Resource placement in three-dimensional tori

In massively parallel processing systems that incorporate hundreds and thousands of processing nodes, the problem of placing a limited number of resources (e.g. printers, disks, software copies, etc.) over processing nodes is of great importance. In this paper, we investigate resource placement in the three-dimensional torus network which is the underlying interconnection network in most current multicomputer. We show that there exists an irregular distance-1 placement only for tori of the form 2×3i×6j,i,j∈N. In addition, we present a new formal proof for the necessary and sufficient conditions for the existence of irregular distance-d placement in 3D tori.