Near Optimal Broadcasting in Optimal Triple Loop Graphs

Abstract

Triple loop networks (graphs) are generalizations of the ring topology where every vertex v is linked to 6 vertices v a, v b, v c. In this paper, we study the broadcast problem in optimal triple loop graphs. In 1987 for a restricted case a = -(b + c) the (maximum) number of vertices in the sub- optimal Triple loop graph has been proved to be a quadratic function of diameter d. In 1998 the broadcast time of this graph is proved to be d + 3. Recently, in 2003 the Optimal Triple Loop Graph in general was constructed, where its number of vertices is a cubic function of d. In this paper we prove d + 2 lower bound and d + 5 upper bound for broadcasting in general Optimal Triple Loop Graph. We also generalize our upper bound algorithm in Multiple Loop Graphs giving d + 2 k-1 general upper bound where the degree of every vertex is 2 k.