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In this paper, two accelerated divide-and-conquer (ADC) algorithms are proposed for the symmetric tridiagonal eigenvalue problem, which cost O(N2r) flops in the worst case, where N is the dimension of the matrix and r is a modest number depending on the distribution of eigenvalues. Both of these algorithms use hierarchically semiseparable (HSS) matrices to approximate some intermediate eigenvector matrices, which are Cauchy-like matrices and are off-diagonally low-rank. The difference of these two versions lies in using different HSS construction algorithms, one (denoted by ADC1) uses a structured low-rank approximation method and the other (ADC2) uses a randomized HSS construction algorithm. For the ADC2 algorithm, a method is proposed to estimate the off-diagonal rank. Numerous experiments have been carried out to show their stability and efficiency. These algorithms are implemented in parallel in a shared memory environment, and some parallel implementation details are included. Comparing the ADCs with highly optimized multithreaded libraries such as Intel MKL, we find that ADCs could be more than six times faster for some large matrices with few deflations. [ABSTRACT FROM AUTHOR]

Local monitoring is an effective mechanism for the security of wireless sensor networks (WSNs). Existing schemes assume the existence of sufficient number of active nodes to carry out monitoring operations. Such an assumption, however, is often difficult for a large-scale sensor network. In this work, we focus on designing an efficient scheme integrated with good self-monitoring capability as well as providing an infrastructure for various security protocols using local monitoring. To the best of our knowledge, we are the first to present the formal study on optimizing network topology for edge self-monitoring in WSNs. We show that the problem is NP-complete even under the unit disk graph (UDG) model and give the upper bound on the approximation ratio in various graph models. We provide polynomial-time approximation scheme (PTAS) algorithms for the problem in some specific graphs, for example, the monitoring-set-bounded graph. We further design two distributed polynomial algorithms with provable approximation ratio. Through comprehensive simulations, we evaluate the effectiveness of our design. [ABSTRACT FROM AUTHOR]

In this paper, we modify a classical downdating SVD algorithm and reduce its complexity significantly. We use a structured low-rank approximation algorithm to compute an hierarchically semiseparable (HSS) matrix approximation to the eigenvector matrix of a diagonal matrix plus rank-one modification. The complexity of our downdating algorithm is analyzed. We further show that the structured low-rank approximation algorithm is backward stable. Numerous experiments have been done to show the efficiency of our algorithm. For some matrices with large dimensions, our algorithm can be much faster than that using plain matrix–matrix multiplication routine in Intel MKL in both sequential and parallel cases. [ABSTRACT FROM AUTHOR]