Embedding Knödel Graph into Cube-like Architectures: Dilation Optimization and Wirelength Analysis

Abstract

An important tool for the execution of parallel algorithms and the simulation of interconnection networks is graph embedding. The quality of an embedding can be assessed using some cost metrics. The dilation and wirelength are the commonly used parameters. The Knödel graph [Formula: see text] is a minimum linear gossip network and has minimum broadcasting. It has [Formula: see text] vertices, [Formula: see text] edges, where [Formula: see text] is even, and [Formula: see text]log[Formula: see text]. In this study, we solve the dilation problem of embedding the Knödel graph into certain cube-like architectures such as hypercube, folded hypercube, and augmented cube. In [G. Fertin, A. Raspaud, A survey on Knödel graphs, Discrete Applied Mathematics 137 (2004) 173–195], it is proved that the dilation of embedding the Knödel graph [Formula: see text] into the hypercube [Formula: see text] is at most [Formula: see text]. In this study, we obtain an improved upper bound for dilation of embedding the Knödel graph into the hypercube and it is equal to [Formula: see text]. Also, we calculate the wirelength of embedding the Knödel graph into the above-said cube-like architectures using dilation.