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 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     

A framework for designing deadlock-free wormhole routing algorithms

This paper presents a framework to design fully-adaptive, deadlock-free wormhole algorithms for a variety of network topologies. The main theoretical contributions are: (a) design of new wormhole algorithms using store-and-forward algorithms, (b) a sufficient condition for deadlock free routing by the wormhole algorithms so designed, and (c) a sufficient condition for deadlock free routing by these wormhole algorithms with centralized flit buffers shared among multiple channels. To illustrate the theory, several wormhole algorithms based on store-and-forward hop schemes are designed. The hop-based wormhole algorithms can be applied to a variety of networks including torus, mesh, de Brujin, and a class of Cayley networks, with the best known bounds on virtual channels for minimal routing on the last two classes of networks. An analysis of the resource requirements and performances of a proposed algorithm, called negative-hop algorithm, with some of the previously proposed algorithms for torus and mesh networks is presented     

A theory of fault-tolerant routing in wormhole networks

Fault-tolerant systems aim at providing continuous operation in the presence of faults. Multicomputers rely on an interconnection network between processors to support the message-passing mechanism. Therefore, the reliability of the interconnection network is very important for the reliability of the whole system. This paper analyzes the effective redundancy available in a wormhole network by combining connectivity and deadlock freedom. Redundancy is defined at the channel level. We propose a sufficient condition for channel redundancy, also computing the set of redundant channels. The redundancy level of the network is also defined, proposing a theorem that supplies its value. This theory is developed on top of our necessary and sufficient condition for deadlock-free adaptive routing. The new theory also considers the failure of physical channels when virtual channels are used. Finally, we propose a methodology for the design of fault-tolerant routing algorithms, showing its application to n-dimensional meshes     

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.     

A new theory of deadlock-free adaptive routing in wormhole networks

The theoretical background for the design of deadlock-free adaptive routing algorithms for wormhole networks is developed. The author proposes some basic definitions and two theorems. These create the conditions to verify that an adaptive algorithm is deadlock-free, even when there are cycles in the channel dependency graph. Two design methodologies are also proposed. The first supplies algorithms with a high degree of freedom, without increasing the number of physical channels. The second methodology is intended for the design of fault-tolerant algorithms. Some examples are given to show the application of the methodologies. Simulations show the performance improvement that can be achieved by designing the routing algorithms with the new theory     

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。     

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     

Mesh网络耐故障虫孔路由

耐故障是互连网络设计中的一个重要问题。本文提出了一种新的耐故障路由算法,并将其应用于使用虫孔交换技术的Mesh网络。由于使用了较低的路由限制,这一算法具有很强的自适应性,可以在各种不同故障域的Mesh网络中保持路由的连通性和无死锁性;由于使用了最小限度的虚拟通道,这一算法所需的缓冲器资源很少,非常适宜构建低成本的耐故障互连网络;由于根据本地故障信息进行绕行故障节点的决策,这一算法的路由决策速度较快并且易于在互连网络中实现。最后网络仿真试验显示,这一算法具有良好的平滑降级使用的性能。     

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．     

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.     

A Comparison of Router Architectures for Virtual Cut-Through and Wormhole Switching in a NOW Environment

Most multicomputer interconnection networks use wormhole switching, leading to fast and compact routers. Current routers incorporate virtual channels and even fully adaptive routing. Networks of workstations (NOWs) inherited multicomputer technology. Most commercial routers designed for NOWs implement wormhole switching. However, wormhole switching is not well suited for NOWs. The long wires required in this environment lead to large buffers to prevent buffer overflow during flow control signaling. Moreover, wire length is limited by buffer size. Virtual cut-through (VCT) achieves a higher throughput than wormhole switching. However, buffer requirements and packetizing overhead prevented its widespread use in multicomputers. Nevertheless, wormhole and VCT switching require similar buffer capacity in NOWs. Moreover, some messaging layers such as Illinois Fast Messages (FM) and BIP split messages into packets for increased performance. Therefore, the traditional disadvantages of VCT switching disappear in NOWs. In this paper, we show that VCT routers can be simpler than wormhole routers, while still achieving the advantages of using virtual channels and adaptive routing. We also propose a fully adaptive routing algorithm for VCT switching in a NOW environment. Moreover, we show that VCT routers outperform wormhole routers in a NOW environment at a lower cost. Also, VCT routers require buffer capacity independent of wire length, making them suitable for networks of workstations.     

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     

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     

虚网叠加构造自适应路由算法的有效框架

大规模并行处理机系统中路由算法对互联网络通信性能和系统性起着重要作用。     

Performance analysis of mesh interconnection networks withdeterministic routing

This paper develops detailed analytical performance models for k-ary n-cube networks with single-hit or infinite buffers, wormhole routing, and the nonadaptive deadlock-free routing scheme proposed by Dally and Seitz (1987). In contrast to previous performance studies of such networks, the system is modeled as a closed queueing network that: includes the effects of blocking and pipelining of messages in the network; allows for arbitrary source-destination probability distributions; and explicitly models the virtual channels used in the deadlock-free routing algorithm. The models are used to examine several performance issues for 2-D networks with shared-memory traffic. These results should prove useful for engineering high-performance systems based on low-dimensional k-ary n-cube networks     

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     

Proof Pearl: A Formal Proof of Dally and Seitz’ Necessary and Sufficient Condition for Deadlock-Free Routing in Interconnection Networks

Avoiding deadlock is crucial to interconnection networks. In ’87, Dally and Seitz proposed a necessary and sufficient condition for deadlock-free routing. This condition states that a routing function is deadlock-free if and only if its channel dependency graph is acyclic. We formally define and prove a slightly different condition from which the original condition of Dally and Seitz can be derived. Dally and Seitz prove that a deadlock situation induces cyclic dependencies by reductio ad absurdum. In contrast we introduce the notion of a waiting graph from which we explicitly construct a cyclic dependency from a deadlock situation. Moreover, our proof is structured in such a way that it only depends on a small set of proof obligations associated to arbitrary routing functions and switching policies. Discharging these proof obligations is sufficient to instantiate our condition for deadlock-free routing on particular networks. Our condition and its proof have been formalized using the ACL2 theorem proving system.     

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.