Tolerating a faulty edge in a multi-dimensional mesh

The author examines the problem of tolerating faulty edges in the multidimensional mesh architecture. This problem can be briefly defined as follows. Given a mesh M, find a mesh G which satisfies the following conditions: (1) G and M have the same number of nodes, (2) for any subset of k edges in G, there is a submesh in G isomorphic to M which excludes these k edges, and (3) G has the least possible number of edges. The mesh G which satisfies these conditions is called a symmetrically optimal k-edge-fault-tolerant (k-EFT) extension of M. It is shown that, even when k=1, finding such a mesh G is a very difficult problem. A necessary and sufficient condition is developed for characterizing the class of symmetrically optimal 1-EFT extensions of any given mesh. Two new methods are proposed based on this characterization for finding symmetrically optimal, or symmetrically near-optimal, 1-EFT extensions. The first method finds an optimal solution, but is useful only for meshes whose number of dimensions is relatively small. The second method finds only a near optimal solution by decomposing a mesh with a large number of dimensions into several meshes with a smaller number of dimensions that can be solved by the first method