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     

A necessary and sufficient condition for deadlock-free routing incut-through and store-and-forward networks

This paper develops the theoretical background for the design of deadlock-free adaptive routing algorithms for virtual cut-through and store-and-forward switching. This theory is valid for networks using either central buffers or edge buffers. Some basic definitions and three theorems are proposed, developing conditions to verify that an adaptive algorithm is deadlock-free, even when there are cyclic dependencies between routing resources. Moreover, we propose a necessary and sufficient condition for deadlock-free routing. Also, a design methodology is proposed. It supplies fully adaptive, minimal and non-minimal routing algorithms, guaranteeing that they are deadlock-free. The theory proposed in this paper extends the necessary and sufficient condition for wormhole switching previously proposed by us. The resulting routing algorithms are more flexible than the ones for wormhole switching. Also, the design methodology is much easier to apply because it automatically supplies deadlock-free routing algorithms     

A necessary and sufficient condition for deadlock-free adaptiverouting in wormhole networks

Deadlock avoidance is a key issue in wormhole networks. A first approach by W.J. Dally and C.L. Seitz (1987) consists of removing the cyclic dependencies between channels. Many deterministic and adaptive routing algorithms have been proposed based on that approach. Although the absence of cyclic dependencies is a necessary and sufficient condition for deadlock-free deterministic routing, it is only a sufficient condition for deadlock-free adaptive routing. A more powerful approach by J. Duato (1991) only requires the absence of cyclic dependencies on a connected channel subset. The remaining channels can be used in almost any way. In this paper, we show that the previously mentioned approach is also a sufficient condition. Moreover, we propose a necessary and sufficient condition for deadlock-free adaptive routing. This condition is the key for the design of fully adaptive routing algorithms with minimum restrictions, An example shows the application of the new theory     

A general theory for deadlock-free adaptive routing using a mixedset of resources

This paper presents a theoretical framework for the design of deadlock-free fully adaptive routing algorithms for a general class of network topologies and switching techniques in a single, unified theory. A general theory is proposed that allows the design of deadlock avoidance-based as well as deadlock recovery-based wormhole and virtual cut-through adaptive routing algorithms that use a homogeneous or a heterogeneous (mixed) set of resources. The theory also allows channel queues to be allocated nonatomically, utilizing resources efficiently. A general methodology for the design of fully adaptive routing algorithms applicable to arbitrary network topologies is also proposed. The proposed theory and methodology allow the design of efficient network routers that require minimal resources for handling infrequent deadlocks     

Adaptive Multicast Wormhole Routing in 2D Mesh Multicomputers

The issues of adaptive multicast wormhole routing in 2D mesh multicomputers are studied. Three adaptive multicast wormhole routing strategies are proposed and evaluated. The methods include minimal partially adaptive, minimal fully adaptive, and nonminimal adaptive routing. All the algorithms are shown to be deadlock-free. A study has been conducted that compares the performance of these multicast algorithms. The results show that the minimal fully adaptive routing method creates the least traffic; however, double vertical channels are required in order to avoid deadlock. The nonminimal routing algorithm exhibits the best adaptivity, although it creates more network traffic than the other methods.     

多计算机互连网通信中关于死锁问题的一种理论

如何获得死锁而且通信性能良好的选路算法始终是人们十分关心的问题。本文提出了纯分流点的概念并证明了选路算法无死锁的充要条件，从理论上解决了死锁关系问题，为死锁的判定，消除和设计无死锁的算法提供了有力的依据。     

Deadlock-free adaptive routing in multicomputer networks usingvirtual channels

The use of adaptive routing in a multicomputer interconnection network improves network performance by using all available paths and provides fault tolerance by allowing messages to be routed around failed channels and nodes. Two deadlock-free adaptive routing algorithms are described. Both algorithms allocate virtual channels using a count of the number of dimension reversals a packet has performed to eliminate cycles in resource dependency graphs. The static algorithm eliminates cycles in the network channel dependency graph. The dynamic algorithm improves virtual channel utilization by permitting dependency cycles and instead eliminating cycles in the packet wait-for graph. It is proved that these algorithms are deadlock-free. Experimental measurements of their performance are presented     

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.     

A Necessary and Sufficient Condition for Deadlock-Free Wormhole Routing

An important open problem in wormhole routing has been to find a necessary and sufficient condition for deadlock-free adaptive routing. Recently, Duato has solved this problem for a restricted class of adaptive routing algorithms. In this paper, a necessary and sufficient condition is proposed that can be used for any adaptive or nonadaptive routing algorithm for wormhole routing, as long as only local information is required for routing. The underlying proof technique introduces a new type of dependency graph, thechannel waiting graph, which omits most channel dependencies that cannot be used to create a deadlock configuration. The necessary and sufficient condition can be applied in a straightforward manner to most routing algorithms. This is illustrated by proving deadlock freedom for a partially adaptive nonminimal mesh routing algorithm that does not require virtual channels and a fully adaptive minimal hypercube routing algorithm with two virtual channels per physical channel. Both routing algorithms are more adaptive than any previously proposed routing algorithm with similar virtual channel requirements.     

High-performance routing in networks of workstations with irregulartopology

Networks of workstations are rapidly emerging as a cost-effective alternative to parallel computers. Switch-based interconnects with irregular topology allow the wiring flexibility, scalability, and incremental expansion capability required in this environment. However, the irregularity also makes routing and deadlock avoidance on such systems quite complicated. In current proposals, many messages are routed following nonminimal paths, increasing latency and wasting resources. In this paper, we propose two general methodologies for the design of adaptive routing algorithms for networks with irregular topology. Routing algorithms designed according to these methodologies allow messages to follow minimal paths in most cases, reducing message latency and increasing network throughput. As an example of application, we propose two adaptive routing algorithms for ANI (previously known as Autonet). They can be implemented either by duplicating physical channels or by splitting each physical channel into two virtual channels. In the former case, the implementation does not require a new switch design. It only requires changing the routing tables and adding links in parallel with existing ones, taking advantage of spare switch ports. In the latter case, a new switch design is required, but the network topology is not changed. Evaluation results for several different tapologies and message distributions show that the new routing algorithms are able to increase throughput for random traffic by a factor of up to 4 with respect to the original up*/down* algorithm, also reducing latency significantly. For other message distributions, throughput is increased more than seven times. We also show that most of the improvement comes from the use of minimal routing     

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     

Adaptive Wormhole Routing in Hypercube Multicomputers

Both adaptive unicast routing and efficient multicast communication have been shown to be important to the performance of distributed-memory multiprocessors, or multicomputers. In this paper, we propose a uniform adaptive routing strategy for wormhole-routed hypercube networks that accommodates both unicast and multicast communication. Based on a node labeling method, the resultant routing algorithms are shown to be deadlock-free without requiring virtual channels. We present an optimal ordering algorithm that minimizes the traffic generated under the proposed paradigm. A greedy algorithm with less time complexity is also proposed.     

不规则网络中基于拐弯模型的路由算法的设计

由虫孔路由交换器连接而成的不规则拓扑网络，越来越多地用于构建工作站机群系统（NOWs），以实现高性能价格比的并行处理．采用虫孔路由技术，网络中容易发生死锁．交换器之间连接的不规则性，使路由避免死锁问题变得更加复杂．本文给出了在不规则网络中，设计基于拐弯模型的无死锁路由算法的一般方法，并采用扩展链路方向的方法得到多种路由策略，确定了up-first与down-last两种性能较优的路由算法．最后通过模拟实验，评价了算法的性能．     

超立方体上路径算法的无死锁性

本文对超立方体上路径算法的无死锁性问题进行了研究，提出了超立方体上的两类最小无死锁受限条件，证明了路径算法的无死锁和对称性两者之间关系。     

The message flow model for routing in wormhole-routed networks

In this paper, we introduce a new approach to deadlock-free routing in wormhole-routed networks called the message flow model. This method may be used to develop deterministic, partially-adaptive, and fully-adaptive routing algorithms for wormhole-routed networks with arbitrary topologies. We first establish the necessary and sufficient condition for deadlock free routing, based on the analysis of the message flow on each channel. We then use the model to develop new adaptive routing algorithms for 2D meshes     

超立方体上基于缓冲机制的无死锁路径算法

本文研究了超立方体上基于单缓冲和双缓冲技术的无死锁受限条件，提出了相应的无死锁路径算法．性能分析表明，路径算法的效率和算法的自适应能力及算法的复杂性相关．     

Compressionless wormhole routing: an analysis for hypercube with virtual channels

Several recent studies have shown that adaptive routing algorithms based on deadlock recovery have superior performance characteristics than those based on deadlock avoidance. Most of these studies, however, have relied on software simulation due to the lack of analytical modelling tools. In an effort towards filling this gap, this paper presents a new analytical model of compressionless routing in wormhole-routed hypercubes. This routing algorithm exploits the tight coupling between wormhole routers for flow control to detect and recover from potential deadlock situations. The advantages of compressionless routing include deadlock-free adaptive routing with no extra virtual channels, simple router design, and order-preserving message transmission. The proposed analytical model computes message latency by determining the message transmission time, blocking delay at each router, multiplexing delay at each network channel, and waiting time in the source before entering the network. The validity of the model is demonstrated by comparing analytical results with those obtained through simulation experiments.     

Deadlock-free multicast wormhole routing in 2-D mesh multicomputers

Multicast communication services, in which the same message is delivered from a source node to an arbitrary number of destination nodes, are being provided in new-generation multicomputers. Broadcast is a special case of multicast in which a message is delivered to all nodes in the network. The nCUBE-2, a wormhole-routed hypercube multicomputer, provides hardware support for broadcast and a restricted form of multicast in which the destinations form a subcube. However, the broadcast routing algorithm adopted in the nCUBE-2 is not deadlock-free. In this paper, four multicast wormhole routing strategies for 2-D mesh multicomputers are proposed and studied. All of the algorithms are shown to be deadlock-free. These are the first deadlock-free multicast wormhole routing algorithms ever proposed. A simulation study has been conducted that compares the performance of these multicast algorithms under dynamic network traffic conditions in a 2-D mesh. The results indicate that a dual-path routing algorithm offers performance advantages over tree-based, multipath, and fixed-path algorithms     

Adaptive and deadlock-free routing for irregular faulty patterns inmesh multicomputer

Message routing achieves the internode communication in parallel computers. A reliable routing is supposed to be deadlock-free and fault-tolerant. While many routing algorithms are able to tolerate a large number of faults enclosed by rectangular faulty blocks, there is no existing algorithm that is capable of handling irregular faulty patterns for wormhole networks. In this paper, a two-staged adaptive and deadlock-free routing algorithm called “Routing for Irregular Faulty Patterns” (RIFP) is proposed. It can tolerate irregular faulty patterns by transmitting messages from sources or to destinations within faulty blocks via multiple “intermediate nodes.” A method employed by RIFP is first introduced to generate intermediate nodes using the local failure information. By its aid, two communicating nodes can always exchange their data or intermediate results if there is at least one path between them. RIFP needs two virtual channels per physical link in meshes     

An Efficient Fault-Tolerant Routing Methodology for Meshes and Tori

In this paper we present a methodology to design fault-tolerant routing algorithms for regular direct interconnection networks. It supports fully adaptive routing, does not degrade performance in the absence of faults, and supports a reasonably large number of faults without significantly degrading performance. The methodology is mainly based on the selection of an intermediate node (if needed) for each source-destination pair. Packets are adaptively routed to the intermediate node and, at this node, without being ejected, they are adaptively forwarded to their destinations. In order to allow deadlock-free minimal adaptive routing, the methodology requires only one additional virtual channel (for a total of three), even for tori. Evaluation results for a 4 x 4 x 4 torus network show that the methodology is 5-fault tolerant. Indeed, for up to 14 link failures, the percentage of fault combinations supported is higher than 99.96%. Additionally, network throughput degrades by less than 10% when injecting three random link faults without disabling any node. In contrast, a mechanism similar to the one proposed in the BlueGene/L, that disables some network planes, would strongly degrade network throughput by 79%.