Emulating Direct Products by Index-Shuffle Graphs

In the theoretical framework of graph embedding and network emulations, we show that the index-shuffle graph (a bounded-degreehypercube-like interconnection network, recently introduced by [ Baumslag and Obreni c (1997): Index-Shuffle Graphs, :::]) efficiently approximates the hypercube in general computations, by emulating the directproduct structure of the hypercube. In the direct product G = G 1 # G 2 #####G k let any factor G i be an instance of any of the three following graphs: cycle, complete binary tree, X- tree. Given an N -node index-shuffle graph Yn , where N = 2 n , and any collection of 2 ` copies of G, such that: jG i j#2 n i ; for i = 1;:::k,where ` + P k i=1 n i # n and 2 dlog 2 ke # #max 1#i#k n i # # n; Yn emulates any factor G i in all copies of G in this collection with slowdown O#log k + log n i #=O#log log N#. As a consequenceof these and previous results, the indexshuffle graph emerges as a uniquely "universal" boundeddegree hypercube substitute. Th..