less space and less time than the current best
near-linear space double sieve.Comment: 12 pages + references, 2 figures. Subsumed/merged into Cryptology
ePrint Archive 2017/228, available at https://ia.cr/2017/122
Sparse Learning over Infinite Subgraph Features
We present a supervised-learning algorithm from graph data (a set of graphs)
for arbitrary twice-differentiable loss functions and sparse linear models over
all possible subgraph features. To date, it has been shown that under all
possible subgraph features, several types of sparse learning, such as Adaboost,
LPBoost, LARS/LASSO, and sparse PLS regression, can be performed. Particularly
emphasis is placed on simultaneous learning of relevant features from an
infinite set of candidates. We first generalize techniques used in all these
preceding studies to derive an unifying bounding technique for arbitrary
separable functions. We then carefully use this bounding to make block
coordinate gradient descent feasible over infinite subgraph features, resulting
in a fast converging algorithm that can solve a wider class of sparse learning
problems over graph data. We also empirically study the differences from the
existing approaches in convergence property, selected subgraph features, and
search-space sizes. We further discuss several unnoticed issues in sparse
learning over all possible subgraph features.Comment: 42 pages, 24 figures, 4 table
Combinatorial algorithm for counting small induced graphs and orbits
Graphlet analysis is an approach to network analysis that is particularly
popular in bioinformatics. We show how to set up a system of linear equations
that relate the orbit counts and can be used in an algorithm that is
significantly faster than the existing approaches based on direct enumeration
of graphlets. The algorithm requires existence of a vertex with certain
properties; we show that such vertex exists for graphlets of arbitrary size,
except for complete graphs and , which are treated separately. Empirical
analysis of running time agrees with the theoretical results
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
ADAM: Analysis of Discrete Models of Biological Systems Using Computer Algebra
Background: Many biological systems are modeled qualitatively with discrete
models, such as probabilistic Boolean networks, logical models, Petri nets, and
agent-based models, with the goal to gain a better understanding of the system.
The computational complexity to analyze the complete dynamics of these models
grows exponentially in the number of variables, which impedes working with
complex models. Although there exist sophisticated algorithms to determine the
dynamics of discrete models, their implementations usually require
labor-intensive formatting of the model formulation, and they are oftentimes
not accessible to users without programming skills. Efficient analysis methods
are needed that are accessible to modelers and easy to use. Method: By
converting discrete models into algebraic models, tools from computational
algebra can be used to analyze their dynamics. Specifically, we propose a
method to identify attractors of a discrete model that is equivalent to solving
a system of polynomial equations, a long-studied problem in computer algebra.
Results: A method for efficiently identifying attractors, and the web-based
tool Analysis of Dynamic Algebraic Models (ADAM), which provides this and other
analysis methods for discrete models. ADAM converts several discrete model
types automatically into polynomial dynamical systems and analyzes their
dynamics using tools from computer algebra. Based on extensive experimentation
with both discrete models arising in systems biology and randomly generated
networks, we found that the algebraic algorithms presented in this manuscript
are fast for systems with the structure maintained by most biological systems,
namely sparseness, i.e., while the number of nodes in a biological network may
be quite large, each node is affected only by a small number of other nodes,
and robustness, i.e., small number of attractors
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