The k-server problem is a fundamental online problem where k mobile servers should be scheduled to answer a sequence of requests for points in a metric space as to minimize the total movement cost. While the determini...
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The k-server problem is a fundamental online problem where k mobile servers should be scheduled to answer a sequence of requests for points in a metric space as to minimize the total movement cost. While the deterministic competitive ratio is at least k, randomized k-server algorithms have the potential of reaching o (k)-competitive ratios. Prior to this work only few specific cases of this problem were solved. For arbitrary metric spaces, this goal may be approached by using probabilistic metric approximation techniques. This paper gives the first results in this direction, obtaining o(k)competitive ratio for a natural class of metric spaces, including d-dimensional grids, and wide range of k. (c) 2004 Elsevier Inc. All rights reserved.
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 paralleliza...
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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.
The eigensystem realization algorithm (ERA) is a data-driven approach for subspace system identification and is widely used in many areas of engineering. However, the computational cost of the ERA is dominated by a st...
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The eigensystem realization algorithm (ERA) is a data-driven approach for subspace system identification and is widely used in many areas of engineering. However, the computational cost of the ERA is dominated by a step that involves the singular value decomposition (SVD) of a large, dense matrix with block Hankel structure. This paper develops computationally efficient algorithms for reducing the computational cost of the SVD step by using randomized subspace iteration and exploiting the block Hankel structure of the matrix. We provide a detailed analysis of the error in the identified system matrices and the computational cost of the proposed algorithms. We demonstrate the accuracy and computational benefits of our algorithms on two test problems: the first involves a partial differential equation that models the cooling of steel rails, and the second is an application from power systems engineering.
Given a set of strings of equal length and an integer , the closest string problem (CSP) requires the computation of a string of length such that for each , where is the Hamming distance between and . The problem is N...
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Given a set of strings of equal length and an integer , the closest string problem (CSP) requires the computation of a string of length such that for each , where is the Hamming distance between and . The problem is NP-hard and has been extensively studied in the context of approximation algorithms and fixed-parameter algorithms. Fixed-parameter algorithms provide the most practical solutions to its real-life applications in bioinformatics. In this paper we develop the first randomized fixed-parameter algorithms for CSP. Not only are the randomized algorithms much simpler than their deterministic counterparts, their time complexities are also significantly better than the previously best known (deterministic) algorithms.
We consider the well-known RAS algorithm for the problem of positive matrix scaling. We give a new bound on the number of iterations of the method for scaling a given d-dimensional matrix A to a prescribed accuracy. A...
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We consider the well-known RAS algorithm for the problem of positive matrix scaling. We give a new bound on the number of iterations of the method for scaling a given d-dimensional matrix A to a prescribed accuracy. Although the RAS method is not a polynomial-time algorithm even for d = 2, our bound implies that for any dimension d the method is a fully polynomial-time approximation scheme. We also present a randomized variant of the algorithm whose (expected) running time improves that of the deterministic method by a factor of d.
Several recent randomized linear algebra algorithms rely upon fast dimension reduction methods. A popular choice is the subsampled randomized Hadamard transform (SRHT). In this article, we address the efficacy, in the...
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Several recent randomized linear algebra algorithms rely upon fast dimension reduction methods. A popular choice is the subsampled randomized Hadamard transform (SRHT). In this article, we address the efficacy, in the Frobenius and spectral norms, of an SRHT-based low-rank matrix approximation technique introduced by Woolfe, Liberty, Rohklin, and Tygert. We establish a slightly better Frobenius norm error bound than is currently available, and a much sharper spectral norm error bound (in the presence of reasonable decay of the singular values). Along the way, we produce several results on matrix operations with SRHTs (such as approximate matrix multiplication) that may be of independent interest. Our approach builds upon Tropp's in "Improved Analysis of the Subsampled randomized Hadamard Transform" [Adv. Adaptive Data Anal., 3 (2011), pp. 115-126].
A randomized (Las Vegas) algorithm is given for finding the Gallai-Edmonds decomposition of a graph. Let n denote the number of vertices, and let M(n) denote the number of arithmetic operations for multiplying two n X...
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A randomized (Las Vegas) algorithm is given for finding the Gallai-Edmonds decomposition of a graph. Let n denote the number of vertices, and let M(n) denote the number of arithmetic operations for multiplying two n X n matrices. The sequential running time (i.e., number of bit operations) is within a poly-logarithmic factor of M(n). The parallel complexity is O((log n)(2)) parallel time using a number of processors within a poly-logarithmic factor of M(n). The same complexity bounds suffice for solving several other problems: (i) finding a minimum vertex cover in a bipartite graph, (ii) finding a minimum X --> Y vertex separator in a directed graph, where X and Y are specified sets of vertices, (iii) finding the allowed edges (i.e., edges that occur in some maximum matching) of a graph, and (iv) finding the canonical partition of the vertex set of an elementary graph. The sequential algorithms for problems (i), (ii), and (iv) are Las Vegas, and the algorithm for problem (iii) is Monte Carlo. The new complexity bounds are significantly better than the best previous ones, e.g., using the best value of M(n) currently known, the new sequential running time is O(n(2.38)) versus the previous best O(n(2.5)/(log n)) or more.
The subject of this paper is the design and analysis of Monte Carlo algorithms for two basic matching techniques used in model-based recognition: alignment, and geometric hashing. We first give analyses of our Monte C...
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The subject of this paper is the design and analysis of Monte Carlo algorithms for two basic matching techniques used in model-based recognition: alignment, and geometric hashing. We first give analyses of our Monte Carlo algorithms, showing that they are asymptotically faster than their deterministic counterparts while allowing failure probabilities that are provably very small. We then describe experimental results that bear out this speed-up, suggesting that randomization results in significant improvements in running time. Our theoretical analyses are not the best possible;as a step to remedying this we define a combinatorial measure of self-similarity for point sets, and give an example of its power. (C) 1999 Elsevier Science B.V. All rights reserved.
We study the numerical integration problem for functions with infinitely many variables. The function spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms related to the ANOVA dec...
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We study the numerical integration problem for functions with infinitely many variables. The function spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms related to the ANOVA decomposition of the integrands. The weights model the relative importance of different groups of variables. We investigate randomized quadrature algorithms and measure their quality by estimating the randomized worst-case integration error. In particular, we provide lower error bounds for a very general class of randomized algorithms that includes non-linear and adaptive algorithms. Furthermore, we propose new randomized changing dimension algorithms (also called multivariate decomposition methods) and present favorable upper error bounds. For product weights and finite-intersection weights our lower and upper error bounds match and show that our changing dimension algorithms are optimal in the sense that they achieve convergence rates arbitrarily close to the best possible convergence rate. As more specific examples, we discuss unanchored Sobolev spaces of different degrees of smoothness and randomized changing dimension algorithms that use as building blocks interlaced scrambled polynomial lattice rules. Crown Copyright (C) 2014 Published by Elsevier Inc. All rights reserved.
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