Algebraic analysis of the topological properties of a banyan network and its application in fault-tolerant switching networks

In this paper, we introduce abstract algebraic analysis of the topological structure of a banyan network, which has become the baseline for most switching networks. The analysis provides the following key results: (1) The switching elements of a switching stage are arranged in order, that is, each stage of a banyan network consists of a series of a cyclic group. (2) The links between switching stages implement a homomorphism relationship in terms of self-routing. Therefore, we can recover the misrouting of a detour fault link by providing adaptive self-routing. (3) The cyclic group of a stage is a subgroup of that of the next stage, so that every stage and its adjacent stage make up a factor group. Based on this analysis, we introduce a cyclic banyan network that is more reliable than other switching networks. We present mathematical analysis of the reliability of the switching network to allow quantitative comparison against other switching networks.     

A method to evaluate the spatial extensibility of a switching unit and network

Switching units and networks have been analyzed as extensible fabrics,mostly in terms of their scheduling algorithms.The traditional literature on switching extensibility has provided complexity theory only relating to the total numbers of inputs(or outputs)and exchange lines.This paper analyzes switching extensibility in terms of not only the scheduling algorithm and also the fabric itself.It is found that determining extensibility from soft complexity related to the number of inputs(or outputs)of the scheduling algorithm and the fabric extensibility in previous studies without quantization is a flawed conception.A method is thus proposed to express the spatial extensibility of a switching unit or network in terms of the connections of a switching resource and capacity.The method calculates parameter ES(the efciency of switching)of an m×n switching unit and obtains two functions of the switching unit to describe spatial extensibility along with the number of unilateral inputs or outputs.It is found that the range of ES is(0,1]and three types of switching unit and two types of crosspoint networks have ES=1.ES is calculated for banyan,Clos,parallel packet,fully interconnected and recirculation switching networks.The ES value for the banyan switching network is larger than that for other networks,and switching networks are classified into three types that have absolute/linear/denied spatial extensibility according to the limES value.It is demonstrated that a switching network has the largest ES value when it contains only the five types of switching unit for which ES=1.Finally,a group-switching-first self-routing banyan switching network with lower blocking probability and time delay is deduced,and the ES method is contrasted with two other methods of evaluating spatial extensibility in terms of their mathematical expressions and intuitive graphics,for the five types of switching network listed above.     

The composite banyan network

A new multipath multistage interconnection network called the composite banyan network is proposed. The network incorporates both the banyan and the reverse banyan networks and is constructed by superimposing the two. The basic building blocks in the composite banyan network are 3×3 switching elements with log₂N stages. A major advantage of the composite banyan network over existing networks with 3×3 SEs is an efficient and fast control algorithm that sets up a path between any source and destination pair. Instead of complex numerical calculations, the network can easily generate a primary routing tag and alternate tags through simple binary operations. Also, the network has a lot of favorable features, including regularity, symmetry, and easy rerouting capability under faults and conflicts. It is shown that at least two totally disjoint paths exist between any source and destination pair, which increase the degree of fault-tolerance. A deterministic permutation routing algorithm is also developed for the 8×8 composite banyan network, Using a simple tabular method, it is shown that the algorithm always finds a set of conflict-free tags     

互连网络拓扑等价的图分析法

提出了描述互连网络拓扑等价的图分析法。获得了全交叉网络与基准，逆基准，Ｏｍｅｇａ，ｆｌｉｐ，Ｓ＝Ｆ＝２ＳＷ榕树，简化数据变换等多级互连网络拓扑等价的逻辑名结构。阐明了用光学全交叉网络模拟实现上述网络的互连函数的原理及其多处理机，电信交换等领域的潜在应用。     

基于概率模型的E-2D Mesh网络容错性分析

研究了太比特路由器核心交换网络拓扑的一种新结构-E-2D Mesh.提出一种计算E-2D Mesh网络连通率的新方法.证明了当网络结点失效率控制在0.66%以下时,具有四万多个结点的E-2D Mesh网络可保持不低于99%的连通率,且在同等规模条件下,E-2D Mesh网络结点容错率至少是Mesh网络的11.09倍.研究结果表明,该方法在计算E-2D Mesh网络连通率时显示出较强的生命力且能够用于研究其它层次的网络和其它网络通信问题.     

On fault tolerance of 3-dimensional mesh networks

In this paper, the concept of k-submesh and k-submesh connectivity fault tolerance model is proposed. And the fault tolerance of 3-D mesh networks is studied under a more realistic model in which each network node has an independent failure probability. It is first observed that if the node failure probability is fixed, then the connectivity probability of 3-D mesh networks can be arbitrarily small when the network size is sufficiently large. Thus, it is practically important for multicomputer system manufacturer to determine the upper bound for node failure probability when the probability of network connectivity and the network size are given. A novel technique is developed to formally derive lower bounds on the connectivity probability for 3-D mesh networks. The study shows that 3-D mesh networks of practical size can tolerate a large number of faulty nodes thus are reliable enough for multicomputer systems. A number of advantages of 3-D mesh networks over other popular network topologies are given.     

Node grouping in system-level fault diagnosis

With the popularization of network applications and multiprocessor systems,dependability of systems has drawn considerable attention.This paper presents a new technique of node grouping for system-level fault diagnosis to simplify the complexity of large system diagnosis.The technique transforms a complicated system to a group network,where each group may consist of many nodes that are either fault-free or faulty.It is proven that the transformation leads to a unique group network to ease system diagnosis.Then it studies systematically one-step t-faults diagnosis problem based on node groupling by means of the concept of independent point sets and gives a simple sufficient and necessary condition.The paper presents a diagnosis procedure for t-diagnosable systems.furthermore,an efficient probabilistic diagnosis algorithm for practical applications is proposed based on the belief that most of the nodes in a system are fault-free.The result of software simulation shows that the probabilistic diagnoisis provides high probability of correct diagnosis and low diagnosis cost,is suitable for systems of any kind of topology.     

A new self-routing multicast network

In this paper, we propose a design for a new self-routing multicast network which can realize arbitrary multicast assignments between its inputs and outputs without any blocking. The network design uses a recursive decomposition approach and is based on the binary radix sorting concept. All functional components of the network are reverse banyan networks. Specifically, the new multicast network is recursively constructed by cascading a binary splitting network and two half-size multicast networks. The binary splitting network, in turn, consists of two recursively constructed reverse banyan networks. The first reverse banyan network serves as a scatter network and the second reverse banyan network serves as a quasisorting network. The advantage of this approach is to provide a way to self-route multicast assignments through the network and a possibility to reuse part of network to reduce the network cost. The new multicast network we design is compared favorably with the previously proposed multicast networks. It uses O(n log² n) logic gates, and has O(log² n) depth and O(log² n) routing time where the unit of time is a gate delay. By reusing part of the network, the feedback implementation of the network can further reduce the network cost to O(n log n)     

Fault-Tolerant Routing in MIN-Based Supercomputers

We study methods for routing data in supercomputers that use multistage interconnection networks (MINs), in the presence of faulty components in the network. These methods are applicable to multiprocessors such as IBM GF11 and RP3. These methods are based on the concept of dynamic full access (DFA), which refers to the ability of the network to route data from any processor in the system to any other processor in a finite number of passes through the network. We introduce a graph model called the DFA graph of a MIN and show how it can be used to determine the DFA capability of the MIN under a given set of network faults. When the faults in the network satisfy certain special properties, we present algorithms for routing

• •any arbitrary permutation in a faulty Benes network, and
• •any Omega permutation in a faulty Omega network.

These algorithms are simple and operate in a distributed fashion. These techniques allow a supercomputer to efficiently realize permutations of data needed in a parallel computing environment despite the presence of faults in the network.     

Probabilistic analysis on mesh network fault tolerance

Mesh networks are among the most important interconnection network topologies for large multicomputer systems. Mesh networks perform poorly in tolerating faults in the view of worst-case analysis. On the other hand, such worst cases occur very rarely in realistic situations. In this paper, we study the fault tolerance of 2-D and 3-D mesh networks under a more realistic model in which each network node has an independent failure probability. We first observe that if the node failure probability is fixed, then the connectivity probability of these mesh networks can be arbitrarily small when the network size is sufficiently large. Thus, it is practically important for multicomputer system manufacture to determine the upper bound for node failure probability when the probability of network connectivity and the network size are given. We develop a novel technique to formally derive lower bounds on the connectivity probability for 2-D and 3-D mesh networks. Our study shows that these mesh networks of practical size can tolerate a large number of faulty nodes thus are reliable enough for multicomputer systems. For example, it is formally proved that as long as the node failure probability is bounded by 0.5%$0.5\%$, a 3-D mesh network of up to a million nodes remains connected with a probability larger than 99%$99\%$.     

Load-Balancing in Multistage Interconnection Networks under Multiple-Pass Routing

Multistage interconnection networks (MINs) have been widely used in multiprocessor systems, and recently they have been adopted as a way to construct ATM switches for broadband networks. In such systems, the fault-tolerant ability is an important issue. Many researchers have proposed ways to enhance the reliability of MINs, among them a low-cost and efficient way is to use multipass routing schemes in MINs in which thedynamic full access(DFA) property exists. The performance of multipass routing, however, has been largely ignored by researchers in the past. In this paper, we show that multipass routing may degrade the system performance if the communication loads are not well balanced among processors; congestion may appear in some processors and the useful communication bandwidth is badly affected. We propose methods to design DFA routing schemes that are load-balanced and thus can utilize system resources (i.e., the bandwidth) more efficiently.     

Software-based rerouting for fault-tolerant pipelined communication

This paper presents a software-based approach to fault-tolerant routing in networks using wormhole or virtual cut-through switching. When a message encounters a faulty output link, it is removed from the network by the local router and delivered to the messaging layer of the local node's operating system. The message passing software can reroute this message, possibly along nonminimal paths. Alternatively, the message may be addressed to an intermediate node, which will forward the message to the destination. A message may encounter multiple faults and pass through multiple intermediate nodes. The proposed techniques are applicable to both obliviously and adaptively routed networks. The techniques are specifically targeted toward commercial multiprocessors where the mean time to repair (MTTR) is much smaller than the mean time between router failures (MTBF), i.e., it is sufficient to tolerate a maximum of three failures. This paper presents requirements for buffer management, deadlock freedom, and livelock freedom. Simulation results are presented to evaluate the degradation in latency and throughput as a function of the number and distribution of faults. There are several advantages of such an approach. Router designs are minimally impacted, and thus remain compact and fast. Only messages that encounter faulty components are affected, while the machine is ensured of continued operation until the faulty components can be replaced. The technique leverages existing network technology, and the concepts are portable across evolving switch and router designs. Therefore, we feel that the technique is a good candidate for incorporation into the next generation of multiprocessor networks     

Mesh网络路由算法容错性的概率分析

该文基于k-Mesh子网的概念提出了两个简单的基于局部信息和分布式的Mesh网络容错路由算法，并对其容错性进行概率分析；在每个结点具有独立的出错概率的假设条件下，推导出路由算法成功返回由正确结点组成的路径的概率．该文运用严格的数学推理，证明了Mesh网络结点出错概率只要控制在1．87％以内，则对于多达几十万个结点的Mesh网络，提出的路由算法具有99％的概率确保找到正确结点组成的路径．路由算法的时间复杂性是线性的．模拟结果表明路由算法所构造的路由路径长度非常接近于两结点之间的最优路径长度．     

基于分块策略的E-3DMesh网络容错性概率分析

对E-3DMesh网络中具有大量失效节点模式进行了研究,提出了基于分块策略的概率分析方法。基于该方法研究了在给定网络连通概率的情况下,E-3DMesh网络对网络节点出错概率p的要求。证明了要使多达上百万个节点的E-3DMesh网络连通概率保持在99%以上,网络节点出错概率必须控制在3.86%以下。新方法能够用于研究其他层次结构的网络和其他网络通信问题。     

Mesh网络连通性的概率分析

Mesh网络是大型多处理器并行计算机系统中极为重要的拓扑结构．本文提出了一种计算Mesh网络连通概率的新方法，该方法在给定网络规模和结点出错概率时，计算出Mesh网络连通概率的一个下界，或者对于要求的Mesh网络连通概率，该方法能计算出对结点出错概率的要求．例如，本文运用严格数学推导证明了当网络结点出错概率控制在0．12％以下，则多达四万个结点的Mesh网络仍可保持高达99％的连通概率．理论计算和实验结果表明，该方法在计算Mesh网络连通概率下界时是一种强有力的技术．     

Routing permutations on baseline networks with node-disjoint paths

Permutation is a frequently-used communication pattern in parallel and distributed computing systems and telecommunication networks. Node-disjoint routing has important applications in guided wave optical interconnects where the optical "crosstalk" between messages passing the same switch should be avoided. In this paper, we consider routing arbitrary permutations on an optical baseline network (or reverse baseline network) with node-disjoint paths. We first prove the equivalence between the set of admissible permutations (or semipermutations) of a baseline network and that of its reverse network based on a step-by-step permutation routing. We then show that an arbitrary permutation can be realized in a baseline network (or a reverse baseline network) with node-disjoint paths in four passes, which beats the existing results [M. Vaez et al., (2000)], [G. Maier et al., (2001)] that a permutation can be realized in an n /spl times/ n banyan network with node-disjoint paths in O(n/sup 1/2/) passes. This represents the currently best-known result for the number of passes required for routing an arbitrary permutation with node-disjoint paths in unique-path multistage networks. Unlike other unique path MINs (such as omega networks or banyan networks), only baseline networks have been found to possess such four-pass routing property. We present routing algorithms in both self-routing style and central-controlled style. Different from the recent work in [Y. Yang et al., (2003)], which also gave a four-pass node-disjoint routing algorithm for permutations, the new algorithm is efficient in transmission time for messages of any length, while the algorithm in [Y. Yang et al., (2003)] can work efficiently only for long messages. Comparisons with previous results demonstrate that routing in a baseline network proposed in this paper could be a better choice for routing permutations due to its lowest hardware cost and near-optimal transmission time.     

Effect of fault tolerance and communication delay on response time in a multiprocessor system with a bus topology

The effect of interprocessor communication and fault tolerance on the response time of N processors (nodes) interconnected through a bus type communication medium is discussed. Deterministic as well as probabilistic approaches are considered. Four correction methods to handle the unprocessed data by the faulty processor(s) are studied and compared. It is found that the effect of interprocessor communication and fault tolerance on the response time for communication-extensive programs (I/O bound) is more than that for computation-extensive programs (CPU bound). It is also found that the effect of fault tolerance on the response time is significant, and cannot be ignored when evaluating the performance of multiprocessor systems. We have shown that the work presented in this paper for a bus topology can be generalized and readily adopted by other multiprocessor network topologies.     

Proof that pyramid networks are 1-Hamiltonian-connected with high probability

Wu revealed in 2001 that pyramid networks are Hamiltonian-connected. This investigation demonstrates that a pyramid network with one faulty node or one faulty edge is Hamiltonian-connected, excluding some special faulty cases by building a Hamiltonian path between any two distinct nodes in it. Although a pyramid network with one fault is not Hamiltonian-connected, this study indicates that a pyramid network is 1-Hamiltonian-connected with a very high probability.     

A fast diagnosis algorithm for interconnection network in high performance computers

In high performance computers, a popular interconnection network, the folded hypercube (FHC), possesses smaller diameter, larger connectivity, better reliability and fault tolerance capability as compared with a hypercube counterpart. This paper addresses the fault identification of FHC multiprocessor interconncection systems under the MM^* model. The pessimistic one-step diagnosability of FHC networks is first determined. On the basis, a pessimistic one-step diagnosis algorithm tailored for FHC multiprocessor systems is proposed. The presented algorithm can isolate all faulty nodes to within a set which has at most one fault-free node, and can run in linear time.     

On a deadlock and performance analysis of ALBR and DAR algorithm on X-Torus topology by optimal utilization of Cross Links and minimal lookups

Interconnection networks play a major role in differentiating modern multiprocessor architectures. They can be categorized according to a number of criteria such as topology, routing strategy and switching technique. They are built up of switching elements; the topology is packaged such that it is cost effective along with its ability to achieve good performance. In this paper, we have studied an existing X-Torus topology (Gu et al. in ICCSA, LNCS, vol. 3984, pp. 149–157, 2006) which is an enhancement of a Torus network by adding Cross Links, and hence contributes to shorter diameter, shorter average distance and larger bisection bandwidth. Furthermore, we proposed the Adaptive Load Balanced Routing (ALBR) algorithm and Dual Adaptive Routing (DAR) algorithm.