Topics in toroidal interconnection networks

A multicomputer is a machine having multiple processing elements that communicate with each other by sending messages. The messages are transmitted through an interconnection network, which can be classified as a direct or an indirect network depending on the style of connection among the processing elements. The pattern of interconnection is called the topology of the network, and a popular topology for a direct interconnection network is the torus. The torus can be characterized as a graph that is the cross product of cycles. The graphs of the k-ary n-cube ($Q\sbsp{n}{k})$ and the hypercube ($Q\sb{n})$ are special cases of a torus graph ($T\sb{\bf K}),$ and a network based on one of these topologies is referred to a toroidal interconnection network. This thesis considers various topological characteristics of a toroidal inter-connection network using Lee distance, a metric from the field of Error-Correcting codes. Using Lee distance, the torus is defined, and the number and length of edge disjoint paths between two nodes is given. In addition, five Lee distance Gray codes are given; and these Gray codes are applied to finding both a Hamiltonian cycle and a cycle of any even length in a torus of certain dimensions. For the k-ary n-cube, formulae for the volume and surface area of a sphere of radius d are derived, and methods of decomposing a $Q\sbsp{n}{k}$ into n disjoint Hamiltonian cycles are considered. In addition to topological characteristics, communication algorithms are considered. A one-to-all, an all-to-all, and a redundant fault tolerant (up to n $-$ 1 faults) one-to-all broadcasting algorithm for the general torus are presented, and a non-redundant fault tolerant (up to n $-$ 1 faults) one-to-all broadcasting algorithm for the k-ary n-cube are given.