Combinatorial properties of hierarchical cubic networks

An n-dimensional hierarchical cubic network (denoted by HCN(n)) contains 2ⁿ n-dimensional hypercubes. The diameter of an HCN(n), which is equal to n+[(n+1)/3]+1, is about two-thirds the diameter of a comparable hypercube, although it uses about half as many links per node. In this paper, a maximal number of node-disjoint paths are constructed between every two distinct nodes of an HCN(n). Their maximal length has an upper bound of n+[n/3]+4, which is nearly optimal. The (n+1)-wide diameter and n-fault diameter of an HCN(n) are shown to be n+[n/3]+3 or n+[n/3]+4, which are about two-thirds those of a comparable hypercube. Our results reveal that an HCN(n) has shorter node-disjoint paths, wide diameter, and fault diameter than a comparable hypercube