A dynamic programming algorithm for simulation of a multi-dimensional torus in a crossed cube

The torus is a popular interconnection topology and several commercial multicomputers use a torus as the basis of their communication network. Moreover, there are many parallel algorithms with torus-structured and mesh-structured task graphs have been developed. If one network can embed a mesh or torus network, the algorithms with mesh-structured or torus-structured can also be used in this network. Thus, the problem of embedding meshes or tori into networks is meaningful for parallel computing. In this paper, we prove that for n ? 6 and 1 ? m ? ⌈n/2⌉ − 1, a family of 2^m disjoint k-dimensional tori of size 2^s₁×2^s₂×?×2^s_k each can be embedded in an n-dimensional crossed cube with unit dilation, where each s_i ? 2, , and max_1?i?k{s_i} ? 3 if n is odd and ; otherwise, max_1?i?k{s_i} ? n − 2m − 1. A new concept, cycle skeleton, is proposed to construct a dynamic programming algorithm for embedding a desired torus into the crossed cube. Furthermore, the time complexity of the algorithm is linear with respect to the size of desired torus. As a consequence, a family of disjoint tori can be simulated on the same crossed cube efficiently and in parallel.