Transpose on vertex symmetric digraphs

In [2] (and earlier in [3]), we defined several global communication tasks (universal exchange, universal broadcast, universal summation) on directed or vertex symmetric networks. In that paper, we mainly focused on universal broadcast. In [4], we discussed universal sum. Here we continue this work and focus on universal exchange (transpose) in directed vertex symmetric graphs. Introduction. In universal exchange, processor i has a vector of data ij a , where the element ij a is a packet needed by processor j. Thus we start with a matrix A where each processor has a separate row and we end the task with each processor having a separate column, in other words, we are computing the transpose. This gives rise to two fundamental questions: Given a directed vertex symmetric graph G (or any directed graph for that matter), how many time steps ) (G τ does it take to perform a transpose? Given two directed vertex symmetric graphs (or general directed graphs for that matter), how do we compare them with respect to their ability to perform a transpose? Graph symmetries. One important property of vertex symmetric graphs is that the number of edges directed into and out of a given vertex is a constant d. In our model of communication, we assume that on every time step all the edges in the graph can be used simultaneously, that is, each processor can simultaneously exchange a single packet of data with all of its neighbors. From here on out, we consider only directed Cayley coset graphs (equivalently, directed vertex symmetric graphs) ) , , ( H G ∆ Γ = . Briefly, a Cayley coset graph G has ] : [ H P Γ = cosets of the subgroup H in the group Γ as vertices and degree ∆ = d where ∆ is a collection of elements in Γ and ∆ ∪ H generates Γ . (Note that we allow multi-edges here but we insist that the graph be connected.) The edges are given by ) , ( H g gH δ for ∆ ∈ δ . A necessary and