83,142 research outputs found
The Parallel Complexity of Growth Models
This paper investigates the parallel complexity of several non-equilibrium
growth models. Invasion percolation, Eden growth, ballistic deposition and
solid-on-solid growth are all seemingly highly sequential processes that yield
self-similar or self-affine random clusters. Nonetheless, we present fast
parallel randomized algorithms for generating these clusters. The running times
of the algorithms scale as , where is the system size, and the
number of processors required scale as a polynomial in . The algorithms are
based on fast parallel procedures for finding minimum weight paths; they
illuminate the close connection between growth models and self-avoiding paths
in random environments. In addition to their potential practical value, our
algorithms serve to classify these growth models as less complex than other
growth models, such as diffusion-limited aggregation, for which fast parallel
algorithms probably do not exist.Comment: 20 pages, latex, submitted to J. Stat. Phys., UNH-TR94-0
Parallel Randomized Tucker Decomposition Algorithms
The Tucker tensor decomposition is a natural extension of the singular value
decomposition (SVD) to multiway data. We propose to accelerate Tucker tensor
decomposition algorithms by using randomization and parallelization. We present
two algorithms that scale to large data and many processors, significantly
reduce both computation and communication cost compared to previous
deterministic and randomized approaches, and obtain nearly the same
approximation errors. The key idea in our algorithms is to perform randomized
sketches with Kronecker-structured random matrices, which reduces computation
compared to unstructured matrices and can be implemented using a fundamental
tensor computational kernel. We provide probabilistic error analysis of our
algorithms and implement a new parallel algorithm for the structured randomized
sketch. Our experimental results demonstrate that our combination of
randomization and parallelization achieves accurate Tucker decompositions much
faster than alternative approaches. We observe up to a 16X speedup over the
fastest deterministic parallel implementation on 3D simulation data
Deterministic parallel algorithms for bilinear objective functions
Many randomized algorithms can be derandomized efficiently using either the
method of conditional expectations or probability spaces with low independence.
A series of papers, beginning with work by Luby (1988), showed that in many
cases these techniques can be combined to give deterministic parallel (NC)
algorithms for a variety of combinatorial optimization problems, with low time-
and processor-complexity.
We extend and generalize a technique of Luby for efficiently handling
bilinear objective functions. One noteworthy application is an NC algorithm for
maximal independent set. On a graph with edges and vertices, this
takes time and processors, nearly
matching the best randomized parallel algorithms. Other applications include
reduced processor counts for algorithms of Berger (1997) for maximum acyclic
subgraph and Gale-Berlekamp switching games.
This bilinear factorization also gives better algorithms for problems
involving discrepancy. An important application of this is to automata-fooling
probability spaces, which are the basis of a notable derandomization technique
of Sivakumar (2002). Our method leads to large reduction in processor
complexity for a number of derandomization algorithms based on
automata-fooling, including set discrepancy and the Johnson-Lindenstrauss
Lemma
New bounds for truthful scheduling on two unrelated selfish machines
We consider the minimum makespan problem for tasks and two unrelated
parallel selfish machines. Let be the best approximation ratio of
randomized monotone scale-free algorithms. This class contains the most
efficient algorithms known for truthful scheduling on two machines. We propose
a new formulation for , as well as upper and lower bounds on
based on this formulation. For the lower bound, we exploit pointwise
approximations of cumulative distribution functions (CDFs). For the upper
bound, we construct randomized algorithms using distributions with piecewise
rational CDFs. Our method improves upon the existing bounds on for small
. In particular, we obtain almost tight bounds for showing that
.Comment: 28 pages, 3 tables, 1 figure. Theory Comput Syst (2019
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