Fault tolerance of hypercubes and folded hypercubes

Let \(G = (V,E)\) be a connected graph. The conditional edge connectivity \(\lambda _\delta ^k(G)\) is the cardinality of the minimum edge cuts, if any, whose deletion disconnects \(G\) and each component of \(G - F\) has \(\delta \ge k\) . We assume that \(F \subseteq E\) is an edge set, \(F\) is called edge extra-cut, if \(G - F\) is not connected and each component of \(G - F\) has more than \(k\) vertices. The edge extraconnectivity \(\lambda _\mathrm{e}^k(G)\) is the cardinality of the minimum edge extra-cuts. In this paper, we study the conditional edge connectivity and edge extraconnectivity of hypercubes and folded hypercubes.     

平衡超立方体的故障容错性

故障容错是衡量多处理器互连网络可靠性的重要方式之一。其中g-限制边连通度和g-限制连通度保证了剩下每个分支之间不连通且每个分支中节点的邻居数目不少于 g,能够更加精准地测量多处理器和多信道系统的容错性和可靠性。平衡超立方体是超立方体的一个变形,它特有的良好拓扑性质能够更好地满足多处理器系统和多种新型网络的需要。提出了n维平衡超立方体的{1,2}-限制边连通度和{1,2}-限制连通度,能够丰富以平衡超立方体为拓扑结构的网络容错性和可靠性的评价体系,并为平衡超立方体的故障诊断算法打下良好基础。     

Structure fault tolerance of balanced hypercubes

The Journal of Supercomputing - Let H be a connected subgraph of a given graph G. The H-structure connectivity of G is the cardinality of a minimal set $${mathcal {F}}$$ of subgraphs of G such...     

Cycles embedding in hypercubes with node failures

The hypercube has been widely used as the interconnection network in parallel computers. The n-dimensional hypercube Qn is a graph having n² vertices each labeled with a distinct n-bit binary strings. Two vertices are linked by an edge if and only if their addresses differ exactly in the one bit position. Let fv denote the number of faulty vertices in Qn. For n?3, in this paper, we prove that every fault-free edge and fault-free vertex of Qn lies on a fault-free cycle of every even length from 4 to n²−2fv inclusive even if fv?n−2. Our results are optimal.     

Broadcasting schemes for hypercubes with background traffic

Optimal broadcasting schemes for interconnection networks (INs) are most essential for the efficient interprocess communication amongst parallel computers. In this paper two novel broadcasting schemes are proposed for hypercube computers with bursty background traffic and a single-port mode of message passing communication. The schemes utilize a maximum entropy (ME) based queue-by-queue decomposition algorithm for arbitrary queueing network models (QNMs) [D.D. Kouvatsos, I. Awan, Perform. Eval. 51 (2003) 191] and are based on binomial trees and graph theoretic concepts. It is shown that the overall cost of the one-to-all broadcasting scheme is given by max{ω₁,ω₂,…,ω_2ⁿ/2}, where ω_i, i=1,2,…,2ⁿ/2 is the total weight at each leaf node of the binomial tree and n is the degree of the hypercube. Moreover, the upper bound of the total cost of the neighbourhood broadcasting scheme is determined by ∑_i=1^Fmax{ω_i}, where F is an upper bound of the number of steps and is equal to 1.33⌈log₂(n−1)⌉+1. Evidence based on empirical studies indicates the suitability of the schemes for achieving optimal broadcasting costs.     

Incomplete crossed hypercubes

In this paper, a new interconnection network called the incomplete crossed hypercube is proposed for connecting processors of parallel computing systems. The incomplete crossed hypercube architecture denoted by CI _n−m ⁿ is made by combining two complete crossed hypercubes CQ _n and CQ _n−m for 1≤m≤n. Several topological properties of CI _n−m ⁿ are analyzed. In particular, accurate mean internode distance formulas of both CQ _n and CI _n−m ⁿ are given. Compared with the incomplete enhanced hypercube EI _n−m ⁿ , CI _n−m ⁿ has shorter mean internode distance for large n. An optimal routing algorithm for CI _n−m ⁿ is also described which guarantees the generation of a shortest path from a node to another in CI _n−m ⁿ .     

Long paths in hypercubes with conditional node-faults

Let F be a set of f?2n-5 faulty nodes in an n-cube Q_n such that every node of Q_n still has at least two fault-free neighbors. Then we show that Q_n-F contains a path of length at least 2ⁿ-2f-1 (respectively, 2ⁿ-2f-2) between any two nodes of odd (respectively, even) distance. Since the n-cube is bipartite, the path of length 2ⁿ-2f-1 (or 2ⁿ-2f-2) turns out to be the longest if all faulty nodes belong to the same partite set. As a contribution, our study improves upon the previous result presented by [J.-S. Fu, Longest fault-free paths in hypercubes with vertex faults, Information Sciences 176 (2006) 759-771] where only n-2 faulty nodes are considered.     

贝叶斯网络的非忠实性分布

贝叶斯网络是图论和概率论有机融合的概率图形模型,主要用于统计推理和智能数据分析．理论上通常假设由基于分布的独立关系可推出基于图结构的d-分割,即贝叶斯网络上的分布是忠实的．针对布尔域上的贝叶斯网络,研究了非忠实分布的构成,提出了贝叶斯网络上分布延拓的概念,得到忠实分布与非忠实分布的平凡延拓均是非忠实分布．     

Cost-sensitive classification with respect to waiting cost

Extant multiple-cost-sensitive learning algorithms are usually designed for dealing with misclassification cost (MC) and test cost (TC) together. This paper outlines a new learning algorithm, called cost-time sensitive classification, designed for minimizing tangible costs (which includes TC and waiting cost (WC)) as well as maximizing the decrease of the intangible costs (also called MC). The proposed algorithm induces decision trees from training datasets with missing data, in which the costs are measured in different units. Firstly, a split criterion is proposed for building cost-time sensitive decision trees, aiming at possibly reducing the intangible cost. Then a hybrid test strategy, which can handle missing values in test datasets, is designed for combining the sequential test with the batch test strategy. To evaluate the efficiency of the proposed method, extensive experiments were conducted on the UCI datasets at different missing rates. The experimental results show that the proposed algorithm achieves better than the existing ones in terms of reducing the intangible costs when taking into account waiting costs.     

Optimization with respect to covariance sequence parameters

A general problem involving optimization of a covariance sequence is considered in the paper. One difficulty with this class of problems is to ensure that the covariance sequence is nonnegative definite (in other words, realizable). It is suggested that this difficulty can be overcome by reformulating the optimization problem in terms of the partial autocorrelation coefficients (PAC). One need only constrain these coefficients to lie in the range (−1, 1) to guarantee that the corresponding covariance sequence is nonnegative definite. The synthesis of a signal realizing the optimizing covariance sequence is also discussed. Special emphasis is given to the case when some of the PACs are either +1 or −1.     

Design optimization with respect to ergonomic properties

This paper introduces the idea of engineering design optimization with respect to ergonomic properties. The optimization is based on a detailed inverse dynamic analysis of the motion and forces of the human body. The problem of muscle recruitment calculation in inverse dynamics is introduced and solved via a min/max optimality criterion. The implementation of this criterion in the body modelling system AnyBody is described, and the use of the system is demonstrated on the sample problem of designing a hand saw. We conclude that the methodology is applicable but dependent on anthropometrically accurate body models.     

Longest fault-free paths in hypercubes with vertex faults

The hypercube is one of the most versatile and efficient interconnection networks (networks for short) so far discovered for parallel computation. Let f denote the number of faulty vertices in an n-cube. This study demonstrates that when f ? n − 2, the n-cube contains a fault-free path with length at least 2ⁿ − 2f − 1 (or 2ⁿ − 2f − 2) between two arbitrary vertices of odd (or even) distance. Since an n-cube is a bipartite graph with two partite sets of equal size, the path is longest in the worst-case. Furthermore, since the connectivity of an n-cube is n, the n-cube cannot tolerate n − 1 faulty vertices. Hence, our result is optimal.     

Path partitions of hypercubes

A path partition of a graph G is a set of vertex-disjoint paths that cover all vertices of G. Given a set of pairs of distinct vertices of the n-dimensional hypercube Qn, is there a path partition of Qn such that ai and bi are endvertices of Pi? Caha and Koubek showed that for 6m?n, such a path partition exists if and only if the set P is balanced in the sense that it contains the same number of vertices from both classes of bipartition of Qn.In this paper we show that this result holds even for 2m−e<n, where e is the number of pairs of P that form edges of Qn. Moreover, our bound is optimal in the sense that for every n?3, there is a balanced set P in Qn such that 2m−e=n, but no path partition with endvertices prescribed by P exists.     

Smooth interpolation over hypercubes

Multivariate interpolation problems, which occur often in scientific research, can sometimes be approached using Coons' patches. Coons' patches are smooth, local interpolants to lower dimensional data sets. (For example, they interpolate curves of data when considering 3-dimensional problems.) However, they do not interpolate as desired unless all the mixed partial derivatives, the ‘twists’, are equal. The twists are not equal in many cases of practical importance, such as for wire frame data. Gregory (1983) has developed a compatibly corrected Coons' patch for 3-dimensional surfaces. We generalize this interpolant, ‘Gregory's square’, to the case of functions of n variables. The interpolant we propose is built up inductively from one to n-dimensions, requiring, at each step, only one additional term to be defined. This is the key to the whole process and involves the definition of a general ‘twist operator’.     

Fast compaction in hypercubes

Compaction relocates active subcubes in a fragmented hypercube so as to produce a contiguous free region and eliminate the adverse impact of fragmentation on performance. The overhead of compaction is often contributed primarily by task migration, which makes use of disjoint paths for transmitting migrated data. Since task migration usually involves transmitting a large amount of data, the time required for migration with single paths is long, making compaction an undesirably lengthy process. This paper considers fast compaction through the use of all disjoint paths in existence for migration simultaneously from a source subcube to its target subcube, effectively reducing the size of data transmitted over a path and shortening the migration time. This approach leads to considerable savings in the compaction time for hypercubes which support circuit switching or wormhole routing, when compared with that using single migration paths     

The diagnosability of hypercubes with arbitrarily missing links

We study the problem of determining diagnosability for incomplete hypercubes that have arbitrarily distributed missing links, under the classic PMC diagnostic model and its variant, the BGM model. Based on the result proved in this paper, for both models, in most cases the diagnosability of an incomplete hypercube can be determined by simply checking the link degree of each node.     

Edge-fault-tolerant edge-bipancyclicity of hypercubes

In this paper, we consider the problem embedding a cycle into the hypercube Qn with existence of faulty edges and show that for any edge subset F of Qn with |F|?n−1 every edge of Qn−F lies on a cycle of every even length from 6 to n² inclusive provided n?4 and all edges in F are not incident with the same vertex. This result improves some known results.