Cayley Graphs and Interconnection Networks

Due to recent developments of parallel and distributed computing, the design and analysis of various interconnection networks has been a main topic of research for the past few years and is still stimulated by the new technologies of communication networks such as optic bers. There are many advantages in using Cayley (di)graphs as models for interconnection networks. This work rst surveys some classes of Cayley graphs which are well studied as models of interconnection networks. Results and problems related to routings in networks are then presented, with emphasis on loads of nodes and links in routings. 2 Chapter 1 Introduction As deened by D.F. Hsu in the introduction of a special issue of the journal Networks ((97]), \an interconnection network consists of hardware and software entities that are interconnected to facilitate eecient computation and communication; these entities can be in the form of processors, processes, memory modules or computer systems". Due to recent developments in parallel and distributed computing, the design and analysis of various interconnection networks has been a main topic of research for the past few years. An essential component of a supercomputer based on large-scale parallel processing is the interconnection network. It provides communication among the processors and the memories. Interconnection networks also play a key role in the design and implementation of communication networks and the recent advent of optic technology adds more design problems. This explains that a growing number of research articles and congresses (such as IWIN every two years since 1991) are devoted to interconnection networks, and that special issues of journals have already been published on the subject (for example 97], 17] and 107]). Interconnection networks are often modeled by nite graphs (or digraphs). The vertices of the graph represent the nodes of the network, that is, processing elements, memory modules or switches, and the edges correspond to communication lines. What qualities are required in order for a graph to be a good model for interconnection networks ? A basic constraint in many network design problems is that a bound on the maximum node degree is imposed by cost and fundamental engineering limitations. That is, the nodes of the network can be connected by at most a xed number of communication lines to other nodes. At the same time other properties are crucial for many applications which need eecient network communications. This leads to some well studied design problems. We can distinguish …