J.C. Bermond and C. Peyrat. De bruijn and kautz networks: a competitor for the hypercube? in Proc. of the 1st Europ. Workshop on Hypercubes and Distributed Computers, pp. 279293, North-Holland, 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 ....

J.C. Bermond and C. Peyrat. De Bruijn and Kautz networks: a competitor for the hypercube? In F. Andre and J. P. Verjus, editors, Proceedings of the 1st European Workshop on Hypercube and Distributed Computers, pages 279--293, Amsterdam, 1989. North-Holland.

.... results completely settling several of the relevant subcases, thereby improving previous results of Andreae, Nolle, Schreiber [1] and Heydemann, 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 ....

J.C. Bermond and C. Peyrat, "De Bruijn and Kautz networks: a competitor for the hypercube?", in Hypercube and Distributed Computers, F. Andre and J. Verjus, Eds., Amsterdam, 1989, pp. 279--294, North Holland, Elsevier Science Publisher.

....one interconnection network into another one is of significant importance in the context of parallel computers. Among the network topologies which have been proposed and investigated in the literature, toroidal grids (briefly: tori) and de Bruijn graphs are among the most popular ones; see, e.g. [1, 2, 3, 5, 6, 7]. In the present paper, we deal with the problem of constructing good embeddings of a 2 dimensional quadratic torus into a de Bruijn graph B(d; D) with the same number of vertices, where the quality of an embedding is measured in terms of its load, expansion, dilation, and clocked congestion. ....

J.C. Bermond and C. Peyrat, De Bruijn and Kautz networks: a competitor for the hypercube?, in Hypercube and Distributed Computers, F. Andre and J. Verjus, Eds., Amsterdam, 1989, pp. 279--294, North Holland, Elsevier Science Publisher.

....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 ....

J.C. Bermond and C. Peyrat. De Bruijn and Kautz networks: a competitor for the hypercube? In F. Andre and J. P. Verjus, editors, Proceedings of the 1st European Workshop on Hypercube and Distributed Computers, pages 279--293, Amsterdam, 1989. North-Holland. 14

....DB d , is a directed graph of 2 d vertices and 2 d 1 arcs. The vertices are labelled by the 2 d binary d tuples. There is an arc from vertex x 1 : x d to vertex y 1 : y d i x 2 : x d = y 1 : y d 1 . As a result, vertices 00: 0 and 11: 1 have self loops. Undirected de Bruijn graphs [1], UDB d , are formed from directed de Bruijn graphs by ignoring the orientations of the edges and deleting the two self loops, which are irrelevant in determining the crossing number. UDB d has 2 d vertices, 2 d 1 2 edges, and maximum degree 4. Figure 4(g) displays UDB 3 which is planar. UBD d ....

J.-C. Bermond, C. Peyrat, De Bruijn and Kautz networks: a competitor for the hypercube?, in: Hypercube and Distributed Computers (ed. F. Andre and J.P. Verjus), Elsevier Science Publ., Amsterdam, 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 ....

J.-C. Bermond and C. Peyrat (1989): The de Bruijn and Kautz networks: a competitor for the hypercube? In Hypercube and Distributed Computers (F. Andre and J.P. Verjus, eds.) North-Holland, Amsterdam, 279-293.

....between processors (for multicomputers) and between processors and memory modules 1 This research was supported by an NSF grant No. MIP 9310082. 1 (for multiprocessors) dominate the cost of the machine, the power budget, the hardware (wiring, packaging, etc. and the overall performance[1, 2, 3, 4]. Many topologies have been explored for parallel computers, including multistage interconnection networks such as omega, baseline, banyan, crossover, etc, and point to point interconnection networks such as hypercube, mesh, ring, bus, star, etc. 3, 5] Currently, two of the most popular ....

....a major emphasis on scalable networks with small diameters and bounded node degrees[14] As an alternative to the hypercube and the mesh topologies, the de Bruijn topology[15, 16] has recently been receiving much attention. Its properties and applications have been studied by several researchers[2, 17, 18, 19, 20]. Its topological properties show that the de Bruijn network is a good candidate for interconnection networks of the next generation of parallel computers after the hypercube. The de Bruijn network behaves like the hypercube, and retains most of the same desired properties (logarithmic diameter, ....

J.-C. Bermond and C. Peyrat, "De Bruijn and Kautz Networks: a Competitor for the Hypercube?," in Proceedings of the First European Workshop on Hypercube and Distributed Computers, Rennes, France, pp. 279--293, Elsevier Science Publishers, Oct. 4-6 1989.

....efficient way. We remark again that the hypercube presents very simple and efficient communication procedures. Unfortunately, though, existing parallel topologies satisfy at most two out of the above conditions, as we briefly showed for the hypercube. Interconnection networks such as the de Bruijn[3] or the star graph[4] show very good theoretical characteristics from a graph theoretic point of view, but fail to give a potential user the feeling that they can easily built or programmed. Therefore, two or three dimensional grids, as in the Paragon of Intel and the Cray T3D[5] have been ....

J.-C. Bermond and C. Peyrat. De Bruijn and Kautz network: a competitor for the hypercube? Hypercube and Distributed Computers, 1989.

.... to, for example, 2] and [9] De Bruijn graphs, both directed and undirected, have received considerable attention since de Bruijn graphs can accommodate more vertices while the maximum degree and the diameter are given, in comparison with other prevailing topologies such as hypercubes and meshes [4,6]. In this paper, we focus on study of undirected de Bruijn graphs, and henceforth, undirected de Bruijn graphs are referred to as de Bruijn graphs. The multiprocessor system which can be modeled by a de Bruijn graph is called de Bruijn network. A de Bruijn graph UB(d; n) is a simple graph ....

J.C. Bermond and C. Peyrat, de Bruijn and Kautz networks: a competitor for the hypercube?, in Hypercube and Distributed Computers, F. Andr'e and J.P. Verjus, eds., Elsevier Science Publishers, pp. 279--293, 1989.

....full (resp. half) duplex constant time 1 port model. The time depends also on the communication network. Here, we are mainly interested in de Bruijn, Kautz, and shuffle exchange networks, and in their generalization. Several definitions of De Bruijn and Kautz networks as digraphs can be given (see [6, 9]) Here Z d will demote the ring of integers modulo d. The de Bruijn digraph B(d; n) of out degree d and diameter n is defined as a digraph on the set of vertices made of words of length n on Z d , each vertex x 1 : x n being joined by the arc denoted ff , ff 2 Z d to the vertex x 2 : ....

J-C. Bermond and C. Peyrat. De Bruijn and Kautz networks: a competitor for the hypercube? In North-Holland, editor, Hypercube and Distributed Computers, pages 279--294, 1989.

....and degree 2d. A node represented as d 1 d 2 : d n , 0 d i d Gamma 1; 1 i n is connected to nodes d 2 d 3 : d n p, and p d 1 d 2 : d n Gamma1 , where 0 p d Gamma 1. Greedy routing corresponds to correcting dimensions from left to right, but it does not produce a shortest path route [28]. Aleliunas extended Valiant s probabilistic algorithm 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 ....

....[265] Problem 18 Is it possible to bridge the gap between the lower bound and the upper bound for sorting on the MIMD n dimensional star graph 5. 5 Other proposed graphs Many graphs with good topological properties have been proposed, e.g. distributed loop networks [27] Kautz graph [28], Cayley graphs [29, 126, 168] and expanders, such as fattrees [184] Although these graphs have asymptotically optimal diameter, very little is known about their routing properties. Another interesting graph is a Cartesian product of two fully connected graphs, called the k ary n cube [61] In ....

Bermond, J. C., and Peyrat, C. De Bruijn and Kautz networks: A competitor for the hypercube? Proc. 1st Euro Workshop Hypercube Distrib. Comput. 1989, pp. 279--293.

....[4] In general, NPcomplete problems are very hard to solve unless the problem size is small enough. Using genetic algorithms (GAs) we solved the above modified colorability problem to get efficient signal distinction schemes for the mesh, the hypercube, and the binary de Bruijn networks [5]. GAs were developed to study the adaptive process of natural systems and to develop artificial systems that mimic the adaptive mechanism of natural systems [6] GAs can also be applied to various optimization problems such as the traveling salesman problem. In a given problem, a set of potential ....

J.-C. Bermond and C. Peyrat, "De Bruijn and Kautz Networks: a Competitor for the Hypercube ?," in Proceedings of the First European Workshop on Hypercube and Distributed Computers, Rennes, France, pp. 279--293, Elsevier Science Publishers, Oct. 4-6 1989.

....of edges per node, a low diameter, a fast shortest path routing and defines a natural connection structure for FFT applications. Especially this last property makes it a good candidate for the interconnection network of parallel computers. Many results concerning the de Bruijn network are given in [Lei92,SP89,BP89]. Nevertheless for the bisection width of the de Bruijn there is still a huge gap between known upper and lower bounds. If we define dB(n) to be the de Bruijn network of dimension n (with 2 n nodes) we already know that 2 n n fi(dB(n) 4 Delta 2 n n : The upper bound is derived by ....

J.C. Bermond and C. Peyrat. De bruijn and kautz networks: a competitor for the hypercube? In Proceedings of the 1st European Workshop on Hypercubes and Distributed Computers, pages 279--293. North-Holland, 1989.

....[1, 2, 4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 22, 23, 25, 26, 28, 29, 30] and minimal, scalable, deadlock free algorithms are known for many important networks including meshes [26] tori [7] trees [23] and hypercubes [26] 1 . However, no such algorithm is known for the de Bruijn [3, 21] or shuffle exchange [27, 21] networks (although a nonminimal, scalable, deadlock free algorithm is known for the shuffle exchange [26] These networks are known to be very powerful in a synchronous setting, but the lack of efficient, deadlock free routing algorithms for these networks has ....

....networks has hindered their use in asynchronous, MIMD, parallel computers. The creation of such an algorithm for the de Bruijn network is particularly important, as it has constant degree and small diameter, it can support many hypercube algorithms efficiently, and it can be packaged efficiently [3, 5, 21]. It is known that a scalable, deadlock free routing algorithm exists for every connected network [23] However, this algorithm forces some messages to take nonminimal paths. Furthermore, this algorithm uses a spanning tree of the network in order to construct the deadlock free routing paths, so ....

J-C. Bermond and C. Peyrat, "de Bruijn and Kautz Networks: A Competitor for the Hypercube", in Hypercube and Distributed Computers, F. Andr'e and J.P. Verjus (eds.), North-Holland, pp. 279--293, 1989.

....Grant Agency. Preprint submitted to Elsevier Preprint 23 February 1 Introduction The de Bruijn and Kautz digraphs were originally studied as asymptotically largest digraphs w.r.t. degree and diameter [4] and were proposed as promising topologies for massively parallel computer architectures [3]. Several graphtheoretic properties and algorithmic design problems have been widely studied for these graphs [1,2,13,17] In this paper, we study vertex and edge bisection width of the k ary n dimensional de Bruijn and Kautz digraphs, respectively. The edge bisection width of a graph G = V; ....

J.-C. Bermond and C. Peyrat, De Bruijn and Kautz networks: A competitor for the hypercube?, in: Proc. Hypercube and Distributed Computing (Elsevier, Amsterdam, 1989) 279--274.

No context found.

J-C. Bermond and C. Peyrat, "De Bruijn and Kautz networks: a competitor for the hypercube ?", In Hypercube and Distributed Computers, J. P. Verjus and F. Andr'e, (Eds.), pp. 279--294, North-Holland, 1989.

.... corresponding to the dipath 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 ....

J-C. Bermond and C. Peyrat, "De Bruijn and Kautz networks: a competitor for the hypercube ?", In Hypercube and Distributed Computers, J. P. Verjus and F. Andr'e, (Eds.), pp. 279--294, North-Holland, 1989.

....as a bus network [17] In [52] Pradhan proposed a generalization of the shuffle exchange network. The dual of this network gives a hypergraph of degree 2, rank r, diameter 2k Gamma 1 (k is a positive integer) It has r k 1 =2 vertices. 6 Kautz graphs are obtained from Kautz digraphs [15] by replacing the directed edges with undirected ones. Kautz graphs of maximum degree r (r is even) and diameter D have (r=2) D (r=2) D Gamma1 vertices. The dual hypergraph of the Kautz graph of diameter D Gamma 1, and maximum degree r, is a (2; D; r) hypergraph with (r=2) D (r=2) ....

....G is 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 ....

[Article contains additional citation context not shown here]

J.-C. Bermond and C. Peyrat. The de Bruijn and Kautz networks: a competitor for the hypercube? In Hypercube and Distributed Computers, proc. First European Workshop on Hypercube and Distributed Computers, Rennes, Oct. 1989, pages 279--293. Elsevier (NorthHolland) , 1989.

.... corresponding to the dipath 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 ....

J-C. Bermond and C. Peyrat, "De Bruijn and Kautz networks: a competitor for the hypercube ?", In Hypercube and Distributed Computers, J. P. Verjus and F. Andr'e, (Eds.), pp. 279--294, North-Holland, 1989.

....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 ....

J.-C. Bermond and C. Peyrat. The de Bruijn and Kautz networks: a competitor for the hypercube? In Hypercube and Distributed Computers, proc. First European Workshop on Hypercube and Distributed Computers, Rennes, Oct. 1989, pages 279--293. Elsevier (NorthHolland) , 1989.

.... corresponding to the dipath 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 ....

J-C. Bermond and C. Peyrat, "De Bruijn and Kautz networks: a competitor for the hypercube ?", In Hypercube and Distributed Computers, J. P. Verjus and F. Andr'e, (Eds.), pp. 279--294, North-Holland, 1989.

....However for the de Bruijn and related networks, the order of b(G) is still to be found. This work has been supported by the French action RUMEUR of the GDR PRC PRS and by the European HCM project MAP. Let us recall the definition and basic properties of the de Bruijn and Kautz networks (see [3, 13] for more details) The de Bruijn digraph B(d; D) of out degree d and diameter D has as vertices the words of length D on an alphabet of d letters. Vertex x 1 : xD is joined by an arc to the vertices x 2 : xD ff where ff is any letter from the alphabet. Between any pair of vertices x 1 : ....

....G, with an arc from e to f in L(G) if and only if in G the initial vertex of f is the final vertex of e. In others words, e represents an arc of the form (x; y) and f an arc of the form (y; z) It is well known that B(d; D 1) resp. K(d; D 1) is isomorphic to L(B(d; D) resp. L(K(d; D) see [3, 13]) For a given digraph G, we denote UG the underlying graph associated to G (obtained by removing the orientation) The underlying de Bruijn (resp. Kautz) graph will therefore be denoted UB(d; D) resp. UK(d; D) Let us recall the previous bounds known on the broadcast time for de Bruijn and ....

J-C. Bermond and C. Peyrat. De Bruijn and Kautz networks: a competitor for the hypercube? In Hypercube and Distributed Computers, pages 279--294, NorthHolland 1989

No context found.

J.C. Bermond and C. Peyrat. De bruijn and kautz networks: a competitor for the hypercube? in Proc. of the 1st Europ. Workshop on Hypercubes and Distributed Computers, pp. 279293, North-Holland, 1989.

No context found.

Jean-Claude Bermond and C. Peyrat. De bruijn and kautz networks: a competitor for the hypercube, 1989.

No context found.

J.C. Bermond and C. Peyrat. De bruijn and kautz networks: a competitor for the hypercube? Proc. of the 1st Europ. Workshop on Hypercubes and Distributed Computers, pp. 279293, North-Holland, 1989.

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