k元船方体网络的可靠性

[k]元[n]方体[Qkn]是设计大规模多处理机系统时最常用的互连网络拓扑结构之一。对于[1≤m≤n-1],设[F]是[Qkn]中的一个由非空点集[VF]和非空边集[EF]构成的故障集,满足[Qkn-F]中不存在[Qkn-m]且[VF]破坏的[Qkn-m]的集合与[EF]破坏的[Qkn-m]的集合互不包含。设[f＊（n,m）]是破坏[Qkn]中的所有子立方[Qkn-m]所需要的故障集[F]的最小基数。证明了对于奇数[k≥3],[fk（n,1）]为[k＋1],[fk（n,n-1）]为[kn-1-1＋n],[f＊（n,m）]的上下界分别为[Cm-1n-1km＋Cm-1n-2km-1]和[km]。举例说明了上界[Cm-1n-1km＋Cm-1n-2km-1]是最优的。     

k元n方体的条件强匹配排除

为了度量发生故障时k元n方体对其可匹配性的保持能力,通过剖析条件故障下使得k元n方体中不存在完美匹配或几乎完美匹配所需故障集的构造,研究了条件故障下使得k元n方体不可匹配所需的最小故障数。当k ≥ 4为偶数且n ≥ 2时,得出了k元n方体这一容错性参数的精确值并对其所有相应的最小故障集进行了刻画;当k ≥ 3为奇数且n ≥ 2时,给出了该k元n方体容错性参数的一个可达下界和一个可达上界。结果表明,选取k为奇数的k元n方体作为底层互连网络拓扑设计的并行计算机系统在条件故障下对其可匹配性有良好的保持能力;进一步地,该系统在故障数不超过2n时仍是可匹配的,要使该系统不可匹配至多需要4n-3个故障元。     

概率故障条件下k元（n-m）方体子网络的可靠性

k元n方体具有许多优良特性,已成为多处理器系统最常用的互连网络拓扑结构之一。当系统互连网络中发生故障时,系统子网络的保持能力对系统实际应用至关重要。为了精确度量k元n方体中任意规模子网络的容错能力,研究了有故障发生时k元n方体中k元（n-m）方体子网络的可靠性。当k(k≥3)为奇整数时,在概率故障条件下得出了k元n方体中存在无故障k元（n-m）方体子网络的概率的上界和下界,并给出了该可靠性的一种近似评估方法。实验结果表明,随着顶点可靠性的降低,k元（n-m）方体子网络可靠性的上下界趋于一致;当顶点可靠性较高时,利用近似评估方法得出的结果更为准确。     

k元n方体互联网络性能分析与研究

从网络的拓扑、路由器、通道3方面分析了k元n方体互联网络的体系结构特征,建立了网络性能模型,并讨论了网络体系结构,应用程序和运行环境对网络性能的影响,以及网络性能的改进措施。     

基于独立元的k近邻故障检测策略

k近邻故障检测(fault detection based on k nearest neighbors,FD–k NN)方法能够提高具有非线性和多模态特征过程的故障检测率.由于系统故障通常由潜隐变量异常变化引起,而该类型故障并不能被观测数据直观表现,因此直接在观测变量上执行FD–k NN方法,其故障检测率降低.本文旨在提高FD–k NN方法针对潜隐变量故障的检测能力,提出基于独立元的k近邻故障检测方法.首先,通过对观测数据应用独立元分析(independent component analysis,ICA)方法,获得独立元矩阵;接下来在独立元矩阵中应用FD–k NN方法进行故障检测.这等同于直接监控过程潜隐变量的变化,可以提高过程故障检测率.通过非线性实例仿真实验,证明本文方法检测潜隐变量故障是有效的;同时,在半导体蚀刻工艺过程的仿真实验中,与主元分析(principal component analysis,PCA)方法、核主元分析(kernel principal component analysis,KPCA)方法、基于主元分析的k近邻故障检测(principal component–based k nearest neighbor rule for fault detection,PC–k NN)方法和FD–k NN方法进行对比,实验结果进一步验证了本文方法的有效性.     

k元n立方网络的k圈排除问题的递归算法

为了度量以k元n立方网络为底层网络拓扑的并行计算机系统的容错能力,通过构造k元n立方网络中使得所有的k元1立方子网都发生故障的最小节点集合的方法,提出求解其k元1立方子网排除点割集的一种递归算法;证明了要使k元n立方网络中所有k元1立方子网都发生故障至少需要破坏掉kn-1个节点。结果表明,在不超过kn-1-1个节点被破坏的情况下,以k元n立方网络为底层拓扑构建的并行计算机系统中依然存在无故障的k元1立方子网。     

k元n方体的可靠性评估

并行计算机系统功能的实现很大程度上依赖于系统互连网络的性能。为了精确度量以k元n方体为底层拓扑结构的并行计算机系统的容错能力,研究了点故障模型下k元n方体中k元（n-1）方体子网络的可靠性。当k ≥ 3且为奇数时,分别在固定划分模式和灵活划分模式下对k元n方体中不同数目的k元（n-1）方体子网络保持无故障状态的平均失效时间进行了分析,并得出了这一子网络可靠性评估参数的计算公式。结果表明,当基于k为奇数的k元n方体构建的并行计算机系统指派子网络执行用户任务时,在点故障模型下灵活划分模式相比固定划分模式有着更好的容错能力。     

k元n树互联网络的2-终点可靠性研究

k元n树是一种用于大规模并行处理的高性能互联网络的拓扑结构。该文提出计算k元n树的2-终点可靠性的递归算法,其计算复杂度为O(n)。结合市场上网络元器件可靠性的实际情况,对k元n树的2-终点可靠性进行了分析。当n趋于无穷大时,计算出k元n树的2-终点可靠性的下限。     

边故障[k]元[n]立方体的超级哈密顿交织性

[k]元[n]立方体（记为[Qkn]）是优于超立方体的可进行高效信息传输的互连网络之一。[Qkn]是一个二部图当且仅当[k]为偶数。令[G[V0,V1]]是一个二部图,若（1）任意一对分别在不同部的顶点之间存在一条哈密顿路,且（2）对于任意一点[v∈Vi],其中[i∈{0,1}],[V1-i]中任意一对顶点可以被[G[V0,V1]-v]中的一条哈密顿路相连,则图[G[V0,V1]]被称为是超级哈密顿交织的。因为网络中的元件发生故障是不可避免的,所以研究网络的容错性就尤为重要。针对含有边故障的[Qkn],其中[k4]是偶数且[n2],证明了当其故障边数至多为[2n-3]时,该故障[Qkn]是超级哈密顿交织图,且故障边数目的上界[2n-3]是最优的。     

边故障K元3立方体的二不交路覆盖

研究具有故障边的[k]元3立方体的非指定二不交路覆盖问题。证明了在具有至多3条故障边的[k]元3立方体[Qk3]中,任意给定两个源点和两个汇点,则存在两条顶点不交的路[P1]和[P2],分别连接一个源点和汇点,且[V(P1)?V(P2)=V(Qk3)]。     

Fault-free Hamiltonian cycles in crossed cubes with conditional link faults

The crossed cube, which is a variation of the hypercube, possesses some properties superior to the hypercube. In this paper, assuming that each node is incident with at least two fault-free links, we show that an n-dimensional crossed cube contains a fault-free Hamiltonian cycle, even if there are up to 2n − 5 link faults. The result is optimal with respect to the number of link faults tolerated. We also verify that the assumption is practically meaningful by evaluating its occurrence probability, which is very close to 1.     

Embedding paths of variable lengths into hypercubes with conditional link-faults

Faults in a network may take various forms such as hardware failures while a node or a link stops functioning, software errors, or even missing of transmitted packets. In this paper, we study the link-fault-tolerant capability of an n-dimensional hypercube (n-cube for short) with respect to path embedding of variable lengths in the range from the shortest to the longest. Let F be a set consisting of faulty links in a wounded n-cube Q_n, in which every node is still incident to at least two fault-free links. Then we show that Q_n-F has a path of any odd (resp. even) length in the range from the distance to 2ⁿ-1 (resp. 2ⁿ-2) between two arbitrary nodes even if |F|=2n-5. In order to tackle this problem, we also investigate the fault diameter of an n-cube with hybrid node and link faults.     

Hamiltonian paths and cycles with prescribed edges in the 3-ary n-cube

The k-ary n-cube has been one of the most popular interconnection networks for massively parallel systems. Given a set P of at most 2n − 2 (n ? 2) prescribed edges and two vertices u and v, we show that the 3-ary n-cube contains a Hamiltonian path between u and v passing through all edges of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths, none of them having u or v as internal vertices or both of them as end-vertices. As an immediate result, the 3-ary n-cube contains a Hamiltonian cycle passing through a set P of at most 2n − 1 prescribed edges if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths.     

Strong Menger connectivity with conditional faults on the class of hypercube-like networks

In this paper, we study the Menger property on a class of hypercube-like networks. We show that in all n-dimensional hypercube-like networks with n−2 vertices removed, every pair of unremoved vertices u and v are connected by min{deg(u),deg(v)} vertex-disjoint paths, where deg(u) and deg(v) are the remaining degree of vertices u and v, respectively. Furthermore, under the restricted condition that each vertex has at least two fault-free adjacent vertices, all hypercube-like networks still have the strong Menger property, even if there are up to 2n−5 vertex faults.     

Embedding Hamiltonian cycles in alternating group graphs under conditional fault model

In this paper, assuming that each node is incident with two or more fault-free links, we show that an n-dimensional alternating group graph can tolerate up to 4n − 13 link faults, where n ? 4, while retaining a fault-free Hamiltonian cycle. The proof is computer-assisted. The result is optimal with respect to the number of link faults tolerated. Previously, without the assumption, at most 2n − 6 link faults can be tolerated for the same problem and the same graph.     

Fast barrier synchronization in wormhole k-ary n-cube networks with multidestination worms

This paper presents a new approach to implement fast barrier synchronization in wormhole k-ary n-cubes. The novelty lies in using multidestination messages instead of the traditional single destination messages. Two different multidestination worm types, gather and broadcasting, are introduced to implement the report and wake-up phases of barrier synchronization, respectively. Algorithms for complete and arbitrary set barrier synchronization are presented using these new worms. It is shown that complete barrier synchronization in a k-ary n-cube system with e-cube routing can be implemented with 2n communication start-ups as compared to 2nlog₂k start-ups needed with unicast-based message passing. This leads to an asymptotic improvement by a factor of log₂k. Simulation results for different system and architectural parameters indicate that the new framework can reduce barrier synchronization cost considerably compared to the unicast-based scheme. For arbitrary set barrier, an interesting trend is observed where the synchronization cost keeps on reducing beyond a certain number of participating nodes. The framework demonstrates potential for supporting fast barrier synchronization in large wormhole-routed systems.     

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.     

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.