k元n方体网络的子网络可靠性

并行计算机系统互连网络的拓扑性质对系统功能的实现起着重要的作用。为了衡量基于[k]元[n]方体网络构建的并行计算机系统的容错能力,研究了边故障模型下[k]元[n]方体网络中[k]元[(n-1)]方体子网络的可靠性。当[k(k≥3)]为奇数时,分别在固定划分模式和灵活划分模式下得出了[k]元[n]方体网络中不同数目的[k]元[(n-1)]方体子网络保持无故障状态的平均失效时间的计算公式,并通过仿真实验验证了理论结果的精确性。研究表明,当[k]为奇数的[k]元[n]方体网络中有边故障发生时,相比固定划分模式,在灵活划分模式下不同数目的[k]元[(n-1)]方体子网络保持无故障状态的平均失效时间更大。     

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)]。     

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。     

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。     

外1-平面图的均匀边染色

图[G]的[s]-均匀边[k]-染色是指用[k]种颜色对图的边进行染色,使得图[G]的每个顶点所关联的任何两种颜色的边的条数至多相差[s]。使得对于每个不小于[k]的整数[t],图[G]都具有[s]-均匀边[t]-染色的最小整数[k]称为图[G]的[s]-均匀边色数阈值。文中证明了外1-平面图的1-均匀边色数阈值最多为5,不含有相邻的3圈的外1-平面图的均匀边色数阈值最多为4,外1-平面图的2-均匀边色数阈值恰好为1。     

在PMC模型下单向k元n立方体的诊断度

图的连通度和诊断度是与互连网络的可靠性密切相关的两个参数,而[g]好邻连通度和[g]好邻诊断度是比连通度和诊断度更精确的指标。[k]元[n]立方体是多处理机系统的最常用网络之一,而单向[k]元[n]立方体是指具有单向边的[k]元[n]立方体。证明了当[k≥3,n≥3]时,单向[k]元[n]立方体在PMC模型下的[1]好邻连通度是[k(n-1)],诊断度是[n]且[1]好邻诊断度是[kn-1]。     

单向k-元n-立方体网络

单向[k]-元[n]-立方体是指具有单向边的[k]-元[n]-立方体互连网络拓扑。当网络包含的顶点数目较大时,比起传统的双向[k]-元[n]-立方体,单向?[k]-元[n]-立方体对通信硬件复杂性的要求更低一些。提出了[k]-元[n]-立方体的一个定向,使得定向后的单向[k]-元[n]-立方体[UQkn]有一些良好的性质。证明了[UQkn]是正则的,极大弧连通的,具有迭代结构的且[UQkn]的直径是小的。此外,提出了一个简单的多项式时间路由算法。     

超方体Qn的路分解及其算法

在超方体[Qn]的路分解的研究中,证明了[Qn]存在[{Pn+1}]-分解的定理;分别给出了[Qn]存在[{P4}]-分解的充分必要条件和存在[{P3,P4}]-分解的充分条件;结合超方体的性质和路分解结论,设计出超方体的路分解算法程序。     

BSCC(4,k)的Hamilton圈分解

冒泡排序连通圈网络BSCC(n)是一类重要的互连网络。2010年师海忠提出了如下猜想:冒泡排序连通圈网络BSCC(n)(n≥4)可分解为边不交的Hamilton圈和完美对集的并。记BSCC(n)为BSCC(n,0),对BSCC(n,0)的每个顶点用一个三角形代替,得到新网络BSCC(n,1),对BSCC(n,1)的每个顶点用三角形代替得到BSCC(n,2),类似迭代k次得新网络BSCC(n,k)。师海忠进一步提出猜想2:BSCC(n,k)可分解为边不交的一个Hamilton圈和一个完美对集的并。证明了BSCC(4,k)可分解成边不交的一个Hamilton圈和一个完美对集的并。     

k元n维冒泡排序网络的子网排除

在并行计算机系统中,元器件和线路故障普遍存在,而系统的容错能力可以通过其底层基础网络的拓扑性质衡量。为了精确度量以k元n维冒泡排序网络为底层拓扑结构的并行计算机系统的容错能力,结合其层次结构和子网划分特征,分别提出了节点故障模型和线路故障模型下攻击该网络中所有k－m元n－m维冒泡排序子网络的算法,确定了需要攻击的最优节点集合和最优线路集合。根据算法可得:当2≤k≤n－2,m≤k－1时,攻击k元n维冒泡排序网络中所有的k－m元n－m维冒泡排序子网络,在节点故障模型下需要攻击至少C^m_nm!个节点,在边故障模型下需要攻击至少C^m_nm!条线路。     

Embedding Long Paths in k-Ary n-Cubes with Faulty Nodes and Links

Let k ges 4 be even and let n ges 2. Consider a faulty k-ary n-cube Q^k _n in which the number of node faults fv and the number of link faults f_e are such that f_v + f_e les 2n-2. We prove that given any two healthy nodes s and e of Q^k _n, there is a path from s to e of length at least kⁿ - 2 f_v -1 (respectively, kⁿ - 2 f_v -2) if the nodes s and e have different (respectively the same) parities (the parity of a node in Q^k _n is the sum modulo 2 of the elements in the n-tuple over {0,1,...,k 1} representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan et al. (2007) and by Fu (2006). Furthermore, we extend known results, obtained by Kim and Park (2000), for the case when n=2.     

Embedding Hamiltonian paths in faulty arrangement graphs with thebacktracking method

The arrangement graph, denoted by A_n,k, is a generalization of the star graph. A recent work by S.Y. Hsieh et al. (1999) showed that when n-k⩾4 and k=2 or n-k⩾4+[k/2] and k⩾3, A_n,k with k(n-k)-2 random edge faults, can embed a Hamiltonian cycle. In this paper, we generalize Hsieh et al. work by embedding a Hamiltonian path between arbitrary two distinct vertices of the same A_n,k. To overcome the difficulty arising from random selection of the two end vertices, a new embedding method, based on a backtracking technique, is proposed. Our results can tolerate more edge faults than Hsieh et al. results as k⩾7 and 7⩽n-k⩽3+[k/2], although embedding a Hamiltonian path between arbitrary two distinct vertices is more difficult than embedding a Hamiltonian cycle     

Conditional edge-fault-tolerant edge-bipancyclicity of hypercubes

The hypercube has been one of the most popular interconnection networks for parallel computer/communication systems. In this paper, we assume that each node is incident with at least two fault-free links. Under this assumption, we show that every fault-free edge lies on a fault-free cycle of every even length from 6 to 2ⁿ inclusive, even if it has up to 2n − 5 link faults. The result is optimal with respect to the number of link faults tolerated.     

Immucube: Scalable Fault-Tolerant Routing for k-ary n-cube Networks

This work presents Immucube, a scalable and efficient mechanism to improve dependability of interconnection networks for parallel and distributed computers. Immucube achieves better flexibility and scalability than any other previous fault-tolerant mechanism in k-ary n-cubes. The proposal inherits from Immunet several advantages over other previous fault-tolerant routing algorithms: 1) allowing any temporal and spatial fault combination, 2) permitting automatic and application-transparent reconfiguration after any fault, and 3) requiring a negligible overhead in the absence of faults. Immucube introduces new important features, such as: 4) providing graceful performance degradation, even in very large interconnection networks, 5) tolerating transparent resource utilization after transitory faults or partial repair of faulty resources, 6) being able to deal with intermittent faults, and 7) being able to dynamically recover the original network performance when all the failed components have been repaired     

Fault-free Hamiltonian cycles in faulty arrangement graphs

The arrangement graph A_n,k, which is a generalization of the star graph (n-k=1), presents more flexibility than the star graph in adjusting the major design parameters: number of nodes, degree, and diameter. Previously, the arrangement graph has proved Hamiltonian. In this paper, we further show that the arrangement graph remains Hamiltonian even if it is faulty. Let |F_e| and |F_v| denote the numbers of edge faults and vertex faults, respectively. We show that A_n,k is Hamiltonian when 1) (k=2 and n-k⩾4, or k⩾3 and n-k⩾4+[k/2]), and |F_e|⩽k(n-k)-2, or 2) k⩾2, n-k⩾2+[k/2], and |F_e|⩽k(n-k-3)-1, or 3) k⩾2, n-k⩾3, and |F_e |⩽k, or 4) n-k⩾3 and |F_v|⩽n-3, or 5) n-k⩾3 and |F_v|+|F_e|⩽k. Besides, for A_n,k with n-k=2, we construct a cycle of length at least 1) [n!/(n-k!)]-2 if |F_e|⩽k-1, or 2) [n!/(n-k)!]-|F_v |-2(k-1) if |F_v|⩽k-1, or 3) [n!/(n-k)!]-|F_v |-2(k-1) if |F_e|+|F_v|⩽k-1, where [n!/(n-k)!] is the number of nodes in A_n,k     

Resource placement with multiple adjacency constraints in k-aryn-cubes

The problem of placing resources in a k-ary n-cube (k>2) is considered in this paper. For a given j⩾1, resources are placed such that each nonresource node is adjacent to j resource nodes. We first prove that perfect j-adjacency placements are impossible in k-ary n-cubes if nr-1=2n/j. In each case, we describe an algorithm to obtain perfect j-adjacency placements. We also show that these algorithms can be extended under certain conditions to place j distinct types of resources in a such way that each nonresource node is adjacent to a resource node of each type. For the cases when perfect j-adjacency placements are not possible, we consider approximate j-adjacency placements. We show that the number of copies of resources required in this case either approaches a theoretical lower bound on the number of copies required for any j-adjacency placement or is within a constant factor of the theoretical lower bound for large k     

Efficient termination detection for loosely synchronousapplications in multicomputers

We propose a simple algorithm which is based on edge-coloring of system graphs for termination detection of loosely synchronous computations. The proposed algorithm is fully symmetric in that all processors run syntactically identical code and can detect global termination at the same time. Under the 1-port communication model, the algorithm is optimal in terms of termination delay, the difference between the time when a global termination occurs and the time it is detected, in a number of structures-chain, ring of even number of nodes, k-ary n-cube and k-ary n-mesh of low degree, where k is even; and near-optimal for other cases. The optimality analysis is based on results from a related problem, periodic gossiping in edge-colored graphs. This algorithm has been applied to some practical cases in which the overhead due to its execution is found to be insignificant     

On the Construction of Fault-Tolerant Cube-Connected Cycles Networks

This paper presents a new approach to tolerating edge faults and node faults in (CCC) networks of Cube-Connected Cycles in a worst-case scenario. Our constructions of fault-tolerant CCC networks are obtained by adding extra edges to the CCC. The main objective is to reduce the cost of the fault-tolerant network by minimizing the degree of the network. Specifically, we have two main results. (i) We have created a fault tolerant CCC that can tolerate any single fault, either a node fault or an edge fault. When the dimension of the CCC is odd, the degree of the fault tolerant graph is 4. In the even case, there is a single node per cycle that is of degree 5 and the rest are of degree 4. (ii) We have created a fault-tolerant CCC, where every node has degree y + 2, which can tolerate any 2y − 1 cube-edge faults. Our constructions are extremely efficient for the case of edge faults-they result in healthy CCC networks that utilize all of the processors.     

关于传递闭包和函数依赖完备公理系统存在问题研究

数据依赖在数据库设计中起着十分重要的作用．自Ｃｏｄｄ提出函数依赖（ＦＤｓ）、Ｆａｇｉｎ引入多值依赖（ＭＶＤｓ）后，近几年来人们又根据设计中的需要引入多种新的依赖，如在工程数据库设计中所引进的传递闭包依赖（ＣＤｓ）等．对这些依赖一般是按其是否具有完备的公理系统而划分为两大类，因为完备性公理系统往往具有有效的判定算法为先决条件．本文对ＣＤｓ和ＦＤｓ的ｋ元完备公理系统存在问题进行了研究，证明了ＣＤｓ和ＦＤｓ不具有共同的ｋ元完备公理系统这一结论．