F. Annexstein, M. Baumslag, and A. L. Rosenberg, "Group action graphs and parallel architectures," SIAM Journal on Computing, vol. 19, pp. 544--569, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....bit. Edges of the first type are called cycle edges, while edges of the second type are referred to as hypercube edges. Our objective is to determine the edge weights for which the diffusion algorithms FOS and SOS will have the fastest convergence. We use the fact that the CCC(d) is a Cayley graph [1]. It is known that the cycles in the CCC(d) are generated by one generator of the corresponding Cayley graph, while the hypercube edges are generated by some other generator. As a consequence of theorem 2, the CCC(d) s optimal value for the condition number is obtained, iff all cycle edges are of ....

....and CCP, we can define a Butterfly graph with wrap around edges. Two vertices (i; q) and (i ) are adjacent, iff i i = 1 mod d and either q = q th bit. Again, edges of the first kind are Cycle edges while edges of the second kind are cross edges. This type of graph is a Cayley graph [1], so optimal values for the edge weights can be determined. Similar to lemma 5 and theorem 4, the optimal condition number occurs when the weight of the cycle edges equals 1 and the weight of the cross edges are a = 2 1 O(1= d) However, there is no significant improvement of the condition ....

F. Annexstein, M. Baumslag, and A.L.Rosenberg. Group action graphs and parallel architectures. SIAM J. Computing, 19:544--569, 1990.

....Rubik s cube. There has been a great deal of interest recently in Cayley graphs and their generalization, Schreier coset graphs, for their exceptionally nice characteristics both as models for traditional parallel network architectures and as a potential source of new networks for parallel CPU s [1, 3, 5, 7, 8]. Using Cayley graphs, researchers have discovered new regular graphs with more nodes for a given diameter and for a given number of edges per node than were previously known. This allows construction of larger networks, while meeting design criteria of a xed number of nearest neighbors and a xed ....

F. Annexstein, M. Baumslag, A.L. Rosenberg, \Group Action Graphs and Parallel Architectures ", SIAM J. Computing 19 (1990), pp. 544-569.

....elements of ZZ Because there is no hazard of ambiguities, we shall henceforth cease to distinguish between ZZ n . That is, we implicitly use the one to one mapping x 2 ZZ B n x n Gamma1 ; x 0 B : x = P n Gamma1 i=0 x i B i We employ the notation of group action graphs from [17]. The undirected group action Graph G = GAG(V; Pi) is given by a set V of vertices and a set Pi of distinct permutation of V . For each v 2 V; 2 Pi there is an edge labeled connecting v and v. Thus the set of edges of G is given by E(GAG(V; Pi) f(v; v)j8v 2 V; 2 Pig. It is easy to see ....

F. Annexstein, M. Baumslag, and A.L. Rosenberg, "Group action graphs and parallel architectures", SIAM J. Comput., vol. 19, no. 3, pp. 544--569, June 1990.

....x to y. So Gamma is strongly adjacency transitive. We characterize strongly adjacency transitive graphs in terms of action graphs. The action graph Gamma = ActGrph(G; X;S) 1) of a group G acting on a set X, relative to an inverse closed nonempty subset S G, is defined as follows (see [1]) the vertex set of Gamma is X, and two different vertices x; y 2 V ( Gamma) are adjacent in Gamma if and only if y = sx for some s 2 S. We refer the reader to [4] for the notion of action digraphs and its application, and to [5] for an implementation of the action graph construction. 2 ....

F. Annexstein, M. Baumslag, A.L. Rosenberg, Group action graphs and parallel architectures, SIAM Journal on Computing 19 (1990) 544-569.

....action (di)graph corresponds to a Schreier coset (di)graph, with repeated generators and semiedges allowed. However, to think of an action (di)graph actually as a Schreier coset (di)graph is much too rigid in many instances. For similar concepts dealing with (di)graphs and group actions see [1, 2, 3, 8, 12, 17, 18, 21, 22, 28, 43, 45]. Some of them, although conceptually di erent, bare the same name [45] and some of them, quite close to our de nition, are referred to by a variety of other names [1, 3] It seems that the term action (di)graph should be attributed to T. Parsons [43] For a computer implementation of (a variant ....

....in many instances. For similar concepts dealing with (di)graphs and group actions see [1, 2, 3, 8, 12, 17, 18, 21, 22, 28, 43, 45] Some of them, although conceptually di erent, bare the same name [45] and some of them, quite close to our de nition, are referred to by a variety of other names [1, 3]. It seems that the term action (di)graph should be attributed to T. Parsons [43] For a computer implementation of (a variant of) action graphs see [44] Supported in part by Ministrstvo za znanost in tehnologijo Slovenije , proj.no. J1 0496 99. 1 Group actions are compared by morphisms. ....

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F. ANNEXSTEIN, M. BAUMSLAG, A. L. ROSENBERG, Group action graphs and parallel architectures, SIAM J. Comput 19 (1990), 544-569.

....set of G. We also say that S generates G. If G is a group and S generates G, then the Cayley network Cay(G, S) is a network where the nodes are the elements of G, and the edges are all ordered pairs (s, t) where t = sg, for some g G and s S [16] Cayley graphs have been extensively studied [1] [6], 16] as bases for interconnection networks, due to their many desirable properties, including regularity, vertex symmetry and recursive or near recursive substructure. Recently, a number of Cayley networks of degree O(1) have been proposed. 23] 18] 17] Some examples of fixed degree Cayley ....

F. Annexstein, M. Baumslag and A. L. Rosenberg. Group action graphs and parallel architectures. SIAM Journal on Computing, 19(3):544-569, June 1990

....are substantial improvements of earlier results stated in [5] Keywords: Cayley graphs, routing algorithms, Hamilton cycles, embeddings, shuffle exchange permutation network. 1: Introduction Cayley graphs [1] have proved to be a useful basis for interconnection networks for a number of reasons [3]. They are regular (unlike networks with a central hub) and vertex symmetric (unlike two dimensional meshes or DeBruijn networks[6] Some examples of networks with low degree and diameter are shown in Table 1) with diameter results found in [1] and [4] Network Vertices Degree Diameter ....

F. Annexstein, M. Baumslag and A. L. Rosenberg. Group action graphs and parallel architectures. SIAM Journal on Computing, 19(3):544-569, June 1990

....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, ....

F. Annexstein, M. Baumslag, and A. L. Rosenberg. Group action graphs and parallel architectures. SIAM Journal on Computing, 19(3):544--569, June 1990.

....through a node is similar for all nodes. 4.1.1 Cayley Networks The class of Cayley networks is an important subclass of node symmetric networks. Many standard networks belong to this class, for instance the multidimensional arrays (generalized hypercubes) BA84, LMRR94] the cube connectedcycles [ABR90, CCSW85], the wrapped butterflies [ABR90] the bubble sort networks [AK89, LJD93] and the star networks [LJD93] CHAPTER 4. CONSTRUCTION OF SHORTEST PATHS SYSTEMS 25 Definition 4.2 (Cayley network) Let Gamma be a finite algebraic group with identity 1, and suppose Sigma is a set of generators of ....

....4.1.1 Cayley Networks The class of Cayley networks is an important subclass of node symmetric networks. Many standard networks belong to this class, for instance the multidimensional arrays (generalized hypercubes) BA84, LMRR94] the cube connectedcycles [ABR90, CCSW85] the wrapped butterflies [ABR90], the bubble sort networks [AK89, LJD93] and the star networks [LJD93] CHAPTER 4. CONSTRUCTION OF SHORTEST PATHS SYSTEMS 25 Definition 4.2 (Cayley network) Let Gamma be a finite algebraic group with identity 1, and suppose Sigma is a set of generators of Gamma with 1 = 2 Sigma. Then the ....

F. Annexstein, M. Baumslag, and A.L. Rosenberg, Group action graphs and parallel architectures, SIAM Journal on Computing 19/3 (1990) pp. 544--569

....elements known from computational group theory. Generalizations of all of the above are presented for Schreier coset graphs. 1. Introduction There has been a strong interest recently in using Cayley graphs as a model for developing interconnection networks for large interacting arrays of CPU s [1, 3, 7, 9, 10]. Many of the traditional parallel network architectures such as cube connected cycles (hypercubes) have descriptions as Cayley graphs [9] Using Cayley graphs and Schreier coset graphs, researchers have also discovered new regular graphs with more nodes for a given diameter than were previously ....

....node to the node g Gamma1 g 0 yields a path of minimal length from nodes g to g 0 . Certain of our techniques generalize to Schreier coset graphs. Of the parallel network architectures which cannot be described as Cayley graphs, many have natural descriptions as Schreier coset graphs [1]. Our methods can save further space by using special, compact encodings of group elements. Section 2 discusses some group theoretic preliminaries and a compact encoding of general group elements for permutation groups. This is required for the space efficient representation of Cayley graphs. ....

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F. Annexstein, M. Baumslag, A.L. Rosenberg, "Group Action Graphs and Parallel Architectures ", SIAM J. Computing 19 (1990), pp. 544--569.

....Abstract Cayley graphs and elementary group theory are used to efficiently find minimum length permutation routes in a bus interconnection network. Cayley graphs have been used extensively to design interconnection networks and provide a natural setting for studying point to point routing [1, 2, 3, 4], but the technique has not been widely used for the more important problem of permutation routing. This is due to the potentially explosive growth in both the size of the graph and the number of generating permutations, referred to as one step permutation routes, used to define the underlying ....

....if Hg 1 s = Hg 2 . Thus, the image of g 0 2 Hg on C depends only on g and not on the choice of g 0 . So, it suffices to find a shortest word w in S representing an element of Hg. Many routing problems can be formulated in terms of Cayley coset graphs (also referred to as group action graph in [1]) In general, Cayley coset graphs are not vertex symmetric. The second reduction is obtained from the symmetries of the chosen bus interconnection architecture. Recall that a group automorphism is a permutation oe of G such that oe(gh) oe(g)oe(h) and oe(g Gamma1 ) oe(g) Gamma1 for all ....

F. Annexstein, M. Baumslag, A.L. Rosenberg, "Group Action Graphs and Parallel Architectures ", SIAM J. Computing 19 (1990), pp. 544--569.

....computations; it efficiently substitutes the hypercube in the on line leveled hypercube algorithms. In each of these five computation classes, the index shuffle graph is shown to be asymptotically exponentially faster than any of the other four standard bounded degree hypercube substitutes. cf. [3, 4, 7, 9, 11, 13, 18, 20, 23]. The only candidate exception is Schwabe s [22] embedding of the mesh of trees into its like sized shuffle like graph with constant dilation. This embedding, however, assigns two guest nodes to a single host node. This modification in the formal setting may induce non trivial changes in the ....

F. Annexstein, M. Baumslag, and A. L. Rosenberg. Group action graphs and parallel architectures. SIAM J. Comput., 19:544--569, 1990.

....2 Lemma 7.2 The proof is complete. 2 8 De Bruijn Graphs The next family of graphs we study epitomize the shuffle oriented 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 ....

F.S. Annexstein, M. Baumslag, A.L. Rosenberg (1990): Group action graphs and parallel architectures. SIAM J. Comput. 19, 544-569.

....demands that two network topologies be interchangeable for all computational communicational applications. Such equivalence has been established most often using the notion of quasi isometry the inter embeddability of two families of networks with small dilation. One finds in sources such as [2, 8, 14, 17] proofs of the strong equivalence of network topologies such as: paths and rings; meshes and tori; de Bruijn and shuffle exchange networks; butterfly, FFT, and Cube connected cycle networks. Equivalence on important classes of problems. This form of equivalence has received less attention than ....

F.S. Annexstein, M. Baumslag, A.L. Rosenberg (1990): Group action graphs and parallel architectures. SIAM J. Comput. 19, 544-569.

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F. Annexstein, M. Baumslag, and A. L. Rosenberg, "Group action graphs and parallel architectures," SIAM Journal on Computing, vol. 19, pp. 544--569, 1990.

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F. Annexstein, M. Baumslag, A. L. Rosenberg, Group action graphs and parallel architectures, SIAM J. Comput. 19 (1990) 544-569.

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Annexstein, F., Baumslag, M. and Rosenberg, A. L. (1990) Group action graphs and parallel architecture. SIAM J. Comp., 19, 544--569.

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F. Annexstein, M. Baumslag, and A.L. Rosenberg, "Group action graphs and parallel architecture," SIAM J. Computing, 19(3)8 June 1990.

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F. Annexstein, M. Baumslag, and A. L. Rosenberg. Group action graphs and parallel architectures. SIAM J. Comput., 19(3):544--569, 1990.

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F. Annexstein, M. Baumslag, and A. L. Rosenberg, "Group action graphs and parallel architectures," SIAM Journal on Computing, vol. 19, pp. 544--569, 1990.

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Annexstein, F., Baumslag, M., and Rosenberg, A. L. Group action graphs and parallel architectures. SIAM Journal on Computing 19, 3 (1990), 544--569.

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F. Annexstein, M. Baumslag, and A.L.Rosenberg. Group action graphs and parallel architectures. SIAM J. Computing, 19:544--569, 1990.

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F. Annexstein, M. Baumslag, and A.L.Rosenberg. Group action graphs and parallel architectures. SIAM J. Computing, 19:544--569, 1990.

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F. Annexstein, M. Baumslag and A. L. Rosenberg, "Group Action Graphs and Parallel Architectures," SIAM Journal on Computing, 19 (1990), pp. 544-569.

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F. Annexstein, M. Baumslag, and A. L. Rosenberg, "Group Action Graphs and Parallel Architectures," SIAM Journal on Computing, vol. 19, pp. 544--569, June 1990.

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