Incomplete k-ary n-cube and its derivatives

Incomplete or pruned k-ary n-cube, ngreater than or equal to3, is derived as follows. All links of dimension n - 1 are left in place and links of the remaining n - I dimensions are removed, except for one, which is chosen periodically from the remaining dimensions along the intact dimension n - 1. This leads to a node degree of 4 instead of the original 2n and results in regular networks that are Cayley graphs, provided that n - 1 divides k. For n = 3 (n = 5), the preceding restriction is not problematic, as it only requires that k be even (a multiple of 4). In other cases, changes to the basis network to be pruned, or to the pruning algorithm, can mitigate the problem. Incomplete k-ary n-cube maintains a number of desirable topological properties of its unpruned counterpart despite having fewer links. It is maximally connected, has diameter and fault diameter very close to those of k-ary n-cube, and an average internode distance that is only slightly greater. Hence, the cost/performance tradeoffs offered by our pruning scheme can in fact lead to useful, and practically realizable, parallel architectures. We study pruned k-ary n-cubes in general and offer some additional results for the special case n = 3. (C) 2003 Elsevier Inc. All rights reserved.