Fault-Tolerant Cube Graphs and Coding Theory

Abstract

Hypercubes, meshes, tori and Omega networks are well known interconnection networks for parallel computers. The structure of those graphs can be described in a more general framework called cube graphs. The idea is to assume that every node in a graph with q to the power of l (letter l) nodes is represented by a unique string of l (letter l) symbols over GF(q). The edges are specified by a set of offsets, those are vectors of length l (letter l) over GF(q), where the two endpoints of an edge are an offset apart. We study techniques for tolerating edge faults in cube graphs that are based on adding redundant edges. The redundant graph has the property that the structure of the original graph can be maintained in the presence of edge faults. Our main contribution is a technique for adding the redundant edges that utilizes constructions of error-correcting codes and generalizes existing ad-hoc techniques.