Product-shuffle networks: toward reconciling shuffles and butterflies

Arnold L Rosenberg
1992 Discrete Applied Mathematics  
Rosenberg, A.L., Product-shuffle networks: toward reconciling shuffles and butterflies, Discrete Applied Mathematics 37/38 (1992) 465-488. We study product-shtrff/e (PS) networks, which are direct products of de Bruijn networks, as interconnection networks for parallel architectures. PS networks can be viewed as generalizing both butterfly-oriented network-: (such as the butterfly and cube-connected cycles networks) and shuffle-oriented networks (such as the de Bruijn and shuffle-exchange
more » ... ks), in the sense that l PS netvrorks can emulate both butterfly-oriented and shuffle-oriented netuorks of any size, via emulations that are work preserving, i.e., preserve the processor-time product; l PS networks share many computationally valuable structural features of various butterflyand shuffle-oriented networks, including pancyclicity, logmithrttic diameter, and large cotnplefe binary tree subnet works; l PS networks overcome certain: computational deficiencies of b;:tterfly-and shuffle-oriented networks, by containing as subnetworks moderate-size meshes and r?i&Cs o,f trees. networks which butterfly-and shuffle-oriented networks cannot emulate efficiently. Finally, PS networks attain their communication power at modest cost: they are 8-valent, and they enjoy VLSI iayouts that consume only modestly more area than the best layout5 of like-Gzed butterfly-and shuffle-oriented networks. Science Publishers B.V. All rights reserved 466 A.L. Rosenberg shuffle-exchange and de Bruijn networks, are among today's dominant interconnection networks for massively parallel architectures. Indeed, architectures based on these networks have been built in both industry and academia. Among these interconnection networks, the hypercube is the clear favorite because of its efficiency on a broad class of algorithm [6,&l 1,211 and its structural uniformity that simplifies programming [19]. The major shortcoming of the hypercube is its high valence.' The technological difficulties attendant to implementing such high-valence networks have led to the development of several butterflyoriented bounded-degree "approximations" of the hypercube, most notably the butterfly and CCC networks [16] . These networks were constructed with a certain important genre of hypercube algorithm, called ascend-descend algorithms [ 161, in mind and so can emulate the hypercube with little or no slowdown on a large, important class of computational problems. Yet, in a sense, butterfly-oriented networks just replace one implementational problem with another, since the) use No log2 N nodes (processors) to emulate the N-node hypercube. Further, algebraic transformations [3] of these large networks yield the smaller, shuffle-oriented bounded-degree "approximations" of the hypercube, most notably the shuffleexchange' [22] and de Bruijn [9,20] networks. Shuffle-oriented networks have only as many nodes as does the hypercube, yet they avoid its large valence; and, on certain computational tasks (including ascend-descend algorithms) they afford one computational efficiency (roughly) equal to that of the butterfly and CCC. Butterfly-and shuffle-oriented networks are roughly equivalent approximations of the hypercube on a broad class of ccmputational tasks, but it is not clear whether or not one of these network families majorizec the other on general computations, Confusingly enough, there is evidence that butterfly-and shuffle-oriented networks have incomparable strengths and weaknesses, and there is countervaiiling evidence that the two families of networks are equiAent in power. Distinguishing the two families are properties such as the following. The N-node de Bruijn network has the computationally useful properties of being pancyclic [24], of having diameter exactly log2 N, and of containing an (N-I)-node complete binary tree as a subnetwork; butterfly-oriented networks enjoy none of these. In contrast, the butterfly network enjoys both node-trslnsitivity and a recursive decomposition structure; neither is shared by shuffle-oriented networks. The symmetry and decomposability of butterfly networks are quite useful in developing efficient algorithms. For instance, the efficient routing and sorting algorithms for the butterfly network, in [17] and [Ml, respectively, exploit the symmetry and recursive structure of the butterfly, hence are not easily transported to any shuffle-oriented network. Further, circumstantial evidence separating the two families (and, indred. suggesting that 'The N-node hypercube bq-*,a1 UJ vnlence (-= maximum node-degree) log-, N. 'The shuffle-exchange network can also be derived directly from the hypercube, via a geometric transformation. 'That is, the network contains cycles of all lengths IN. ' More precisely, an emulation of an N-processor architecture :G by an (MS N)-processor architecture .;I/' is work preserving if .W can emulate any sequence of Tcomputational steps of 8 in at most c(N/M)T steps, for some fixed overhead constant c.
doi:10.1016/0166-218x(92)90152-z fatcat:maihv5dfzjf55d33frhcvb5wea