The design and analysis of an efficient load balancing algorithm employing the symmetric balanced incomplete block design

In order to maintain load balancing in a distributed network, each node should obtain workload information from all the nodes in the network. To accomplish this, this processing requires O(v²) communication complexity, where v is the number of nodes. First, we present a new synchronous dynamic distributed load balancing algorithm on a (v, k + 1, 1)-configured network applying a symmetric balanced incomplete block design, where v = k² + k + 1. Our algorithm designs a special adjacency matrix and then transforms it to (v, k + 1, 1)-configured network for an efficient communication. It requires only communication complexity and each node receives workload information from all the nodes without redundancy since each link has the same amount of traffic for transferring workload information. Later, this algorithm is revised for distributed networks and is analyzed in terms of efficiency of load balancing.     

On-line restricted assignment of temporary tasks with unknown durations

We consider load balancing of temporary tasks on m machines in the restricted assignment model. It is known that the best competitive ratio for this problem is . This bound is not achieved by the greedy algorithm whose competitive ratio is known to be Ω(m2/3). We give an alternative analysis to the greedy algorithm which is better than the known analysis for relatively small values of m. We also show that for a small number of machines, namely m?5, the greedy algorithm is optimal.     

A new approach for approximating node deletion problems

We present a new approach for approximating node deletion problems by combining the local ratio and the greedy multicovering algorithms. For a function , our approach allows to design a 2+max_v∈V(G)logf(v) approximation algorithm for the problem of deleting a minimum number of nodes so that the degree of each node v in the remaining graph is at most f(v). This approximation ratio is shown to be asymptotically optimal. The new method is also used to design a 1+(log2)(k−1) approximation algorithm for the problem of deleting a minimum number of nodes so that the remaining graph contains no k-bicliques.     

Improved generalized Atkin algorithm for computing square roots in finite fields

Recently, S. Müller developed a generalized Atkin algorithm for computing square roots, which requires two exponentiations in finite fields GF(q) when . In this paper, we present a simple improvement to it and the improved algorithm requires only one exponentiation for half of squares in finite fields GF(q) when . Furthermore, in finite fields GF(pm), where and m is odd, we reduce the complexity of the algorithm from O(m³log³p) to O(m²log²p(logm+logp)) using the Frobenius map and normal basis representation.     

On the Euler sequence spaces which include the spaces ?p and ?∞ I

In the present paper, we introduce the Euler sequence space consisting of all sequences whose Euler transforms of order r are in the space ?_p of non-absolute type which is the BK-space including the space ?_p and prove that the spaces and ?_p are linearly isomorphic for 1 ? p ? ∞. Furthermore, we give some inclusion relations concerning the space . Finally, we determine the α-, β- and γ-duals of the space for 1 ? p ? ∞ and construct the basis for the space , where 1 ? p < ∞.     

Regularization parameter estimation for large-scale Tikhonov regularization using a priori information

Solutions of numerically ill-posed least squares problems for A∈R^m×n by Tikhonov regularization are considered. For D∈R^p×n, the Tikhonov regularized least squares functional is given by where matrix W is a weighting matrix and is given. Given a priori estimates on the covariance structure of errors in the measurement data , the weighting matrix may be taken as which is the inverse covariance matrix of the mean 0 normally distributed measurement errors in . If in addition is an estimate of the mean value of , and σ is a suitable statistically-chosen value, J evaluated at its minimizer approximately follows a χ² distribution with degrees of freedom. Using the generalized singular value decomposition of the matrix pair , σ can then be found such that the resulting J follows this χ² distribution. But the use of an algorithm which explicitly relies on the direct solution of the problem obtained using the generalized singular value decomposition is not practical for large-scale problems. Instead an approach using the Golub-Kahan iterative bidiagonalization of the regularized problem is presented. The original algorithm is extended for cases in which is not available, but instead a set of measurement data provides an estimate of the mean value of . The sensitivity of the Newton algorithm to the number of steps used in the Golub-Kahan iterative bidiagonalization, and the relation between the size of the projected subproblem and σ are discussed. Experiments presented contrast the efficiency and robustness with other standard methods for finding the regularization parameter for a set of test problems and for the restoration of a relatively large real seismic signal. An application for image deblurring also validates the approach for large-scale problems. It is concluded that the presented approach is robust for both small and large-scale discretely ill-posed least squares problems.     

Uniform metrical task systems with a limited number of states

We give a randomized algorithm (the “Wedge Algorithm”) of competitiveness for any metrical task system on a uniform space of k points, for any k?2, where , the kth harmonic number. This algorithm has better competitiveness than the Irani-Seiden algorithm if k is smaller than 10⁸. The algorithm is better by a factor of 2 if k<47.     

Embedding paths and cycles in 3-ary n-cubes with faulty nodes and links

The k-ary n-cube, denoted by , is one of the most important interconnection networks for parallel computing. In this paper, we consider the problem of embedding cycles and paths into faulty 3-ary n-cubes. Let F be a set of faulty nodes and/or edges, and n?2. We show that when |F|?2n-2, there exists a cycle of any length from 3 to in . We also prove that when |F|?2n-3, there exists a path of any length from 2n-1 to between two arbitrary nodes in . Since the k-ary n-cube is regular of degree 2n, the fault-tolerant degrees 2n-2 and 2n-3 are optimal.