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 P 6 and 1 6m 6 dn/2e 1, a family of 2 disjoint k-dimensional tori of size 21 22 2k each can be embedded in an n-dimensional crossed cube with unit dilation, where each si P 2, Pk i1⁄41si 1⁄4 n m, and max16i6k{si} P 3 if n is odd and m 1⁄4 n 3 2 ; otherwise, max16i6k{si} P 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. 2010 Elsevier Inc. All rights reserved.