The minimum rainbow subgraph problem arises in bioinformatics. The graph is given as an edge-colored undirected graph. Our goal is to find a subgraph with minimum number of vertices such that there is exactly one edge...
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The minimum rainbow subgraph problem arises in bioinformatics. The graph is given as an edge-colored undirected graph. Our goal is to find a subgraph with minimum number of vertices such that there is exactly one edge from each color class. The currently best approximation ratio achieved by a deterministic approximation algorithm is O(Delta). (Here Delta is the max degree of a graph.) In [4] he proposes a randomized algorithm which achieves an approximation ratio of O (root Delta ln Delta). However, we find that there is a flaw in his probability analysis which renders this approximation ratio invalid. We present a simple example to show why his analysis does not work. Instead, we propose an alternative analysis for his randomized algorithm. Our estimate shows that this randomized algorithm may achieve approximation ratio of O(Delta) in general. However, if the number of colors is Theta(n Delta(r)) for some positive r <= 1, his randomized algorithm can beat the bound of O(Delta). Moreover, through our analysis, we also find that if we impose an extra constraint on the color function, the bound O(root Delta ln Delta) still holds. (C) 2015 Elsevier B.V. All rights reserved.
A randomized algorithm is substantiated for the strongly NP-hard problem of partitioning a finite set of vectors of Euclidean space into two clusters of given sizes according to the minimum-of-the sum-of-squared-dista...
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A randomized algorithm is substantiated for the strongly NP-hard problem of partitioning a finite set of vectors of Euclidean space into two clusters of given sizes according to the minimum-of-the sum-of-squared-distances criterion. It is assumed that the centroid of one of the clusters is to be optimized and is determined as the mean value over all vectors in this cluster. The centroid of the other cluster is fixed at the origin. For an established parameter value, the algorithm finds an approximate solution of the problem in time that is linear in the space dimension and the input size of the problem for given values of the relative error and failure probability. The conditions are established under which the algorithm is asymptotically exact and runs in time that is linear in the space dimension and quadratic in the input size of the problem.
Effective testing is essential for assuring software quality. While regression testing is time-consuming, the fault detection capability may be compromised if some test cases are discarded. Test case prioritization is...
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ISBN:
(纸本)9781467379892
Effective testing is essential for assuring software quality. While regression testing is time-consuming, the fault detection capability may be compromised if some test cases are discarded. Test case prioritization is a viable solution. To the best of our knowledge, the most effective test case prioritization approach is still the additional greedy algorithm, and existing search-based algorithms have been shown to be visually less effective than the former algorithms in previous empirical studies. This paper proposes a novel Proportion-Oriented randomized algorithm (PORA) for test case prioritization. PORA guides test case prioritization by optimizing the distance between the prioritized test suite and a hierarchy of distributions of test input data. Our experiment shows that PORA test case prioritization techniques are as effective as, if not more effective than, the total greedy, additional greedy, and ART techniques, which use code coverage information. Moreover, the experiment shows that PORA techniques are more stable in effectiveness than the others.
Big data analysis has become a crucial part of new emerging technologies such as the internet of things, cyber-physical analysis, deep learning, anomaly detection, etc. Among many other techniques, dimensionality redu...
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Big data analysis has become a crucial part of new emerging technologies such as the internet of things, cyber-physical analysis, deep learning, anomaly detection, etc. Among many other techniques, dimensionality reduction plays a key role in such analyses and facilitates feature selection and feature extraction. randomized algorithms are efficient tools for handling big data tensors. They accelerate decomposing large-scale data tensors by reducing the computational complexity of deterministic algorithms and the communication among different levels of memory hierarchy, which is the main bottleneck in modern computing environments and architectures. In this article, we review recent advances in randomization for computation of Tucker decomposition and Higher Order SVD (HOSVD). We discuss random projection and sampling approaches, single-pass and multi-pass randomized algorithms and how to utilize them in the computation of the Tucker decomposition and the HOSVD. Simulations on synthetic and real datasets are provided to compare the performance of some of best and most promising algorithms.
This manuscript describes a technique for computing partial rank-revealing factorizations, such as a partial QR factorization or a partial singular value decomposition. The method takes as input a tolerance epsilon an...
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This manuscript describes a technique for computing partial rank-revealing factorizations, such as a partial QR factorization or a partial singular value decomposition. The method takes as input a tolerance epsilon and an mxn matrix A and returns an approximate low-rank factorization of A that is accurate to within precision epsilon in the Frobenius norm (or some other easily computed norm). The rank k of the computed factorization (which is an output of the algorithm) is in all examples we examined very close to the theoretically optimal epsilon-rank. The proposed method is inspired by the Gram-Schmidt algorithm and has the same O(mnk) asymptotic flop count. However, the method relies on randomized sampling to avoid column pivoting, which allows it to be blocked, and hence accelerates practical computations by reducing communication. Numerical experiments demonstrate that the accuracy of the scheme is for every matrix that was tried at least as good as column-pivoted QR and is sometimes much better. Computational speed is also improved substantially, in particular on GPU architectures.
Traditional low-rank approximation is a powerful tool for compressing large data matrices that arise in simulations of partial differential equations (PDEs), but suffers from high computational cost and requires sever...
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Traditional low-rank approximation is a powerful tool for compressing large data matrices that arise in simulations of partial differential equations (PDEs), but suffers from high computational cost and requires several passes over the PDE data. The compressed data may also lack interpretability thus making it difficult to identify feature patterns from the original data. To address these issues, we present an online randomized algorithm to compute the interpolative decomposition (ID) of large-scale data matrices in situ. Compared to previous randomized IDs that used the QR decomposition to determine the column basis, we adopt a streaming ridge leverage score-based column subset selection algorithm that dynamically selects proper basis columns from the data and thus avoids an extra pass over the data to compute the coefficient matrix of the ID. In particular, we adopt a single-pass error estimator based on the non- adaptive Hutch++ algorithm to provide real-time error approximation for determining the best coefficients. As a result, our approach only needs a single pass over the original data and thus is suitable for large and high-dimensional matrices stored outside of core memory or generated in PDE simulations. A strategy to improve the accuracy of the reconstructed data gradient, when desired, within the ID framework is also presented. We provide numerical experiments on turbulent channel flow and ignition simulations, and on the NSTX Gas Puff Image dataset, comparing our algorithm with the offline ID algorithm to demonstrate its utility in real-world applications.
In this paper, we propose novel quaternion matrix UTV (QUTV) and quaternion tensor UTV (QTUTV) decomposition methods, specifically designed for color image and video processing. We begin by defining both QUTV and QTUT...
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In this paper, we propose novel quaternion matrix UTV (QUTV) and quaternion tensor UTV (QTUTV) decomposition methods, specifically designed for color image and video processing. We begin by defining both QUTV and QTUTV decompositions and provide detailed algorithmic descriptions. To enhance computational efficiency, we introduce randomized versions of these decompositions using random sampling from the quaternion normal distribution, which results in cost-effective and interpretable solutions. Extensive numerical experiments demonstrate that the proposed algorithms significantly improve computational efficiency while maintaining relative errors comparable to existing decomposition methods. These results underscore the strong potential of quaternion-based decompositions for real-world color image and video processing applications. Theoretical findings further support the robustness of the proposed methods, providing a solid foundation for their widespread use in practice.
We present a randomized algorithm to compute a clique of maximum size in the visibility graph G of the vertices of a simple polygon P. The input of the problem consists of the visibility graph G, a Hamiltonian cycle d...
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We present a randomized algorithm to compute a clique of maximum size in the visibility graph G of the vertices of a simple polygon P. The input of the problem consists of the visibility graph G, a Hamiltonian cycle describing the boundary of P, and a parameter delta is an element of (0, 1) controlling the probability of error of the algorithm. The algorithm does not require the coordinates of the vertices of P. With probability at least 1 - delta the algorithm runs in O (vertical bar E(G)vertical bar(2)/omega(G) log(1/delta)) time and returns a maximum clique, where omega(G) is the number of vertices in a maximum clique in G. A deterministic variant of the algorithm takes O (vertical bar E(G)vertical bar(2)) time and always outputs a maximum size clique. This compares well to the best previous algorithm by Ghosh et al. (2007) for the problem, which is deterministic and runs in O(vertical bar V(G)vertical bar(2) vertical bar E(G)vertical bar) time.
The low -rank approximation of big data matrices and tensors plays a pivotal role in many modern applications. Although, a truncated version of the singular value decomposition (SVD) furnishes the best approximation, ...
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The low -rank approximation of big data matrices and tensors plays a pivotal role in many modern applications. Although, a truncated version of the singular value decomposition (SVD) furnishes the best approximation, its computation is challenging on modern, multicore architectures. Recently, the randomized subspace iteration has shown to be a powerful tool in approximating large-scale matrices. In this paper we present a two-sided variant of the randomized subspace iteration. Novel in our work is the exploitation of the unpivoted QR factorization, rather than the SVD, for factorizing the compressed matrix. Hence our algorithm is a randomized rank -revealing URV decomposition. We prove the rank-revealingness of our algorithm by establishing bounds for the singular values as well as the other blocks of the compressed matrix. We further provide bounds on the error of the low -rank approximations of the proposed algorithm, in both 2- and Frobenius norm. In addition, we employ the proposed algorithm to efficiently compute low rank tensor decompositions: we present two randomized algorithms, one for the truncated higher -order SVD, and the other for the tensor SVD. We conduct numerical tests on (i) various classes of matrices, and (ii) synthetic tensors and real datasets to demonstrate the efficacy of the proposed algorithms.
In this paper, a randomized parallel algorithm is proposed to solve the distributed optimal consensus problem of multi-agent systems. Involving both the transient response and the final consensus state, the problem is...
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In this paper, a randomized parallel algorithm is proposed to solve the distributed optimal consensus problem of multi-agent systems. Involving both the transient response and the final consensus state, the problem is described as a constrained non-separable optimization problem. Inspired by the randomized Jacobi proximal alternating direction method of multipliers, the proposed algorithm makes it possible for only a fraction of agents to solve their private subproblems in parallel at each iteration, which greatly saves computational resources and enhances running efficiency. The convergence analysis of the algorithm gives fully distributed convergence conditions. A trade-off between the convergence speed and resource savings is then obtained, where the convergence rate is estimated to be at least O (1 ). Furthermore, the algorithm can be accelerated to enjoy a convergence ( 1 ) t rate of O t2 by adaptively adjusting the auxiliary parameters properly. Numerical simulations demonstrate the effectiveness of the theoretical results.
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