The maximum degree and diameter-bounded subgraph in the mesh
Discrete Applied Mathematics
| Hi-index | 0.00 |
We consider the degree-diameter problem for Cayley graphs of Abelian groups (Abelian graphs) for both directed graphs and undirected graphs. The problem is closely related to that of finding efficient lattice coverings of Euclidean space by shapes such as octahedra and tetrahedra; we exploit this relationship in both directions. For two generators (dimensions), these methods yield optimal Abelian graphs with a given diameter k. (The results in two dimensions are not new; they are given in the literature of distributed loop networks.) We find an undirected Abelian graph with three generators and a given diameter k, which we conjecture to be as large as possible; for the directed case, we obtain partial results. These results are connected to efficient lattice coverings of ${\bf R}^3$ by octahedra or by tetrahedra; computations on Cayley graphs lead us to such lattice coverings, which we conjecture to be optimal. (The problem of finding such optimal coverings can be reduced to a finite number of nonlinear optimization problems.) We discuss the asymptotic behavior of the Abelian degree-diameter problem for large numbers of generators. The graphs obtained here are substantially better than traditional toroidal meshes, but, in the simpler undirected cases, retain certain desirable features such as good routing algorithms, easy constructibility, and the ability to host mesh-connected numerical algorithms without any increase in communication times.