17,280 research outputs found
The Contest Between Simplicity and Efficiency in Asynchronous Byzantine Agreement
In the wake of the decisive impossibility result of Fischer, Lynch, and
Paterson for deterministic consensus protocols in the aynchronous model with
just one failure, Ben-Or and Bracha demonstrated that the problem could be
solved with randomness, even for Byzantine failures. Both protocols are natural
and intuitive to verify, and Bracha's achieves optimal resilience. However, the
expected running time of these protocols is exponential in general. Recently,
Kapron, Kempe, King, Saia, and Sanwalani presented the first efficient
Byzantine agreement algorithm in the asynchronous, full information model,
running in polylogarithmic time. Their algorithm is Monte Carlo and drastically
departs from the simple structure of Ben-Or and Bracha's Las Vegas algorithms.
In this paper, we begin an investigation of the question: to what extent is
this departure necessary? Might there be a much simpler and intuitive Las Vegas
protocol that runs in expected polynomial time? We will show that the
exponential running time of Ben-Or and Bracha's algorithms is no mere accident
of their specific details, but rather an unavoidable consequence of their
general symmetry and round structure. We define a natural class of "fully
symmetric round protocols" for solving Byzantine agreement in an asynchronous
setting and show that any such protocol can be forced to run in expected
exponential time by an adversary in the full information model. We assume the
adversary controls Byzantine processors for , where is an
arbitrary positive constant . We view our result as a step toward
identifying the level of complexity required for a polynomial-time algorithm in
this setting, and also as a guide in the search for new efficient algorithms.Comment: 21 page
Polynomial Response Surface Approximations for the Multidisciplinary Design Optimization of a High Speed Civil Transport
Surrogate functions have become an important tool in multidisciplinary design optimization to deal with noisy functions, high computational cost, and the practical difficulty of integrating legacy disciplinary computer codes. A combination of mathematical, statistical, and engineering techniques, well known in other contexts, have made polynomial surrogate functions viable for MDO. Despite the obvious limitations imposed by sparse high fidelity data in high dimensions and the locality of low order polynomial approximations, the success of the panoply of techniques based on polynomial response surface approximations for MDO shows that the implementation details are more important than the underlying approximation method (polynomial, spline, DACE, kernel regression, etc.). This paper surveys some of the ancillary techniques—statistics, global search, parallel computing, variable complexity modeling—that augment the construction and use of polynomial surrogates
Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers
In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration
Security problems with a chaos-based deniable authentication scheme
Recently, a new scheme was proposed for deniable authentication. Its main
originality lied on applying a chaos-based encryption-hash parallel algorithm
and the semi-group property of the Chebyshev chaotic map. Although original and
practicable, its insecurity and inefficiency are shown in this paper, thus
rendering it inadequate for adoption in e-commerce.Comment: 8 pages, 1 figure, latex forma
The Computational Complexity of Generating Random Fractals
In this paper we examine a number of models that generate random fractals.
The models are studied using the tools of computational complexity theory from
the perspective of parallel computation. Diffusion limited aggregation and
several widely used algorithms for equilibrating the Ising model are shown to
be highly sequential; it is unlikely they can be simulated efficiently in
parallel. This is in contrast to Mandelbrot percolation that can be simulated
in constant parallel time. Our research helps shed light on the intrinsic
complexity of these models relative to each other and to different growth
processes that have been recently studied using complexity theory. In addition,
the results may serve as a guide to simulation physics.Comment: 28 pages, LATEX, 8 Postscript figures available from
[email protected]
Analysis of a parallel multigrid algorithm
The parallel multigrid algorithm of Frederickson and McBryan (1987) is considered. This algorithm uses multiple coarse-grid problems (instead of one problem) in the hope of accelerating convergence and is found to have a close relationship to traditional multigrid methods. Specifically, the parallel coarse-grid correction operator is identical to a traditional multigrid coarse-grid correction operator, except that the mixing of high and low frequencies caused by aliasing error is removed. Appropriate relaxation operators can be chosen to take advantage of this property. Comparisons between the standard multigrid and the new method are made
An O(log sup 2 N) parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix
An O(log sup 2 N) parallel algorithm is presented for computing the eigenvalues of a symmetric tridiagonal matrix using a parallel algorithm for computing the zeros of the characteristic polynomial. The method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level. Intervals that contain exactly one zero are determined by the zeros of polynomials at the previous level which ensures that different processors compute different zeros. The exact behavior of the polynomials at the interval endpoints is used to eliminate the usual problems induced by finite precision arithmetic
Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity
The computational complexity of internal diffusion-limited aggregation (DLA)
is examined from both a theoretical and a practical point of view. We show that
for two or more dimensions, the problem of predicting the cluster from a given
set of paths is complete for the complexity class CC, the subset of P
characterized by circuits composed of comparator gates. CC-completeness is
believed to imply that, in the worst case, growing a cluster of size n requires
polynomial time in n even on a parallel computer.
A parallel relaxation algorithm is presented that uses the fact that clusters
are nearly spherical to guess the cluster from a given set of paths, and then
corrects defects in the guessed cluster through a non-local annihilation
process. The parallel running time of the relaxation algorithm for
two-dimensional internal DLA is studied by simulating it on a serial computer.
The numerical results are compatible with a running time that is either
polylogarithmic in n or a small power of n. Thus the computational resources
needed to grow large clusters are significantly less on average than the
worst-case analysis would suggest.
For a parallel machine with k processors, we show that random clusters in d
dimensions can be generated in O((n/k + log k) n^{2/d}) steps. This is a
significant speedup over explicit sequential simulation, which takes
O(n^{1+2/d}) time on average.
Finally, we show that in one dimension internal DLA can be predicted in O(log
n) parallel time, and so is in the complexity class NC
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