A lower bound of Omega(root log k/log log k) is proved for the competitive ratio of randomized algorithms for the k-server problem against an oblivious adversary. The bound holds for arbitrary metric spaces (having at...
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A lower bound of Omega(root log k/log log k) is proved for the competitive ratio of randomized algorithms for the k-server problem against an oblivious adversary. The bound holds for arbitrary metric spaces (having at least k + 1 points) and provides a new lower bound for the metrical task system problem as well. This improves the previous best lower bound of (log log k) for arbitrary metric spaces [H. J. Karlo, Y. Rabani, and Y. Ravid, SIAM J. Comput., 23 (1994), pp. 293-312] and more closely approaches the conjectured lower bound of ( log k). For the server problem on k + 1 equally spaced points on a line, which corresponds to a natural motion-planning problem, a lower bound of (log k/log log k) is obtained. The results are deduced from a general decomposition theorem for a simpler version of both the k-server and the metrical task system problems, called the pursuit-evasion game. It is shown that if a metric space M can be decomposed into two spaces M-L and M-R such that the distance between them is sufficiently large compared to their diameter, then the competitive ratio for this game on M can be expressed nearly exactly in terms of the ratios on each of the two subspaces. This yields a divide-and-conquer approach to bounding the competitive ratio of a space.
In this paper, we consider the online version of the following problem: partition a set of input points into subsets, each enclosable by a unit ball, so as to minimize the number of subsets used. In the one-dimensiona...
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In this paper, we consider the online version of the following problem: partition a set of input points into subsets, each enclosable by a unit ball, so as to minimize the number of subsets used. In the one-dimensional case, we show that surprisingly the na < ve upper bound of 2 on the competitive ratio can be beaten: we present a new randomized 15/8-competitive online algorithm. We also provide some lower bounds and an extension to higher dimensions.
The Kaczmarz and Gauss-Seidel methods both solve a linear system X beta - y by iteratively refining the solution estimate. Recent interest in these methods has been sparked by a proof of Strohmer and Vershynin which s...
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The Kaczmarz and Gauss-Seidel methods both solve a linear system X beta - y by iteratively refining the solution estimate. Recent interest in these methods has been sparked by a proof of Strohmer and Vershynin which shows the randomized Kaczmarz method converges linearly in expectation to the solution. Lewis and Leventhal then proved a similar result for the randomized Gauss-Seidel algorithm. However, the behavior of both methods depends heavily on whether the system is underdetermined or overdetermined, and whether it is consistent or not. Here we provide a unified theory of both methods, their variants for these different settings, and draw connections between both approaches. In doing so, we also provide a proof that an extended version of randomized Gauss-Seidel converges linearly to the least norm solution in the underdetermined case (where the usual randomized Gauss-Seidel fails to converge). We detail analytically and empirically the convergence properties of both methods and their extended variants in all possible system settings. With this result, a complete and rigorous theory of both methods is furnished.
It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan-...
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It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan-Raspaud colorings of a given cubic graph and, analyzing the computer results, we try to find and describe the Fan-Raspaud colorings for some selected classes of cubic graphs. The presented algorithms can then be applied to the pair assignment problem in cubic computer networks. Another possible application of the algorithms is that of being a tool for mathematicians working in the field of cubic graph theory, for discovering edge colorings with certain mathematical properties and formulating new conjectures related to the Fan-Raspaud conjecture.
The generalized singular value decomposition (GSVD) is one of the essential tools in numerical linear algebra. This paper proposes a regularization method, combining Tikhonov regularization in general form with the tr...
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The generalized singular value decomposition (GSVD) is one of the essential tools in numerical linear algebra. This paper proposes a regularization method, combining Tikhonov regularization in general form with the truncated GSVD. Then the randomized algorithms are adopted to implement the truncation process. This randomized GSVD for the regularization of the large-scale ill-posed problems can achieve good accuracy with less computational time and memory requirement than the classical regularization methods. Finally, we present the error analyses for the randomized algorithms. Some illustrative numerical examples are provided.
We show that a simple randomized algorithm has an expected constant factor approximation guarantee for fitting bucket orders to a set of pairwise preferences. (C) 2008 Elsevier B.V. All rights reserved.
We show that a simple randomized algorithm has an expected constant factor approximation guarantee for fitting bucket orders to a set of pairwise preferences. (C) 2008 Elsevier B.V. All rights reserved.
We study randomized on-line scheduling on mesh machines. We show that for scheduling independent jobs randomized algorithms can achieve a significantly better performance than deterministic ones;on the other hand with...
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We study randomized on-line scheduling on mesh machines. We show that for scheduling independent jobs randomized algorithms can achieve a significantly better performance than deterministic ones;on the other hand with dependencies randomization does not help. (C) 1996 Academic Press, Inc.
We study approximation of multivariate functions from a general separable reproducing kernel Hilbert space in the randomized setting with the error measured in the L-infinity norm. We consider algorithms that use stan...
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We study approximation of multivariate functions from a general separable reproducing kernel Hilbert space in the randomized setting with the error measured in the L-infinity norm. We consider algorithms that use standard information consisting of function values or general linear information consisting of arbitrary linear functionals. The power of standard or linear information is defined as, roughly speaking, the optimal rate of convergence of algorithms using n function values or linear functionals. We prove under certain assumptions that the power of standard information in the randomized setting is at least equal to the power of linear information in the worst case setting, and that the powers of linear and standard information in the randomized setting differ at most by 1/2. These assumptions are satisfied for spaces with weighted Korobov and Wiener reproducing kernels. For the Wiener case, the parameters in these assumptions are prohibitively large, and therefore we also present less restrictive assumptions and obtain other bounds on the power of standard information. Finally, we study tractability, which means that we want to guarantee that the errors depend at most polynomially on the number of variables and tend to zero polynomially in n(-1) when n function values are used.
This paper investigates the randomized version of the Kaczmarz method to solve linear systems in the case where the adjoint of the system matrix is not exacta situation we refer to as mismatched adjoint. We show that ...
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This paper investigates the randomized version of the Kaczmarz method to solve linear systems in the case where the adjoint of the system matrix is not exacta situation we refer to as mismatched adjoint. We show that the method may still converge both in the over- and underdetermined consistent case under appropriate conditions, and we calculate the expected asymptotic rate of linear convergence. Moreover, we analyze the inconsistent case and obtain results for the method with mismatched adjoint as for the standard method. Finally, we derive a method to compute optimized probabilities for the choice of the rows and illustrate our findings with numerical examples.
An innovative method is introduced in this paper to significantly increase computational speed and to reduce memory usage, when applied to nonlinear methods and algorithms for dimensionality reduction (DR). Due to the...
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An innovative method is introduced in this paper to significantly increase computational speed and to reduce memory usage, when applied to nonlinear methods and algorithms for dimensionality reduction (DR). Due to the incapability of linear DR methods in the preservation of data geometry, the need of effective nonlinear approaches has recently attracted the attention of many researchers, particularly from the Mathematics and Computer Science communities. The common theme of the current nonlinear DR approaches is formulation of certain matrices, called dimensionality reduction kernels (DRK) in terms of the data points, followed by performing spectral decomposition. Hence, for datasets of large size with data points that lie in some high dimensional space, the matrix dimension of the DRK is very large. Typical examples for the need of very high dimensional DRK arise from such applications as processing multispectral imagery data, searching desired text documentary data in the internet, and recognizing human faces from given libraries. For such and many other applications, the matrix dimension of the DRK is so large that computation for carrying out spectral decomposition of the DRK often encounters various difficulties, not only due to the need of significantly large read-only memory (ROM), but also due to computational instability. The main objective of this paper is to introduce the notion of the anisotropic transform (AT) and to develop its corresponding effective and efficient computational algorithms by integrating random embedding with the AT to formulate the randomized anisotropic transform (RAT), in order to reduce the size of the DRK significantly, while preserving local geometries of the given datasets. Illustrations with various examples will also be given in this paper to demonstrate that RAT algorithms dramatically reduce the ROM and CPU requirement, and thus allow fast processing, even on PC and potentially for handhold devices as well.
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