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.
Recently Karger proposed a new randomized algorithm for finding a minimum cut of an n-vertex graph (weighted or unweighted) with probability Omega(n(-2)). In this paper we present a new probabilistic analysis of Karge...
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Recently Karger proposed a new randomized algorithm for finding a minimum cut of an n-vertex graph (weighted or unweighted) with probability Omega(n(-2)). In this paper we present a new probabilistic analysis of Karger's randomized algorithm for a few classes of unweighted graphs. For random graphs whose edges are selected with a given probability p, (log n)/n less than or equal to p less than or equal to 1, we show that the expectation of success probability of the algorithm is Omega(p/n). We also investigate a class of graphs with special structure that consists of two n-cliques and gamma(n - 1) edges between the two cliques, Here gamma is a parameter satisfying 0 < gamma < 1 that makes these gamma(n - 1) edges a unique minimum cut. We show that the algorithm finds the unique minimum cut with probability n(gamma/n(gamma)). (C) 1997 Elsevier Science B.V.
Autonomic systems exhibit self-managing behavior using various algorithms. Case-based reasoning is one the techniques that enable the autonomic manager to learn from past experience. Case-base is partitioned into some...
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Autonomic systems exhibit self-managing behavior using various algorithms. Case-based reasoning is one the techniques that enable the autonomic manager to learn from past experience. Case-base is partitioned into some clusters in order to improve the retrieval efficiency. Deciding an appropriate number of clusters for a case-base is not a trivial problem. This paper proposes a randomized algorithm for determining the number of clusters to be formed of the case-base. Subsequently, a binary search-based case retrieval strategy has been applied to ensure enhanced retrieval time performance. The paper presents two versions of the randomized algorithm. The first version guarantees success but its computational cost is a function of random variable;the other guarantees a deterministic computational cost but success is not guaranteed. The performance of the proposed algorithms has been reported on a simulated case study of the Autonomic Forest Fire Application.
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.
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