Recursive circulants and their embeddings among hypercubes

Abstract

We propose an interconnection structure for multicomputer networks, called recursive circulant. Recursive circulant G(N, d) is defined to be a circulant graph with N nodes and jumps of powers of d. G(N, d) is node symmetric, and has some strong hamiltonian properties. G(N, d) has recursive structure when N = cd(m), 1 less than or equal to c < d. We develop a shortest-path routing algorithm in G(cd(m),d), and analyze various network metrics of G(cd(m),d) such as connectivity, diameter, mean internode distance, and visit ratio. G(2(m),4), whose degree is m, compares favorably to the hypercube Q(m). G(2(m),4) has the maximum possible connectivity, and its diameter is [(3m - 1)/4]. Recursive circulants have interesting relationship with hypercubes in terms of embedding. We present expansion one embeddings among recursive circulants and hypercubes, and analyze the costs associated with each embedding. The earlier version of this paper appeared in Park and Chwa (Proc. Internat. Symp. Parallel Architectures, Algorithms and Networks ISPAN'94, Kanazawa, Japan, December 1994, pp. 73-80). (C) 2000 Elsevier Science B.V. All rights reserved.