Lower bounds on communication loads and optimal placements in torus networks

Fully populated torus-connected networks, where every node has a processor attached, do not scale well since load on edges increases superlinearly with network size under heavy communication, resulting in a degradation in network throughput. In a partially populated network, processors occupy a subset of available nodes and a routing algorithm is specified among the processors placed. Analogous to multistage networks, it is desirable to have the total number of messages being routed through a particular edge in toroidal networks increase at most linearly with the size of the placement. To this end, we consider placements of processors which are described by a given placement algorithm parameterized by k and d: We show formally, that to achieve linear communication load in a d-dimensional k-torus, the number of processors in the placement must be equal to ck^d-1 for some constant c. Our approach also gives a tighter lower bound than existing bounds for the maximum load of a placement for arbitrary number of dimensions for placements with sufficient symmetries. Based on these results, we give optimal placements and corresponding routing algorithms achieving linear communication load in tori with arbitrary number of dimensions