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 (2N0)2k/(2Πi=1sik) nodes, where S = {si|1 ≤ i ≤ k} is the set of integers with each si representing the number of nodes in a super-node at the level i for 1 ≤ i ≤ k, and N0 is the number of nodes in the base network B. The node degree of HDN(B, k,S) is d0 + k, where d0 is the node degree of the base network. The benefit of the HDN is that we can select suitable si 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.