M. Samatham and D. Pradhan. The de bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI. EEE Trans. on Computers, 38(4), 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....attention; see, e.g. the survey article of Monien and Sudborough [10] and the literature mentioned there. De Bruijn graphs are popular communication networks for parallel computers because they feature several nice properties such as fixed node degree and small diameter (for more details cf. [4, 15, 16]) The problem of studying embeddings of hypercubes, grids and tori into de Bruijn graphs was initiated by Heydemann, Opatrny, and Sotteau [7, 8] who obtained a variety of results on embedding hypercubes and 2 dimensional grids into de Bruijn graphs, while grids of higher dimension and tori were ....

M.R. Samatham and D.K. Pradhan. The de Bruijn Multiprocessor Network: A Versatile Parallel Processing and Sorting Network for VLSI. IEEE Trans. on Computers, Vol. 38, pages 567--581, April 1989.

....instead of working with a base 2 de Bruijn graph, Koorde can work with a base O(log n) de Bruijn graph. With such a graph, Koorde has fault tolerance and the number of routing hops is O( log n) log log n) which is optimal. 5 Related work We are not the first to use de Bruijn graphs in routing [1, 3, 13, 14], and concurrent with our work others have noted their application to DHTs [4] Compared to the related work, our primary contribution is how to simulate a lookup using a de Bruijn graph in a sparsely populated identifier space. Koorde s approach of using de Bruijn graphs is different than D2B s ....

SAMATHAM, M., AND PRADHAM, D. The de bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI. IEEE Trans. on Computers 38, 4 (1989), 567--581.

.... Opatrny, Sotteau [6, 7] For general information on interconnection networks and, in particular, on containment and embedding results, we refer to [2, 3, 9, 10] and the literature mentioned there; for applications, e.g. in the field of parallel image processing and pattern recognition, see [5, 11, 12, 13]. All graphs considered in this paper are simple, i.e. have no loops or multiple edges. If G is a graph, then V (G) and E(G) denote the set of vertices and the set of edges of G, respectively. Our terminology is standard; for graph theoretic terminology not explained here, we refer to [4] For ....

M.R. Samatham and D.K. Pradhan, "The de Bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI", IEEE Transactions on Computers, vol. 38, no. 4, pp. 567--581, 1989.

....attention; see, e.g. the survey article of Monien and Sudborough [7] and the literature mentioned there. De Bruijn graphs are popular communication networks for parallel computers because they feature several nice properties such as fixed node degree and small diameter (for more details cf. [3, 11, 12]) The problem of studying embeddings of hypercubes, grids and tori into de Bruijn graphs was initiated by Heydemann, Opatrny, and Sotteau [5, 6] who obtained a variety of results on embedding hypercubes and 2 dimensional grids into de Bruijn graphs, while grids of higher dimension and tori were ....

M.R. Samatham and D.K. Pradhan. The de Bruijn Multiprocessor Network: A Versatile Parallel Processing and Sorting Network for VLSI. IEEE Trans. on Computers, Vol. 38, pages 567--581, April 1989.

....and tori are considered only briefly. In contrast, in the paper [4] the emphasis is on tori. For definitions, see below. De Bruijn graphs are popular communication networks for parallel computers because they feature several nice properties such as fixed node degree and small diameter ( 5] [6], 7] For information on the role of de Bruijn networks in parallel image processing and pattern recognition, see e.g. 8] In the present paper, for Cartesian products G = G 1 Theta : Theta Gm (m 2) of non trivial connected graphs G i and the n dimensional base B de Bruijn graph D = DB ....

M.R. Samatham and D.K. Pradhan, "The de Bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI", IEEE Transactions on Computers, vol. 38, no. 4, pp. 567--581, 1989.

.... x 1 Delta Delta Delta xD x 2 Delta Delta Delta xD y 1 x 3 Delta Delta Delta xD y 1 y 2 Delta Delta Delta y 1 Delta Delta Delta y D : We refer the reader to one of the two recent surveys concerning de Bruijn networks written by Bermond and Peyrat [4] and Samatham and Pradhan [13] or to the recent book of Leighton [10] In this paper, we analyze the mean eccentricity of these graphs. The eccentricity of a vertex X is defined [6] as the distance to the farthest node from this vertex: e(X) maxfd(X; Y ) Y 2 V g. We define the mean eccentricity of a vertex X , denoted ....

M.R. Samatham and D.K. Pradhan, "The de Bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI", IEEE Trans. on Comp., 38, No. 4, pp. 567--581, 1989. 32

....1) is Hamiltonian cycle of K(d; D) and the same holds for B(d; D) De Bruijn (di)graphs, and the binary ones in particular, received more attention from the researchers than Kautz digraphs. Various results on nontrivial embeddings and emulations have been published in last few years (see, e.g. [1, 8, 13, 14]) Neither alphabet nor line digraph definitions allow incrementally expandable digraphs. Congruent arithmetics enables to define generalized K B digraphs for any number of vertices. 2.4 Generalized K B digraphs Let GK(d; n) and GB(d; n) denote the generalized Kautz and de Bruijn digraph, ....

M. R. Samathan and D. K. Pradhan. The de Bruijn multiprocessor network: A versatile parallel processing and sorting network for VLSI. IEEE Transactions on Computers, C38 (4):567--581, Apr. 1989.

....graphs. These graphs are known to be able to emulate the much larger butterfly graph and its butterflyoriented relatives efficiently on a large class of computations [2] 39] hence, these graphs have been widely proposed as interconnection networks for parallel architectures [6] [31], 33] 36] The order n de Bruijn graph D n is usually presented as a directed graph. The digraph D n has node set Nodes(D n ) Z n 2 ; its arcs lead every node fix, where x 2 Z n Gamma1 2 and fi 2 Z 2 to nodes xfi and x fi. Because D n has 2 n nodes, each of indegree and outdegree 2, ....

M.R. Samatham and D.K. Pradhan (1989): The de Bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI. IEEE Trans. Comp. 38, 567-581.

....path. 952 1. Introduction The generalized de Bruijn network is based on the de Bruijn digraph [8] 3] This network received much attention from researchers because it possesses several interesting properties which makes it suitable for many parallel processing applications [3] 6] 8] [9]. For instance, Samathan and Pradhan [9] have shown that the de Bruijn network can efficiently embed several other important architectures such as the linear array, ring, 2 D mesh, cube connected cycles, complete binary tree, and tree machine. Thus, the network can efficiently run the important ....

.... generalized de Bruijn network is based on the de Bruijn digraph [8] 3] This network received much attention from researchers because it possesses several interesting properties which makes it suitable for many parallel processing applications [3] 6] 8] 9] For instance, Samathan and Pradhan [9], have shown that the de Bruijn network can efficiently embed several other important architectures such as the linear array, ring, 2 D mesh, cube connected cycles, complete binary tree, and tree machine. Thus, the network can efficiently run the important algorithm classes best suited for these ....

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M. R. Samathan and D. K. Pradhan, "The de Bruijn Multiprocessor Network: A Versatile Parallel Processing and Sorting Network for VLSI," IEEE Trans. on Computers, pp. 567-581. vol. 38, No. 4, April, 1989.

....of our study. Section 3 presents our emulation strategy in a topology independent fashion. Section 4 describes and analyzes our emulations on three substantively different array topologies that have been used in computer architectures: the hypercube [27] Section 4. 1) the de Bruijn network [22] (Section 4.2) and the extended coterie network, a mesh with a reconfigurable bus which abstracts the UMass CAAPP architecture [30] Section 4.3. In Section 5, we show that the operational performance of our emulation based multigauging is superior to having the host A perform k bit operations ....

M.R. Samatham and D.K. Pradhan, The de Bruijn Multiprocessor Network: A Versatile Parallel Processing and Sorting Network for VLSI, IEEE Trans. Computers, vol. 38, pp. 567581, 1989.

....has as its vertices all those strings of length D over the alphabet f0; 1; dg in which consecutive characters are distinct; 11 there is an arc from a vertex a 1 a 2 a 3 : a D to all vertices a 2 a 3 : a D a with a from f0; 1; dg and distinct from aD . It is easy to see (cf. 5] 16] [24]) that both B(d; D) and K(d; D) are (d; D) digraphs. Moreover, the digraph B(d; D) has d D vertices and the digraph K(d; D) has d D d D Gamma1 vertices. We immediately obtain, via the proof of Proposition 3, large (balanced) 2d 2; D)C digraphs. However, we were able to prove (see [6] ....

M.R. Samatham and D.K. Pradhan, The de Bruijn multiprocessor network: A versatile parallel processing and sorting network for VLSI, IEEE Trans. Comput., vol. C--38, 4, pp. 567--581, 1989.

....constructed as follows [21] For every vertex v 2 V(G) there are two vertices v and v Gamma 2 V( G) The vertices v i and v Gamma j are adjacent in G, if and only if the vertices v i and v j are adjacent in G. Let G be the bipartite double of the de Bruijn graph (see [15] or [53]) of maximum degree r (r is even) and diameter D Gamma 1. Then G is regular of degree r, has 2(r=2) D Gamma1 vertices and diameter D. The dual hypergraph of G is a (2; D; r) hypergraph with 2(r=2) D vertices. An extension of the bipartite double of de Bruijn graphs, is the C s graphs ....

M.R. Samatham and D.K. Pradhan. The de Bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI. IEEE Trans. on Computers, 38:567--581, 1989. (corrections in IEEE Trans. on Computers, vol. 40, pp. 122-123).

.... x 1 Delta Delta Delta xD x 2 Delta Delta Delta xD y 1 x 3 Delta Delta Delta xD y 1 y 2 Delta Delta Delta y 1 Delta Delta Delta y D : We refer the reader to one of the two recent surveys concerning de Bruijn networks written by Bermond and Peyrat [4] and Samatham and Pradhan [13] or to the recent book of Leighton [10] 2 In this paper, we analyze the mean eccentricity of these graphs. The eccentricity of a vertex X is defined [6] as the distance to the farthest node from this vertex: e(X) maxfd(X; Y ) Y 2 V g. We define the mean eccentricity of a vertex X , ....

M.R. Samatham and D.K. Pradhan, "The de Bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI", IEEE Trans. on Comp., 38, No. 4, pp. 567--581, 1989. 34

....As a result, several families of networks with large number of processors for given degree and diameter have been proposed. Surveys on this topic can be found in [1] 2] 6] and [8] Among them, de Bruijn and Kautz networks appear to have many other desirable properties (for details see [7] [17]) Classical definition of the de Bruijn networks is based on alphabets. These networks can be generalized for any number of processors by using arithmetic congruences (see [13] 16] When the bus size is taken into account as an extra parameter, the problem becomes more complicated. There are ....

M.R. Samatham and D.K. Pradhan. The de Bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI. IEEE Trans. on Computers, 38:567--581, 1989. (corrections in IEEE Trans. on Computers, vol. 40, pp. 122-123). 20

.... x 1 Delta Delta Delta xD x 2 Delta Delta Delta xD y 1 x 3 Delta Delta Delta xD y 1 y 2 Delta Delta Delta y 1 Delta Delta Delta y D : We refer the reader to one of the two recent surveys concerning de Bruijn networks written by Bermond and Peyrat [5] and Samatham and Pradhan [14] or to the recent book of Leighton [11] In this paper, we analyze the mean eccentricity of these graphs. The eccentricity of a vertex X is defined [7] as the distance to the farthest node from this vertex: e(X) maxfd(X; Y ) Y 2 V g. We define the mean eccentricity of a vertex X, denoted ....

M.R. Samatham and D.K. Pradhan, "The de Bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI", IEEE Trans. on Comp., 38, No. 4, pp. 567--581, 1989.

.... to dynamic routing on the d way shuffle [9] On the n way shuffle (d = n) deterministic sorting, by embedding Batcher s sorting network, achieves O(n 2 ) delay [229] probabilistic two phase routing is asymptotically optimal [236] and off line BPC, Omega, and Inverse Omega routing is optimal [275]. Wei and Hsu have provided upper and lower bounds for deterministic oblivious permutation routing and sorting on the directed version of the de Bruijn network [112] The undirected n dimensional shuffle exchange has N = 2 n nodes, diameter 2n Gamma 1, and degree 3. A node represented by a ....

Samatham, M. R., and Pradhan, D. K. The de Bruijn multiprocessor network: A versatile parallel processing and sorting network for VLSI. IEEE Trans. Comput. C-38 (4), 1989, pp. 567--581.

....also have more paths between pairs of nodes, which permits more adaptivity and fault tolerance. A class of shuffle networks known as de Bruijn (dB) graphs have become popular recently. They are suitable for large network and can be defined for any number of nodes, including prime numbers [61]. For a specific node degree, dB networks, in most cases, have the smallest diameter compared to the contemporary network topologies. Formally, a unidirectional dB network can be defined as follows [61] Definition 3: An r radix unidirectional de Bruijn digraph dBD(r; r m ) has the total number ....

....suitable for large network and can be defined for any number of nodes, including prime numbers [61] For a specific node degree, dB networks, in most cases, have the smallest diameter compared to the contemporary network topologies. Formally, a unidirectional dB network can be defined as follows [61]. Definition 3: An r radix unidirectional de Bruijn digraph dBD(r; r m ) has the total number of nodes N = r m and the address of a node X is represented as (x m Gamma1 ; xm Gamma2 ; x 0 ) where x i 2 f0; 1; r Gamma 1)g for 0 i m Gamma 1. Its neighboring nodes are (x m Gamma2 ; ....

M. R. Samatham and D. K. Pradhan, "The de Bruijn Multiprocessor Network: A versatile parallel processing and sorting network for VLSI," IEEE Trans. on Computers, C-38, pp. 567-581, Apr. 1989.

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M. Samatham and D. Pradhan. The de bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI. EEE Trans. on Computers, 38(4), 1989.

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D.K. Pradhan and M.R. Samatham, "The de Bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI", IEEE Trans. Computers 38 (1989) pp. 567-581.

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M.R. Samatham and D.K. Pradhan, "The de Bruijn multiprocessor network: A versatile parallel processing and sorting network for VLSI," IEEE Transactions on Computers, 38(4)8 April 1989.

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M.R. Samatham and D.K. Pradhan, The de Bruijn multiprocessor network: A versatile parallel processing and sorting network for VLSI, IEEE Trans. Comput., vol. C--38, 4, pp. 567--581, 1989.

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M.R. Samatham and D.K. Pradhan, "The de Bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI", IEEE Trans. on Comp., 38, No. 4, pp. 567--581, 1989. 34

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M. Samatham and D. Pradhan. The de bruijn multiprocessor network: a versatile parallel processing and sorting network for VLSI. EEE Trans. on Computers, 38(4), 1989.

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M. Samatham and D. K. Pradhan, The De Bruijn multiprocessor networks: A versatile parallel processing and sorting network for VLSI, IEEE Trans. Computers, (1989), vol. 38, no. 4, pp. 567-581.

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M.R. Samatham and D.K. Pradhan, The de Bruijn multiprocessor network: A versatile parallel processing and sorting network for vlsi, IEEE Trans Comput 38 (1989), 567--581.

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