T. Leighton. Methods for message routing in parallel machines. In Proceedings of the 24th ACM Symposium on Theory of Computing, pages 77-96, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is commonly called permutation routing because the packets are being permuted among the hosts. The situation we usually refer to where packets arrive continually from outside the network is known as dynamic packet routing) Examples of research on static packet routing networks are [44] [45], 82] 80] 4] 77] 28] 3] 16] All of these are specific to a particular network and a particular routing scheme. They mostly concentrate on the problem of permutation routing, and use the Chernoff bound approach. Some research on static packet routing networks applies to general ....

....so as to get around particularly bad permutations where many packets need to traverse the same edge at the same time. 4 Therefore, there s also been a lot of research which concentrates on computing delays for the case where the final destinations are random (see for example [44] 80] [45], 36] In 1990 Tom Leighton pioneered the study of dynamic packet routing networks 4 Valiant and Brebner first proposed the now common idea of 2 stage randomized routing in 1981, 82] In the first stage packets are routed to random intermediate destinations, and in the second stage they are ....

Tom Leighton. Methods for message routing in parallel machines. In 24th Annual ACM Symposium on Theory of Computing, pages 77--96, May 1992.

....(i.e. all statements take t l time) In such a world, our job would use P processors, and take (f l f t f r f h )t l time. We define work to be the processor time product in the ideal world W = P (f l f t f r f h )t l (3:2) 4 In fact, even the network delays ts are functions of P [Lei92] However, to a very good approximation, we may assume them to be constant. This is because the time to go across a network (which scales with processors) is dwarfed by the high start up times to access the network (which is a constant) KC94] 34 1e 07 1e 08 1e 09 1e 10 10 100 1000 10000 ....

Tom Leighton. Methods for message routing in parallel machines. In Symposium on Theory of Computing, pages 77 -- 95, 1992.

....with no constraint on the local criterion, we shall seek good approximation algorithms. 1.1. Packet Routing. Our main result is a constant factor approximation algorithm for store and forward packet routing, a fundamental routing problem in interconnection networks (see Leighton s book and survey [14, 15]) furthermore, the queue sizes will all be bounded by a constant. This packet routing problem has received considerable attention for more than 15 years, and is as follows: Definition 1.1 (Store and Forward Packet Routing) We are given an arbitrary N node routing network (directed or ....

....NP hard objective is to select a path for each packet and to coordinate the routing so that the elapsed time by which all packets have reached their destinations is minimized; i.e. we wish to keep this routing time as small as possible. Extensive research has been conducted on this problem: see [14, 15] and the references therein. The most desirable type of algorithm here would, in addition to keeping the routing time and queue sizes low, also be distributed: given a set of incoming packets and their (source, destination) values, any switch (node of G) decides what to do with them next, without ....

F. T. Leighton. Methods for message routing in parallel machines. In Proc. ACM Symposium on the Theory of Computing, pages 77--96, 1992.

....have served as the basis for a number of actual parallel machines [3] 7 1.2.3 Routing and Sorting on Fixed Connection Networks In our study of fixed connection networks, we focus on the problems of routing and sorting. We restrict our attention to store and forward routing techniques; see [67] for an overview of other message routing methods. We define the packet routing problem on a fixed connection network as the problem of rearranging a set of packets in the network such that every packet ends up at the processor specified in its destination address. Here, the address of a ....

F. T. Leighton. Methods for message routing in parallel machines. In Proceedings of the 24th Annual ACM Symposium on the Theory of Computing, pages 77--96, May 1992.

.... is referred to the excellent book of Leighton [23] for an in depth treatment of this area, to a survey paper by Dally [12] for a more practical point of view, to a survey paper by Pippenger [37] for a discussion of circuit switching which will not be covered here, and to survey papers by Leighton [25], and by Valiant [44] 2 Preliminaries We model a communication network by a graph G = V; E) The vertices of the graph represent processors, memory models, or routing switches, and the edges (directed or indirected) represent communication channels. We distinguish between two types of networks ....

F.T. Leighton. Methods for Message Routing in Parallel Machines. In Proc. of the 24th Annual ACM Symp. on Theory of Computing, 1992, pages 77--96.

....are representative of the general routing activities which occur on parallel computer networks. It is also easier to develop analytical models for random traffic. It has been shown that random routing tasks are representative of the average case routing task (over all possible routing tasks) [72]. In fact, Valiant routing is based upon this fact. However, it is not clear that random routing tasks are in any way representative of what actually occurs on a parallel computer network. Many of parallel programming models and languages divide program activities into computation and ....

....Networks The butterfly network is the canonical example of a multistage interconnection network. There are a number of different ways in which randomization can be employed. Recently, researchers have proposed schemes which involve randomizing the wiring of butterflies or related networks [72, 132]. In these schemes, one can use deterministic routing without concern for problematic routing tasks. Unfortunately, there are some complicated issues associated with fabricating such networks. Recent results suggest that a variant of the butterfly, called the multiplexed metabutterfly, 169 000 ....

T. Leighton, Methods for Message Routing in Parallel Machines, in Proc. of the 24th Annual ACM Symp. on the Theory of Computing, ACM Press, May 1992, pp. 77--96.

....respect to the maximum load (i.e. congestion) on communication links. 1 Introduction We study the routing problem, i.e. the problem of assigning paths to packets, in an online and distributed setting on tori. The routing problem is important in, among others, the packet delivering problem (see [14] and references therein) which is of moving packets towards their destinations under the constraint that at most one packet is allowed to move along an edge in a time step. The goal of the delivering problem is to minimize the work span, i.e. the number of time steps required to deliver all ....

....determined on its source node. In our setting, the path of a packet whose source node is s depends only on those packets whose source nodes are s as well. Like many existing routing algorithms and implementations which are based upon the shortest path assumption (see the survey paper by Leighton [14] and, for example, 4] we restrict ourselves to routing algorithms which operate in the shortest path mode (i.e. only paths of shortest length are eligible for assignment) Enforcing the shortest path constraint to routing ensures that the dilation and the total load in the network are ....

T. Leighton. Methods for message routing in parallel machines. In Proceedings of the Twenty-Fourth Annual ACM Symposium on the Theory of Computing, pages 77-96, 1992.

....Figure 6 illustrates the dependency of running time on the number of processors in an experiment we performed. The job involved multiplying two 8 Admittedly, the MasPar MP 2 is an SIMD machine, however it illustrates the point well. 9 In fact, even the network delays ts are functions of P 0 [14]. However, to a very good approximation, we may assume them to be constant. This is because the time to go across a network (which scales with processors) is dwarfed by the high start up times to access the network (which is a constant) 11] 10 1e 07 1e 08 1e 09 1e 10 10 100 1000 10000 ....

T. Leighton. "Methods for Message Routing in Parallel Machines". In Symposium on Theory of Computing, pages 77 -- 95, 1992.

....the QSM has a work efficient emulation. The parallel slackness indicated is needed for the emulation to obtain the given work efficiency; additional parallel slackness will not help the EREW or CRCW results. The last row indicates that the BSP has a work efficient emulation on the hypercube model [42]. In particular, even with slackness = 1, the BSP(g; L) model has a work efficient emulation on the following hypercube model: Each node can send and receive messages from all its neighbors in parallel, all messages are sent point to point (i.e. there is no combining of messages) there is a ....

F. T. Leighton. Methods for message routing in parallel machines. In Proc. 24th ACM Symp. on Theory of Computing, pages 77--96, May 1992. Invited paper.

....packet switching, ATM [HenPat96, Ch. 7] Network protocols TCP IP as an example of a layered protocol [Comer95, Chs. 3 13] Processor Interconnection Networks Topologies Direct (hypercube, torus, ring) and indirect (butterfly) networks [Dally90] Routing Virtual channels, deadlock avoidance [Leighton92] and [Dally90] General Issues in Parallel Computing Applications and technology [HenPat96, Chs. 8.1 8.2, 8.7, and 8.9 8.11] 5 Data Parallel Computing Programming Model Algorithms and linguistic expression [Blelloch96] Vector machines Vector machine design [HenPat96, App. B] memory ....

Leighton, T., "Methods for Message Routing in Parallel Machines," 24th ACM Symposium on Theory of Computing (STOC '92), May, 1992, pp. 77--96. 8

....the ith smallest element to the place indexed with i Gamma 1. For a full h h routing problem one can supply each packet with an index of its destination processor. In this manner the full h h routing problem becomes an h h sorting problem. The model of computation is the conventional one [9, 10], where only nearest neighbors exchange data. In one step a communication channel can transport at most one packet (as an atomic unit) in each direction. Processors may store more than h packets, but the number has to be bounded by a constant that is independent of the number of processors. For ....

T. Leighton. Methods for message routing in parallel machines. In Proceedings of the 24th ACM Symposium on Theory of Computing, pages 77--96, 1992.

....any injection sequence generated by the adversary, the buffers at the nodes are polynomially bounded and that each packet has a polynomiallybounded delivery time. 1 Introduction Packet routing is one of the central issues in the areas of parallel computing and networking (see a survey of Leighton [L]) In this paper, we consider packet routing in the setting Bellcore. e mail:aiello bellcore.com y Dept. of Computer Science, Technion, Haifa, Israel. e mail: eyalk cs.technion.ac.il; Part of this research was done while visiting DIMACS. URL:http www.cs.technion.ac.il eyalk z Bellcore. ....

....that the packets follow depend on the local traffic conditions. Static vs. dynamic routing: Until recently, routing algorithms which afforded worstcase analysis on latency and queue size were limited to the static routing problem. There is a long history of work in this area. See, for example, [L, LMR, LMRi, RT, OR]. The dynamic routing problem can be solved using a static routing algorithm by running the static routing algorithm periodically, storing packets injected during a given run of the algorithm, and using those packets as input for the next run. However, this approach will result in large packet ....

F.T. Leighton. "Methods for message routing in parallel machines". Invited paper in Proc. of 24th STOC, pp. 77-96, 1992.

....routing consists of moving packets of data from each node of a network to the other nodes in the network. The goal is to move all of the packets to their desired locations as quickly as possible. Various routing problems have been extensively studied under different models. We refer the reader to [4] for a survey of the topic. In this paper, we study the packet routing problem under the matching model, which was proposed by Alon, Chung and Graham in [1] The matching model assumes that at each step, a disjoint collection S of links of a network G is selected, and packets at the endpoints of ....

T. Leighton. Methods for message routing in parallel machines. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing, 77-96, 1992.

.... and simulations among them [198, 316] analytical approaches based on queuing theory [30] network reliability and dependability [62, 101] optical networks and high performance broadband networks [15, 325] and issues concerning the design of communication networks in actual parallel systems [114, 177, 185, 259]. 1 Disha means direction in Hindi. 2 Results for General Graphs 2.1 Common Permutations Suppose that an arbitrary packet initially located at node x j x 1 x 2 : x n is destined to the node (x) x 0 j x 0 1 x 0 2 : x 0 n . For the Bit Permute and Complement (BPC) class of ....

Leighton, F. T. Methods for message routing in parallel machines. Theoret. Comput. Sci., North-Holland. 128 (1-2), 1994, pp. 31--62.

....Queuing [9] do not explicitly use message deadlines, but just attempt to keep the worst case message delay small and bounded. Other relevant experimental work includes [16, 22, 14, 21, 5] Of course, routing without timing constraints has been the subject of a large number of works; see [11, 12] for a survey. These works usually attack the problem under a best effort model, and thus focus on optimizing global performance measures such as the overall completion time of a routing problem, or the maximum and or expected delay experienced by any packet. 2 Routing on the tree In this section, ....

F.T. Leighton (1992): Methods for message routing in parallel machines (invited survey). 24th ACM Symp. on Theory of Computing.

....with i Gamma 1. For a full h h routing problem one can supply each packet with an index of its destination processor. In this manner the full h h routing problem becomes an h h sorting problem. The model of computation is the conventional one, where only nearest neighbors exchange data [17, 15]. In one step a communication link can transport at most one packet in each direction. Processors may store more than h packets, but the number has to be bounded by a constant that is independent of the number of processors. For complexity considerations we count only communication steps; we ....

F. T. Leighton. Methods for message routing in parallel machines. Theoretical Computer Science, 128:31--62, 1994.

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T. Leighton. Methods for message routing in parallel machines. In Proceedings of the 24th ACM Symposium on Theory of Computing, pages 77-96, 1992.

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F.T. Leighton. "Methods for message routing in parallel machines". Invited paper in Proc. of 24th STOC, pp. 77-96, 1992.

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F.T. Leighton. Methods for message routing in parallel machines. Invited survey paper in STOC '92.

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F.T. Leighton. "Methods for message routing in parallel machines". Invited paper in Proc. of 24th STOC, pp. 77-96, 1992. 27

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F.T. Leighton (1992): Methods for message routing in parallel machines (invited survey). Proc.24th ACM Symp. on Theory of Computing, pp. 77--96.

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F.T. Leighton. "Methods for message routing in parallel machines". Invited paper in 24th STOC, pp. 77-96, 1992.

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F.T. Leighton. Methods for message routing in parallel machines. In Proc. of the 24th Ann. ACM Symp. on the Theory of Computing, May 1992, pages 77--96.

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T. Leighton, Methods for message routing in parallel machines, 24th Annual ACM Symposium on Theory of Computing (1992) 77-96.

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F.T. Leighton. \Methods for message routing in parallel machines". Invited paper in 24th STOC, pp. 77-96, 1992.

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