Finding a Hamiltonian Cycle in a Hierarchical Dual-Net with Base Network of p -Ary q -Cube

Abstract

We first introduce a flexible interconnection network, called the hierarchical dual-net (HDN), with low node degree and short diameter for constructing a large-scale supercomputer. The HDN is constructed based on a symmetric product graph (base network). A k-level hierarchical dual-net, HDN(B,k,S), contains ${(2N_0)^{2^k}/(2\prod_{i=1}^{k}s_i)}$ nodes, where S = {s _i |1 ≤ i ≤ k} is the set of integers with each s _i representing the number of nodes in a super-node at the level i for 1 ≤ i ≤ k, and N ₀ is the number of nodes in the base network B. The node degree of HDN(B,k,S) is d ₀ + k, where d ₀ is the node degree of the base network. The benefit of the HDN is that we can select suitable s _i to control the growing speed of the number of nodes for constructing a supercomputer of the desired scale. Then we show that an HDN with the base network of p-ary q-cube is Hamiltonian and give an efficient algorithm for finding a Hamiltonian cycle in such hierarchical dual-nets.