Fault-Tolerant Meshes and Hypercubes with Minimal Numbers of Spares

Many parallel computers consist of processors connected in the form of a d-dimensional mesh or hypercube. Twoand three-dimensional meshes have been shown to be efficient in manipulating images and dense matrices, whereas hypercubes have been shown to be well suited to divide-andconquer algorithms requiring global communication. However, even a single faulty processor or communication link can seriously affect the performance of these machines. This paper presents several techniques for tolerating faults in tl-dimensional mesh and hypercube architectures. Our approach consists of adding spare processors and communication links so that the resulting architecture will contain a fault-free mesh or hypercube in the presence of faults. We optimize the cost of the fault-tolerant architecture by adding exactly k spare processors (while tolerating up to k processor and/or link faults) and minimizing the maximum number of links per processor. For example, when the desired architecture is a d-dimensional mesh and !i = 1, we present a fault-tolerant architecture that has the same maximum degree as the desired architecture (namely, 2tl) and has only one spare processor. We also present efficient layouts for fault-tolerant twoand three-dimensional meshes, and show how multiplexers and buses can be used to reduce the degree of fault-tolerant architectures. Finally, we give constructions for fault-tolerant tori, eight-connected meshes, and hexagonal meshes.