L. R. Goke and G. J. Kipovski, "Banyan networks for partitioning multiprocessor systems," Proceedings of 1st Annual Symposium on Computer Architecture, pp. 21-28, 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is defined by C n;d;q = X d;q 1 X n=d;n=d 1 X q;d , where we require of course that d divides n. The delta network [7] D n;d is defined by D d;d = X d;d D n;d = X d;d Theta D n=d;d where n = d for some integer k. The number of stages in the delta network is exactly k. The banyan network [3] Y n;d is defined by Y d;d = X d;d Y n;d = n=d;d ; n=d Delta X d ) d;n=d ; d Delta Y n=d;d ) The banyan network is isomorphic to the delta network, and so is equivalent in all respects. However, it is useful to define it separately as certain properties are more easily proved using the ....

Goke, G. R. and G. J. Lipovski. "Banyan Networks for Partitioning Multiprocessor Systems," 1st Ann. Symp. on Computer Architecture, pp. 21--28, 1973.

....methodology on some example MINs are presented in section 5 before concluding the paper. 2 Topological Classification of MINs We propose in figure 1 a topological classification of MINs. We give in the following some important definitions related to this classification. Definition 1 A banyan[8] MIN is a MIN having the property of the existence of one and only one path be tween each source and destination. Banyan MINs might have the delta property or not. Delta networks, proposed by Patel[20] are built of a x b crossbars. Let oi; i 0, 1, b i be an output of index i of a ....

G.R. Goke and G.J. Lipovski. Banyan networks for partitioning multiprocessor systems. In Proc. 1st Annu. Symp. Comput. Arch., pages 21-28, 1973.

....by its complement the blocking probability, and we shall do the same here. The problem of computing blocking probabilities in regular variants of unique path multistage interconnection networks has been extensively studied. These networks were called Banyan networks byGoke and Lipovsky [9]. Patel [21] and Kruskal and Snir [15] in particular presented expressions for the probability of successful message transmission of delta networks, which are a particular regular variantofBanyan networks. Multiprocessors have been built using such regular Banyan networks for interconnection [22, ....

....a bijective mapping from sources to destinations can always be accomplished without blocking. Non isochronous applications such as shared memory references in a multiprocessor can better tolerate blocking, and thus often use blocking variants of Banyan networks, as presented byGoke and Lipovski [9]. Patel [21]presented a probabilistic analysis of the blocking probability of delta networks, a subset of the more general class of Banyan networks. His work assumed that all sources transmit with uniform probability,and that all destinations are selected with uniform probability.Bhuyan [5] has ....

Goke, L. R., and Lipovski, G. J. "Banyan Networks for Partitioning Multiprocessor Systems," in Proceedings of the First Annual Symposium on Computer Architecture,1973.

....of networks and discuss about what is optimal in a cost performance sense. Finally, in section 5 we give the conclusions and some ideas for future work. 2. The model 2.1. Banyan networks Banyan networks belong to the class of Multistage Interconnection Networks (MINs) They were defined in [GOK 73] and are characterized by the property that there is exactly one path from any input to any output. A simple example of a 8 8 banyan interconnection network (i.e. which has 8 inputs and 8 outputs) is drawn in figure 1. This network consists of nodes and links. Every node is a 2 2 switch, which ....

GOKE L. R., LIPOVSKI G. J., "Banyan Networks for partitioning multiprocessor systems", Proc. 1st Annu. Int. Symp. Comput. Architectures, December 1973, pp. 21--28.

....pattern, not all of the L links may act to add usable bandwidth, but this constraint provides an upper bound. The most commonly discussed and analyzed indirect networks are the multistage networks with a logarithmic number of stages, such as the Omega network [85] the rectangular SW Banyan [51], square delta networks [110] the baseline network [146] and the indirect binary n cube [113] all of which can be shown to be essentially equivalent [146] when the number of inputs and outputs to the network and the degree of the switches is the same. By analogy with the k ary n cube direct ....

L. Rodney Goke and G. J. Lipovski. Banyan networks for partitioning multiprocessor systems. Proceedings of the 1st Annual Symposium on Computer Architecture, pages 21--28, 1973. 156

....for them. Embedding communication structures in a MIN in the space domain has been studied previously. For example, the problem of satisfying a set of resource allocation requests in rectangular SW banyans has been studied with the assumption that all output ports of the network are equivalent [10, 21]. Embedding of unidirectional binary trees in a MIN with broadcasting capabilities has been studied in [16] However, in all the above work, no time domain embedding is involved. Finding a configuration sequence for a CR graph representing a regular communication structure can be regarded as ....

G.R. Goke and G.J. Lipovski, "Banyan networks for partitioning multiprocessor systems," First Annual Symposium on Computer Architecture, pp. 21-28, Dec. 1973.

....output port from a given input port, these fabrics are classified into two groups called singlepath and multiple path networks. 3.8.2.1 Single Path Networks In single path networks, there is only one path to the destination from a given input. These networks are also called Banyan networks [23]. Since there is only one path to the destination port, routing is simple. The disadvantage is that internal blocking can occur as an internal link can be used simultaneously by different groups. Banyan networks can further be classified into sub groups. In L level Banyan networks, only the ....

....First, the commonly used sorting trap butterfly network is discussed. Later, a stacked banyan balancing network is evaluated and the performance of the two networks is compared. A sorting circuit banyan network can be constructed by placing a sorting circuit [37] in front of a banyan network [23]. 5.1 Banyan and Delta Networks Any network for which there is a unique path from each input to each output is called a banyan network. For instance, the log 2 (N) dimensional butterfly is an example of a 2Ninput and 2N output banyan network. In order to facilitate packet routing, the k output ....

Goke, L. R., Lipovski, G. J., "Banyan Networks For Partitioning Multiprocessor Systems," First Annual Symposium on Computer Architecture, 1973, pg. 21-28. 83

....Partial Word Store, Reflect, and Broadcast. All operations are combinable by the network except for Partial Word Store, Reflect, and Broadcast. 2.2.3. Interconnection Network The Ultracomputer network is built from KK switches and has the topology of the omega network [16] and the SW Banyan [9]. It consists of log K N stages, where N is the number of PEs in the system. The Ultra III network has log 2 N stages, each having N 2 22 switch nodes that routes memory requests from PEs to MMs and responses (to these requests) from MMs to PEs. A switch has eight unidirectional ports, four of ....

G. R. Goke and G. J. Lipovski, "Banyan Networks for Partitioning Multiprocessor Systems", Proc. 1st Annual International Symposium on Computer Architecture, pp. 21-28 (1973).

....and routing algorithms, and new parallel application algorithms. Some research results in this area are described in [10] Multistage interconnection networks (MINs) have been an important interconnecting scheme for parallel computing systems. A MIN can be blocking such as a banyan network [6], rearrangeably nonblocking such as a Benes network [1] nonblocking such as crossbar [15] or with variable connecting capabilities, from rearrangeable for permutation to nonblocking for multicast, such as a Clos network [1] 4] 19] depending on the number of stages, the number of switches, ....

....for permutation to nonblocking for multicast, such as a Clos network [1] 4] 19] depending on the number of stages, the number of switches, the switch capability, and the interconnection patterns used between stages. MINs have been extensively studied in the literature, e.g. see [1] [6], 4] 18] 19] However, most of the research reported are for electronic MINs. As optical technology advances, there are a lot of interests in using optical technology for implementing interconnection networks and switches. Although electronic MINs and optical MINs have many similarities, ....

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L.R. Goke and G.J. Lipovski, "Banyan networks for partitioning multiprocessor systems," Proc. of the First Annual Symposium on Computer Architecture, 1973, pp. 21-28.

....design a TBSF by arriving at an approximate relationship between the loss rate L and the number of blocks (L) that constitute the TBSF. Section 9 summarizes the chief contributions of this paper. 2 Characterization of Bi Delta Networks A n stage, d degree, N Theta N Banyan Network consists of [11] (See Figure 2) P1: N = d n labeled source nodes of indegree 0 and outdegree 1, constituting stage 0. P2: n stages of N=d switches (in each stage) of indegree d and outdegree d, constituting stages 1; 2; n. The d outputs, and d inputs of each switch are labeled from 0 to (d Gamma 1) ....

....above arguments are true for all m, so that u ae = u ae [ Pi a ; Pi z ] 2 Several other symmetric traffics apart from the uniform and independent traffic are possible. Indeed permutation traffic is one such [1] 4 Conflict Resolution Policies It is well known that a Banyan Network is blocking [11]. That is even if all the packets in a traffic element e have differing destination addresses, there still would (in general) be conflicts at the outputs of switches. In our scenario, destination addresses are not necessarily distinct, so that would create conflicts too. In any case, a Conflict ....

L.R.Goke & G.J.Lipovski, "Banyan Networks for Partitioning Multiprocessor Systems", Proc. 1st Annual Symposium Computer Arch., Dec. 1973. pp. 21-28.

....either a crossbar [WuBe72] or a simple bus [ThGF88] Hill86] However, in considering a highly parallel machine (with hundreds of PEs, for example) busses become too easily saturated and crossbars become too expensive. In constructing an N PE multiprocessor where N is large, a multistage network [GoLi73] [Pate81] Gott87] can offer a good compromise between hardware cost (Q (N log N) and bandwidth (Q (N) 2 Unfortunately, multistage networks are also subject to saturation especially while supporting programs that display non uniformity in their access patterns [PfNo85] When such hot spot ....

....allows for a more selective use of consistency operations. Caches with stale values are made consistent by the UpdateAll operation (described below) which is issued under software control. Our methods are sufficiently general to be applicable to any multiprocessor configured with a banyan network [GoLi73]. Note that delta [Pate81] and omega networks [Lawr75] are subsets of the banyan class. These networks are all characterized by the property that a set of paths exist from each memory node to all the processors. Furthermore, these paths comprise a tree that fans out through the various network ....

L. Rodney Goke and G.J. Lipovsky, "Banyan Networks for Partitioning Multiprocessor Systems," First Annual Symposium on Computer Architecture, 1973, pp. 21-28.

....direct single cycle access to shared memory characteristic of paracomputers is approximated in the NYU Ultracomputer by indirect access via a multicycle connection network. A message switching network with the topology of Lawrie s [18] W network (equivalently, the SW Banyan of Goke and Lipovsky [6]) is used to connect N = 2 D autonomous PEs to a central shared memory composed of N memory modules (MMs) Figure 1 gives a block diagram of the machine. The Ultracomputer design places few constraints on the processors and memory modules. Naturally, the fetch and add instruction is needed. In ....

L.R. Goke and G.J. Lipovsky, "Banyan Networks for Partitioning Multiprocessor Systems", Proc. First Annual Symp. on Comp. Arch., 1973.

....networks for parallel processing (see, for example, 4] 6] 15] 17] Nevertheless, there seems to be a surprisingly small number of basic designs for interconnection networks that recur under many disguises. A particularly ubiquitous geometry is the multistage shuffle exchange network [1] 5] [7], 11] 12] 14] 16] 19] This geometry provides good performance and simple message routing (or control) Given the paucity of other network geometries, one is tempted to conjecture that these networks are in some sense optimal. In this paper, we characterize the networks with simple message ....

....We shall use switch as a synonym for node. We assume without loss of generality that for each node u of G there is a directed path connecting some input to u and a directed path connecting u to some output. A network with a unique path from each input to each output is called a banyan network [7]. The indegree of a node is the number of edges leading into it, and the outdegree of a node is the number of edges leading out of it. A banyan network is layered if the nodes can be arranged in successive layers, with inputs at the first layer, outputs at the last layer, and edges connecting ....

G. R. GOKE and G. J. LIPOVSKI, Banyan Networks for Partitioning Multiprocessor Systems, 1st Ann. Symp. on Computer Architecture, 21-28, 1973.

....interconnection schemes, the total number of connections, given a fixed framework, has to be determined for them. Layered graphs can represent the underlying structure of networks in diverse disciplines, like computer (interconnection) networking: multistage networks [2] 3] and banyan networks [4], switching) circuit analysis: cascaded networks ( 5] and other network based fields. The theory presented in this document has been applied to neural networks [6] 2. Counting Edges 2.1. Interlayer connections The number of connections between two adjacent layers in a fully interlayer ....

L. Rodney Goke and G. J. Lipovski, Banyan Networks for Partitioning Multiprocessor Systems, Proceedings of the First Annual Symposium on Computer Architectures pages 21--28, December 1973.

.... s) Yous 90] A Banyan network is a multistage interconnection network [Adam 87] its topology is a directed graph with a unique path from every base to every apex 28 where normally the bases and apexis represent processors or memory modules, whereas all other nodes are switching cells (see [Goke 73] or [Sieg 79] for a more detailed discussion of Banyan networks) In a regular Banyan network the indegree s (number of ingoing links) equals the outdegree, and there are s n nodes within each level. In a Banyan Hypercube all nodes in the Banyan network represent processing elements, connected ....

L. R. Goke and G. J. Lipovski. "Banyan networks for partitioning multiprocessor systems". In: Proceedings Annual Symposium on Computer Architecture, pp. 21--28, December 1973.

....Section 4 evaluates and compares the performance of BanyanNet, ShuffleNet, and the bilayered ShuffleNet. Finally, in section 5 we present a summary and conclusions. 2 BanyanNet Goke and Lipovski proposed a general class of dynamic networks for multiprocessor interconnection, called Banyan networks [20, 21]. These networks are essentially made up of superimposed trees. Banyan is the name of a multiply rooted tree in India. Of the general class of Banyan networks, a sub class called regular SW Banyan is of special interest to us. Mathematically, these networks can be defined as: For an N = p m ....

....diameter and hence propagation delay becomes unacceptably large. However, our performance analysis in section 4 shows that when k is moderately large, the network has a good total throughput. The result is a cylindrically and multiply cascaded version of the SW Banyan network by Goke and Lipovski [20, 21]. We called this network, an N = p m Theta k BanyanNet. These additional flexibilities reduce the diameter and increase network performance such as channel efficiency, network and user throughputs as will be discussed in Section 4. As obvious from the above definition, there are 2p ....

L.R. Goke. Banyan Networks For Partitioning Multiprocessor Systems. PhD thesis, University of Florida, Gainesville, Florida, 1976.

....Section 4 evaluates and compares the performance of BanyanNet, ShuffleNet, and the bilayered ShuffleNet. Finally, in section 5 we present a summary and conclusions. 2 BanyanNet Goke and Lipovski proposed a general class of dynamic networks for multiprocessor interconnection, called Banyan networks [20, 21]. These networks are essentially made up of superimposed trees. Banyan is the name of a multiply rooted tree in India. Of the general class of Banyan networks, a sub class called regular SW Banyan is of special interest to us. Mathematically, these networks can be defined as: For an N = p m ....

....diameter and hence propagation delay becomes unacceptably large. However, our performance analysis in section 4 shows that when k is moderately large, the network has a good total throughput. The result is a cylindrically and multiply cascaded version of the SW Banyan network by Goke and Lipovski [20, 21]. We called this network, an N = p m Theta k BanyanNet. These additional flexibilities reduce the diameter and increase network performance such as channel efficiency, network and user throughputs as will be discussed in Section 4. As obvious from the above definition, there are 2p ....

L.R. Goke and G.J. Lipovski. "Banyan Networks For Partitioning Multiprocessor Systems". In Proceedings of the First Annual Computer Architecture Conference, pages 21--28, 1973.

....for all but the last stage, module outputs are connected by links to module inputs in the next stage. Module outputs in the last stage are connected to network outputs. For a banyan network the links can be connected in any way as long as there is exactly one path between every input output pair [3]. Each module consists of m queues, each with d slots connected to an m Theta m crossbar switch. The module inputs connect to the queues; the queue outputs connect to the crossbar; and the crossbar outputs connect to the module outputs. A packet is the unit of communication; it consists of a ....

Goke, L.R., and Lipovski, G.J. Banyan networks for partitioning multiprocessor systems. In Proceedings of the International Symposium on Computer Architecture. 1973, pp. 21-- 28.

....for all but the last stage, module outputs are connected by links to module inputs in the next stage. Module outputs in the last stage are connected to network outputs. For a banyan network the links can be connected in any way as long as there is exactly one path between every input output pair [3]. See [13] for an elementary introduction and [8] a discussion of the topology of banyan networks. Each module consists of 2 d slot queues, connected to a 2 Theta 2 crossbar switch. Module inputs connect to the queues; queue outputs connect to the crossbar; crossbar outputs connect to the module ....

L. R. Goke and G. J. Lipovski, "Banyan networks for partitioning multiprocessor systems, " in Proceedings of the International Symposium on Computer Architecture, 1973, pp. 21--28.

....by its complement the blocking probability, and we shall do the same here. The problem of computing blocking probabilities in regular variants of unique path multistage interconnection networks has been extensively studied. These networks were called Banyan networks by Goke and Lipovsky [9]. Patel [21] and Kruskal and Snir [15] in particular presented expressions for the probability of successful message transmission of delta networks, which are a particular regular variant of Banyan networks. Multiprocessors have been built using such regular Banyan networks for interconnection ....

....a bijective mapping from sources to destinations can always be accomplished without blocking. Non isochronous applications such as shared memory references in a multiprocessor can better tolerate blocking, and thus often use blocking variants of Banyan networks, as presented by Goke and Lipovski [9]. Patel [21] presented a probabilistic analysis of the blocking probability of delta networks, a subset of the more general class of Banyan networks. His work assumed that all sources transmit with uniform probability, and that all destinations are selected with uniform probability. Bhuyan [5] has ....

Goke, L. R., and Lipovski, G. J. "Banyan Networks for Partitioning Multiprocessor Systems," in Proceedings of the First Annual Symposium on Computer Architecture, 1973.

....2 Node 3 Node N Interconnection Network . Node 1 Node 2 Node 3 Node N FIGURE 2. The simplest interconnection network ringlet PAGE 5 OF 25 Constructing Large Scale SCI based Processing Systems by Switch Elements 76] Ston 71] data manipulator [Feng 73] baseline [WuFe 80] SW banyan [GoLi 73], omega [Lawr 75] delta [Pate 79] etc. Among them the crossbar switch is quite popular and suitable for general cases. In a crossbar switch, every input port can be connected to a free output port without blocking. The shuffle exchange network might also be investigated, it requires less switch ....

L.R. Goke and G.J. Lipovski, "Banyan Networks for Partitioning Multiprocessor Systems", Proceedings of 1. Annual Computer Architecture Conference, Dec. 1973, pp. 21-28

....in part by the Louisiana Board of Regents through the Louisiana Education Quality Support Fund, contract number LEQSF (1993 95) RD A 07 and by the National Science Foundation under Grant No. MIP 9410435. i 1 INTRODUCTION Banyan networks, unique path multistage interconnection networks [1], are widely considered for use in communication and parallel computing systems. Performance analyses of such networks are needed both for evaluation of system designs and for understanding the networks themselves. Many banyan network analysis methods have been reported; the bulk of the work was ....

....of a m slot queues, each connected to the crossbar inputs. Links connect SE s in adjacent stages and first and last stage SE s to network inputs and outputs, respectively. The links can be connected in any pattern for which there is exactly one path between all network input output pairs. See [1,2,6] for details. 2.2 MESSAGE STRUCTURE AND FLOW CONTROL Data arrives at network inputs in the form of messages. Each message consists of a number of fixed length packets which pass through the network as a unit. The first packet is called the head packet and the last packet is called the tail packet; ....

L. R. Goke and G. J. Lipovski, "Banyan networks for partitioning multiprocessor systems, " in Proceedings of the International Symposium on Computer Architecture, 1973, pp. 21--28.

....Mapping and Communication y http: www.uni paderborn.de cs ag monien.html To appear in the Proc. of the 8th ACM Symposium on Parallel Algorithms and Architectures (SPAA 96) Padua, Italy, June 1996. A technical report is available from http: www.uni paderborn.de cs diek.html banyan networks [5]. Within this paper we consider banyans with butterfly interconnection structure consisting of (2 Theta 2) switches (as shown in Fig. 1) For simplicity reasons, they will be called MINs in the following. Simulation results have shown that the communication performance of MINs can be increased ....

G.F. Goke, G.J. Lipovski, Banyan Networks for Partitioning Multiprocessor Systems, Proc. 1st Ann. Symp. on Computer Architecture, 1973, pp. 21--28

.... 0 1 Queues Feeder Fed Cycle F1 F2 A B 5 A A c 1 1 6 B A c 1 0 7 B E c 0 1 8 E B c 1 1 9 A E d Gamma 1 1 3 Analysis Overview The following is an outline of the analysis method; a complete description can be found in [9] The network to be modeled is an n stage input buffered banyan network [3] with queues having d slots each. Local flow control is used; that is, a packet cannot enter a queue in a cycle unless the queue has a free slot. The slot occupied by a packet leaving a queue in a cycle is considered free in the next cycle. Offered traffic consists of fixed length packets; each ....

L. R. Goke and G. J. Lipovski, "Banyan networks for partitioning multiprocessor systems, " in Proceedings of the International Symposium on Computer Architecture, 1973, pp. 21--28.

....port of one of the switches in the path. In this paper, the n stages of a delta network are numbered from 0 to n 0 1 (the inputs connect to stage 0) The switches in each stage are numbered from 0 to b n01 0 1. There are many MIN s in this class, including the baseline [23] SW banyan [8], omega [12] ip [3] and indirect binary n cube [18] networks. They are topologically equivalent [2, 15, 23] so the choice between them is mainly a matter of which one is easiest to implement in a particular system. The important property they all have in common is that from any input port in ....

L. R. Goke and G. J. Lipovski. Banyan networks for partitioning multiprocessor systems. In Proceedings of the First Annual Computer Architecture Conference, pages 21--28, December 1973.

....provide multiple communication routes between any given pair of processor nodes. Multiple routes provide low latency, high bandwidth, and reliable interprocessor communication. There are multistage interconnection networks (MIN s) 18, 25] which have a regular structure, such as Omega [15] Banyan [9], and indirect binary n cube [21] networks. Using the inherent knowledge of the interconnection topology, each switch in the network knows which output ports lead a packet to its destination at each stage. Route generation for such networks makes use of the structure in the topology to determine ....

L. R. Goke and G. J. Lipovski. Banyan networks for partitioning multiprocessor systems. In Proc. 1st Ann. Symp. on Computer Architecture, pages 21--28, 1973.

.... that include the omega network [16] the generalized cube network [23] and the indirect binary n cube [22] Wu and Feng [25] defined the class of baseline networks recursively and showed that the inverse omega network, the indirect binary n cube, a restricted version of the SW banyan networks [9], and the baseline network on N inputs are all topologically equivalent. Similar equivalence relations have also been shown in [20] Thus, results on any of the above networks can be extended to the others. Here, we choose the baseline network and the indirect binary n cube network (IBC, for ....

L. Goke and G. Lipovski, Banyan networks for partitioning multiprocessor systems, in Proceedings of the 1st Annual Symposium on Computer Architecture, 1973, pp. 21--28.

....by Lawrie [4] which uses log 2 N stages of switches, each stage consisting of N=2 crossbar switches of size 2 Theta 2. A perfect shuffle permutation [5] is used to connect the switches in adjacent stages. Delta, Omega, indirect binary n Gammacube [6] etc. are subsets of the Banyan network [7]. The Omega network is a blocking type of network where contention arising at a switch results in performance degradation. Performance evaluation of Omega and Delta networks have been reported in the literature [3, 8, 9, 10, 11, 12] using both analytical modeling and simulation techniques. Most of ....

L.R. Goke, "Banyan networks for partitioning multiprocessor systems," First Annual Symposium on Computer Architecture, pp. 21--28, December 9-11, 1973.

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L. R. Goke and G. J. Kipovski, "Banyan networks for partitioning multiprocessor systems," Proceedings of 1st Annual Symposium on Computer Architecture, pp. 21-28, 1973.

No context found.

Goke, G. F. and Lipovski, G. J.: Banyan Network for Partitioning Multiprocessor Systems, Proc. 1st Ann. Symp. on Computer Architecture, 1973, pp. 21-28.

No context found.

L. Rodney Goke and G.J. Lipovsky, "Banyan Networks for Partitioning Multiprocessor Systems", First Annual Symp. on Computer Architecture, pp. 21-28, 1973.

No context found.

G.R. Goke and G.J. Lipovski, "Banyan Networks for Partitioning Multiprocessor Systems", Proc. 1st Annual Intern. Symp. on Comp. Arch., 1973, pp. 21-28.

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