Hyper-ring connection machines

Abstract

A graph G = ( V, E) is called a hyper-ring with N nodes ( N-HR for short) if V = {0,…, N − 1} and E = {{u, v}¦v − u modulo N is a power of 2} . We study constructions, properties, spanners of HRs, and embeddings into HRs. A hypercube with N nodes, a grid of size a × b, and a complete binary tree with N nodes can be embedded as subgraphs into an N-HR. The stretch factors of three types of spanners given in this paper are at most [log 2 N], 2 k − 1 for any 1 ≤ k ≤ [log 2 N], and 2 k − 1 for any 2 ≤ k ≤ [log 2 N] − 1, respectively. The numbers of edges of these types of spanners are N − 1, at most N[ ( log 2 N) k ] , and at most N[ log 2 N] − k) (2k) + Nk , respectively. Some of these spanners are superior in both stretch factors and numbers of edges to corresponding spanners for synchronizer γ of HRs.