Technical Reports (CIS)
Document Type
Technical Report
Date of this Version
August 1988
Abstract
Toruses and meshes include graphs of many varieties of topologies, with lines, rings, and hypercubes being special cases. Given a d-dimensional torus or mesh G and a c-dimensional torus or mesh H of the same size, we study the problem of embedding G in H to minimize the dilation cost. For increasing dimension cases (d < c) in which the shapes of G and H satisfy the condition of expansion, the dilation costs of our embeddings are either 1 or 2, depending on the types of graphs of G and H. These embeddings a,re optimal except when G is a torus of even size and H is a mesh. For lowering dimension cases (d > c) in which the shapes of G and H satisfy the condition of reduction, the dilation costs of our embeddings depend on the shapes of G and H. These embeddings, however, are not optimal in general. For the special cases in which G and H are square, the embedding results above can always be used to construct embeddings of G in H: these embeddings are all optimal for increasing dimension cases in which the dimension of H is divisible by the dimension of G, and all optimal to within a constant for fixed values of d and c for lowering dimension cases. Our main analysis technique is based on a generalization of Gray code for radix-2 (binary) numbering system to similar sequences for mixed-radix numbering systems.
Keywords
torus, mesh, hypercube, ring, interconnection network, embedding, dilation cost, Hamiltonian circuit
Recommended Citation
Eva Ma and Lixin Tao, "Embeddings Among Toruses and Meshes", . August 1988.
Date Posted: 19 September 2007
Comments
University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-88-63.