On the maximal connected component of hypercube with faulty vertices (II)

evaluating the fault tolerance of an interconnection network, it is essential to estimate the size of a maximal connected component of the network at the presence of faulty processors. Hypercube is one of the most popular interconnection networks. In this paper, we prove that for ngreater than or equal to6, an n-dimensional cube with a set F of at most (4n-10) failing processors has a component of size greater than or equal to2"-\F-3. This result demonstrates the superiority of hypercube in terms of the fault tolerance.

Maximum induced subgraph of a recursive circulant

The recursive circulant RC(2(n), 4) enjoys several attractive topological properties. Let max_epsilon(G) (m) denote the maximum number of edges in a subgraph of graph G induced by m nodes. In this paper, we show that max_epsilon(RC(2n,4))(m) = Sigma(i)(r)=(0)(p(i)/2 + i)2(Pi), where p(0) > p(1) > ... > p(r) are nonnegative integers defined by m = Sigma(i)(r)=(0)2(Pi). We then apply this formula to find the bisection width of RC(2(n), 4). The conclusion shows that, as n-dimensional cube, RC(2(n), 4) enjoys a linear bisection width. (c) 2005 Elsevier B.V. All rights reserved.

On a non-autonomous difference equation

Abu-Saris and DeVault proposed two open problems about the difference equation x(n+1) = a(n)x(n)/x(n-1), n = 0, 1, 2,..., where a(n) not equal 0 for n = 0, 1, 2..., x(-1) not equal 0, x(0) not equal 0. In this paper we provide solutions to the two open problems. (c) 2004 Elsevier Inc. All rights reserved.

On the maximal connected component of a hypercube with faulty vertices III

Hypercube is one of the most popular topologies for connecting processors in multicomputer systems. In this paper we address the maximum order of a connected component in a faulty cube. The results established include several known conclusions as special cases. We conclude that the hypercube structure is resilient as it includes a large connected component in the presence of large number of faulty vertices.

On two rational difference equations

In this paper, we study the oscillating property of positive solutions and the global asymptotic stability of the unique equilibrium of the two rational difference equations [GRAPHICS] and [GRAPHICS] where a is a nonnegative constant. (c) 2005 Elsevier Inc. All rights reserved.

Global asymptotic stability in a rational recursive sequence

In this paper, we study the global stability of the difference equation x(n) = a + bx(n-1) + cx(n-1)(2)/d - x(n-2), n = 1,2,....., where a, b greater than or equal to 0 and c, d > 0. We show that one nonnegative equilibrium point of the equation is a global attractor with a basin that is determined by the parameters, and every positive Solution of the equation in the basin exponentially converges to the attractor. (C) 2003 Elsevier Inc. All rights reserved.

Global exponential periodicity of a class of bidirectional associative memory networks with finite distributed delays

In this paper, we study the periodic oscillatory behavior of a class of bidirectional associative memory (BAM) networks with finite distributed delays. A set of criteria are proposed for determining global exponential periodicity of the proposed BAM networks, which assume neither differentiability nor monotonicity of the activation function of each neuron. In addition, our criteria are easily checkable. (c) 2005 Elsevier Inc. All rights reserved.

Global exponential periodicity of a class of neural networks with recent-history distributed delays

In this paper, we propose to study a class of neural networks with recent-history distributed delays. A sufficient condition is derived for the global exponential periodicity of the proposed neural networks, which has the advantage that it assumes neither the differentiability nor monotonicity of the activation function of each neuron nor the symmetry of the feedback matrix or delayed feedback matrix. Our criterion is shown to be valid by applying it to an illustrative system. (c) 2005 Elsevier Ltd. All rights reserved.

Locally twisted cubes are 4-pancyclic

The locally twisted cube is a newly introduced interconnection network for parallel computing. Ring embedding is an important issue for evaluating the performance of an interconnection network. In this paper, we investigate the problem of embedding rings into a locally twisted cube. Our main contribution is to find that, for each integer l is an element of (4,5,...,2(n)}, a ring of length I can be embedded into an n-dimensional locally twisted cube so that both the dilation and the load factor are one. As a result, a locally twisted cube is Hamiltonian. We conclude that a locally twisted cube is superior to a hypercube in terms of ring embedding capability. (C) 2004 Elsevier Ltd. All rights reserved.

A comparison-based diagnosis algorithm tailored for crossed cube multiprocessor systems

Comparison-based diagnosis is an effective approach to system-level fault diagnosis. Under the Maeng-Malek comparison model (NM* model), Sengupta and Dahbura proposed an O(N-5) diagnosis algorithm for general diagnosable systems with N nodes. Thanks to lower diameter and better graph embedding capability as compared with a hypercube of the same size, the crossed cube has been a promising candidate for interconnection networks. In this paper, we propose a fault diagnosis algorithm tailored for crossed cube connected multicomputer systems under the MM* model. By introducing appropriate data structures, this algorithm runs in O(Nlog(2)(2) N) time, which is linear in the size of the input. As a result, this algorithm is significantly superior to the Sengupta-Dahbura's algorithm when applied to crossed cube systems. (C) 2004 Elsevier B.V. All rights reserved.

Pancyclicity of Mobius cubes with faulty nodes

An interconnection network with n nodes is four-pancyclic if it contains a cycle of length l for each integer l with 4 <= l <= n. An interconnection network is fault-tolerant four-pancyclic if the surviving network is four-pancyclic in the presence of faults. The fault-tolerant four-pancyclicity of interconnection networks is a desired property because many classical parallel algorithms can be mapped onto such networks in a communication-efficient fashion, even in the presence of failing nodes or edges. Due to some attractive properties as compared with its hypercube counterpart of the same size, the Mobius cube has been proposed as a promising candidate for interconnection topology. Hsieh and Chen [S.Y. Hsieh, C.H. Chen, Pancyclicity on Mobius cubes with maximal edge faults, Parallel Computing, 30(3) (2004) 407-421.] showed that an n-dimensional Mobius cube is four-pancyclic in the presence of up to n-2 faulty edges. In this paper, we show that an n-dimensional Mobius cube is four-pancyclic in the presence of up to n-2 faulty nodes. The obtained result is optimal in that, if n-1 nodes are removed, the surviving network may not be four-pancyclic. (C) 2005 Elsevier B.V. All rights reserved.

An oblivious shortest-path routing algorithm for fully connected cubic networks

Fully connected cubic networks (FCCNs) are a class of newly proposed hierarchical interconnection networks for multicomputer systems, which enjoy the strengths of constant node degree and good expandability. The shortest path routing in FCCNs is an open problem. In this paper, we present an oblivious routing algorithm for n-level FCCN with N = 8(n) nodes, and prove that this algorithm creates a shortest path from the source to the destination. At the costs of both an O(N)-parallel-step off-line preprocessing phase and a list of size N stored at each node, the proposed algorithm is carried out at each related node in O(n) time. In some cases the proposed algorithm is superior to the one proposed by Chang and Wang in terms of the length of the routing path. This justifies the utility of our routing strategy. (C) 2006 Elsevier Inc. All rights reserved.

Diameter of parallelogramic honeycomb torus

The determination of the diameter of an interconnection network is essential in evaluating the performance of the network. Parallelogramic honeycomb torus is an attractive alternative to classical torus network due to smaller vertex degree, and hence, lower implementation cost. In this paper, we present the expression for the diameter of a parallelogramic, honeycomb torus, which extends a known result about rhombic: honeycomb torus. (c) 2005 Elsevier Ltd. All rights reserved.

A novel radix-3/9 algorithm for type-III generalized discrete Hartley transform

A novel radix-3/9 algorithm for type-III generalized discrete Hartley transform (GDHT) is proposed, which applies to length-3(P) sequences. This algorithm is especially efficient in the case that multiplication is much more time-consuming than addition. A comparison analysis shows that the proposed algorithm outperforms a known algorithm when one multiplication is more time-consuming than five additions. When combined with any known radix-2 type-III GDHT algorithm, the new algorithm also applies to length-2(q)3(P) sequences.

On the recursive sequence Xn = axn-1+bxn-2/c+dxn-1xn-2

In this Paper, we study the invariant intervals, the global attractivity of the equilibrium points, and the asymptotic behavior of the solutions of the difference equation x(n) = ax(n-1) + bx(n-2) / c + dx(n-1)x(n-2), n =1, 2, ..., where a greater than or equal to 0, b, c, d > 0. (C) 2004 Elsevier Inc. All rights reserved.