Characterization of deadlocks in k-ary n-cube networks

A spate of deadlock avoidance-based and deadlock recovery-based routing algorithms have been proposed in recent years without full understanding of the likelihood and characteristics of actual deadlocks in interconnection networks. This work models the interrelationships between routing freedom, message blocking, correlated resource dependencies, and deadlock formation. It is empirically shown that increasing routing freedom, as achieved by allowing unrestricted routing over multiple physical and virtual channels, reduces the probability of deadlocks and the likelihood of other types of correlated message blocking that can degrade performance. Moreover, when true fully adaptive routing is used in k-ary n-cube networks with two or more virtual channels (wormhole OF virtual cut-through switched), it is empirically shown that deadlocks are virtually eliminated in networks with n⩾2. These results indicate that deadlocks are very infrequent when the network and routing algorithm inherently provide sufficient routing freedom, thus increasing the viability of deadlock recovery routing strategies     

广义K-元n-立方网络拓扑性质研究

k-元n-立方( )网络被广泛应用于直接互联网络并行处理系统中。当 某些维上的节点数不相同时,该文利用一维环的交叉乘积定义了广义的K-元n-立方( , )网络,提出 网络表面积和体积的计算公式,利用表面积确定最优的广播结构,并分析了2个节点间最短路由数的计算公式和路由选取方法。     

Alleviating consumption channel bottleneck in wormhole routed k-aryn-cube systems

This paper identifies performance degradation in wormhole routed k-ary n-cube networks due to limited number of router-to-processor consumption channels at each node. Many recent research in wormhole routing have advocated the advantages of adaptive routing and virtual channel flow control schemes to deliver better network performance. This paper indicates that the advantages associated with these schemes cannot be realized with limited consumption capacity. To alleviate such performance bottlenecks, a new network interface design using multiple consumption channels is proposed. To match virtual multiplexing on network channels, we also propose each consumption channel to support multiple virtual consumption channels. The impact of message arrival rate at a node on the required number of consumption channels is studied analytically. It is shown that wormhole networks with higher routing adaptivity, dimensionality, degree of hot-spot traffic, and number of virtual lanes have to take advantage of multiple consumption channels to deliver better performance. The interplay between system topology, routing algorithm, number of virtual lanes, messaging overheads, and communication traffic is studied through simulation to derive the effective number of consumption channels required in a system. Using the ongoing technological trend, it is shown that wormhole-routed systems can use up to two-four consumption channels per node to deliver better system performance     

Fault diameter of k-ary n-cube networks

We obtain the fault diameter of k-ary n-cube interconnection networks (also known as n-dimensional k-torus networks). We start by constructing a complete set of node-disjoint paths (i.e., as many paths as the degree) between any two nodes of a k-ary n-cube. Each of the obtained paths is of length zero, two, or four plus the minimum length except for one path in a special case (when the Hamming distance between the two nodes is one) where the increase over the minimum length may attain eight. These results improve those obtained by B. Bose et al. (1995) where the length of some of the paths has a variable increase (which can be arbitrarily large) over the minimum length. These results are then used to derive the fault diameter of the k-ary n-cube, which is shown to be Δ+1 where Δ is the fault free diameter     

BWR——带缓冲的虫孔路由技术

MPP互联网中通常使用虫孔路由WR（Wormhole Routing)交换技术来提高网络性能,采用该技术,每个结点所需的通信缓冲小;并且当消息长度远远大于微片长度时,消息的传输延迟时间与传输距离无关。但WR技术也具有容易阻塞和产生刹车问题的缺点。该文在WR技术的基础上,提出了带缓冲的虫孔路由BWR（Buffered Wormhole Routing)交换技术,并对采用BWR技术的k-ary n-mesh的消息平均传输延迟进行理论分析与模型模拟。结果均表明BWR技术可以较好地解决WR技术带来的问题,可以较好地提高网络的性能。     

使用多级交换网络进行高性能路由器设计

拓扑结构和路由算法是影响多级交换网络性能的重要因素．在比较多种多级互连拓扑属性的基础上,提出将3D Torus结构应用于大规模交换网络设计．然后针对3D Torus交换网络中报文路由面临的两个关键问题：多路径负载均衡和报文保序,提出一种基于维序的多路径路由算法DMR(dimension-order—based multi—path routing)．该算法可在保证报文顺序的同时在多条路径上平衡负载,提高交换网络吞吐率．最后通过模拟验证了算法的性能,并与维序路由和随机路由算法进行了比较．模拟结果表明,DMR算法的性能优于维序路由算法,能够达到随机路由算法性能水平,同时具有随机路由算法所不具备的报文保序特性．     

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。     

Adaptive-trail routing and performance evaluation in irregularnetworks using cut-through switches

Cut-through switching promises low latency delivery and has been used in new generation switches, especially in high speed networks demanding low communication latency. The interconnection of cut-through switches provides an excellent network platform for high speed local area networks (LANs). For cost and performance reasons. Irregular topologies should be supported in such a switch-based network. Switched irregular networks are truly incrementally scalable and have potential to be reconfigured to adapt to the dynamics of network traffic conditions. Due to the arbitrary topologies of networks, it is critical to develop an efficient deadlock-free routing algorithm. A novel deadlock-free adaptive routing algorithm called adaptive-trail routing is proposed to allow irregular interconnection of cut-through switches. The adaptive routing algorithm is based on two unidirectional adaptive trails constructed from two opposite unidirectional Eulerian trails. Some heuristics are suggested in terms of the selection of Eulerian trails, the avoidance of long routing paths, and the degree of adaptivity. Extensive simulation experiments are conducted to evaluate the performance of the proposed and two other routing algorithms under different topologies and traffic workloads     

Routing and wavelength assignment for 3-ary n-cube communication patterns in linear array optical networks for n communication rounds

k-ary n-cubes are a class of communication patterns that are employed by a number of typical parallel algorithms. This paper addresses the implementation of parallel algorithms with bidirectional 3-ary n-cube communication patterns on a bidirectional linear array WDM optical networks when the information is transmitted one dimension after another. By giving an embedding scheme ?, we prove the optimal number of wavelengths under ? and design a routing and wavelength assignment strategy of it.     

Toward a more realistic performance evaluation of interconnectionnetworks

lnterconnection network design plays a central role in the design of parallel systems. Most of the previous research has evaluated the performance of interconnection networks in isolation. In this study, we investigate the relationship between application program characteristics and interconnection network performance using an execution driven simulation test bed: the Reconfigurable Architecture Workbench (RAW). We simulate five topological configurations of a k-ary n-cube interconnect and four different network link models for a 4,096 node SIMD machine, and quantify the impact of the network on two application programs. We provide experimental evidence that such “in-context” simulation provides a better view of the impact of network design variables on system performance. We show that recent results, indicating that low-dimensional designs provide better ICN performance, ignore application requirements that may favor high-dimensional designs. Furthermore, applications that would appear to favor low dimensional designs may not, in fact, be significantly impacted by the network's dimensionality. We experimentally test the results of published performance models comparing the use of a synthetic load to that of a load generated by a typical application program     

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。     

A general methodology for direction-based irregular routing algorithms

This paper presents a general methodology for generating deadlock-free routing algorithms for irregular networks. Constructing a spanning tree on the given network, assigning directions to the network channels, creating deadlock-free zones, and specifying a logical sequence of the produced deadlock-free zones are the four fundamental steps that the proposed methodology takes to generate deadlock-free and connected routing algorithms. By applying the proposed methodology with two known labeling methods we have generated six irregular routing algorithms: three of them are novel routing algorithms and three of them (the Up/Down, Left/Right, and L-turn routing algorithms) have already been proposed in the literature. Extensive simulation experiments have been performed considering various network topologies, different network sizes (considering different network nodes and network channels), various message lengths, a variety of spanning tree roots, and a wide range of message (traffic) generation rates. Simulation results show that the six routing algorithms can be divided into three pairs. Routing members of each pair show similar behavior in terms of message latencies and saturation generation rates. However, it is worth noting that for a given topology the performance of the six routing algorithms may be totally different and it mainly depends on the network topology.     

The impact of pipelined channels on k-ary n-cube networks

In a pipelined-channel interconnection network, multiple bits may be simultaneously in flight on a single wire. This allows the cycle time of the network to be independent of the wire lengths, significantly affecting the network design trade-offs. This paper investigates the design and performance of pipelined channel k-ary n-cube networks, with particular emphasis on the choice of dimensionality and radix. Networks are investigated under the constant link width, constant node size and constant bisection constraints. We find that the optimal dimensionality of pipelined-channel networks is higher than that of nonpipelined-channel networks, with the difference being greater under looser wiring constraints. Their radix should remain roughly constant as network size is grown, decreasing slightly for some unidirectional tori and increasing slightly for some bidirectional meshes. Pipelined-channel networks are shown to provide lower latency and higher bandwidth than their nonpipelined-channel counterparts, especially for high-dimensional networks. The paper also investigates the effects of switching overhead and message lengths, indicating where results agree with and differ from previous results obtained for nonpipelined-channel networks     

Routing performance enhancement in hierarchical torus network by link-selection algorithm

A hierarchical torus network (HTN) is a 2D-torus network of multiple basic modules, in which the basic modules are 3D-torus networks that are hierarchically interconnected for higher-level networks. The static network performance of the HTN and its dynamic communication performance using the popular dimension-order routing algorithm have already been evaluated and shown to be superior to the performance of other conventional and hierarchical interconnection networks. In this paper, we propose a link-selection algorithm for efficient use of physical links of the HTN, while keeping the link-selection algorithm as simple as the dimension-order routing algorithm. We also prove that the proposed algorithm for the HTN is deadlock-free using three virtual channels. We evaluate the dynamic communication performance of an HTN using dimension-order routing and link-selection algorithms under various traffic patterns. We find that the dynamic communication performance of an HTN using the link-selection algorithm is better than when the dimension-order routing algorithm is used.     

EMPLOYING K-ARY n-CUBES FOR PARALLEL LAGRANGE INTERPOLATION

This paper proposes a parallel algorithm for computing anN( = Kⁿ) point Lagrange interpolation on fc-ary n-cube networks. The algorithm consists of three phases: initialisation, main and final. There is no computation in the initialisation phase. The main phase is composed of N/2 steps, each consisting of four multiplications and four subtractions, and an additional step including one division and one multiplication. Communication in the main phase is based on an all-to-all broadcast algorithm on a Hamiltonian ring embedded in a k-ary n-cube. The final phase is carried out in n x ?k/l? steps, each requiring one addition. A performance evaluation of the proposed algorithm reveals a near to optimum speedup for a typical range of sy:;tem parameters used in current state-of-the-art implementations. Our study also reveals that when implementation cost is taken into account low-dimensional K-ary n-cubes achieve better speedup than their higher-dimensional counterparts.     

Multidestination message passing in wormhole k-ary n-cube networkswith base routing conformed paths

This paper proposes multidestination message passing on wormhole k-ary n-cube networks using a new base-routing-conformed-path (BRCP) model. This model allows both unicast (single-destination) and multidestination messages to co-exist in a given network without leading to deadlock. The model is illustrated with several common routing schemes (deterministic, as well as adaptive), and the associated deadlock-freedom properties are analyzed. Using this model, a set of new algorithms for popular collective communication operations, broadcast and multicast, are proposed and evaluated. It is shown that the proposed algorithms can considerably reduce the latency of these operations compared to the Umesh (unicast-based multicast) and the Hamiltonian path-based schemes. A very interesting result that is presented shows that a multicast can be implemented with reduced or near-constant latency as the number of processors participating in the multicast increases beyond a certain number. It is also shown that the BRCP model can take advantage of adaptivity in routing schemes to further reduce the latency of these operations. The multidestination mechanism and the BRCP model establish a new foundation to provide fast and scalable collective communication support on wormhole-routed systems     

带有条件故障边的k元2方体的圈嵌入

[k]元[n]方体已经成为分布式储存并行系统最常用的网络拓扑结构。研究带有条件故障边的[k]元2方体的圈嵌入问题,证明了在[k≥4]为偶整数的[k]元2方体中,若其故障边数不超过3且每个顶点至少与两条非故障边相关联,那么该[k]元2方体存在长度在4到[k2]间的任意偶长的无故障圈。     

A general theory for deadlock avoidance in wormhole-routed networks

Most machines of the last generation of distributed memory parallel computers possess specific routers which are used to exchange messages between nonneighboring nodes in the network. Among the several technologies, wormhole routing is usually preferred because it allows low channel-setup time and reduces the dependency between latency and internode distance. However, wormhole routing is very susceptible to deadlock because messages are allowed to hold many resources while requesting others. Therefore, designing deadlock-free routing algorithms using few hardware facilities is a major problem for wormhole-routed networks. In this paper, we describe a general theoretical framework for the study of deadlock-free routing functions. We give a general definition of what can be a routing function. This definition captures many specific definitions of the literature (e.g., vertex dependent, input-dependent, source-dependent, path-dependent etc.). Using our definition, we give a necessary and sufficient condition which characterizes deadlock-free routing functions. Our theory embraces, at a high level, most of the theories related to deadlock avoidance in wormhole-routed networks previously derived in the literature. In particular, it applies not only to one-to-one routing, but also to one-to-many routing. The latter paradigm is used to solve the multicast problem with the path-based or tree-based facility     

A theory of deadlock-free adaptive multicast routing in wormholenetworks

A theory for the design of deadlock-free adaptive routing algorithms for wormhole networks, proposed by the author (1991, 1993), supplies sufficient conditions for an adaptive routing algorithm to be deadlock-free, even when there are cyclic dependencies between channels. Also, two design methodologies were proposed. Multicast communication refers to the delivery of the same message from one source node to an arbitrary number of destination nodes. A tree-like routing scheme is not suitable for hardware-supported multicast in wormhole networks because it produces many headers for each message, drastically increasing the probability of a message being blocked. A path-based multicast routing model was proposed by Lin and Ni (1991) for multicomputers with 2D-mesh and hypercube topologies. In this model, messages are not replicated at intermediate nodes. This paper develops the theoretical background for the design of deadlock-free adaptive multicast routing algorithms. This theory is valid for wormhole networks using the path-based routing model. It is also valid when messages with a single destination and multiple destinations are mixed together. The new channel dependencies produced by messages with several destinations are studied. Also, two theorems are proposed, developing conditions to verify that an adaptive multicast routing algorithm is deadlock-free, even when there are cyclic dependencies between channels. As an example, the multicast routing algorithms of Lin and Ni are extended, so that they can take advantage of the alternative paths offered by the network     

Exploiting global knowledge to achieve self-tuned congestion control for k-ary n-cube networks

Network performance in tightly-coupled multiprocessors typically degrades rapidly beyond network saturation. Consequently, designers must keep a network below its saturation point by reducing the load on the network. Congestion control via source throttling-a common technique to reduce the network load-prevents new packets from entering the network in the presence of congestion. Unfortunately, prior schemes to implement source throttling either lack vital global information about the network to make the correct decision (whether to throttle or not) or depend on specific network parameters, or communication patterns. This paper presents a global-knowledge-based, self-tuned, congestion control technique that prevents saturation at high loads across different communication patterns for k-ary n-cube networks. Our design is composed of two key components. First, we use global information about a network to obtain a timely estimate of network congestion. We compare this estimate to a threshold value to determine when to throttle packet injection. The second component is a self-tuning mechanism that automatically determines appropriate threshold values based on throughput feedback. A combination of these two techniques provides high performance under heavy load, does not penalize performance under light load, and gracefully adapts to changes in communication patterns.