Fault Tolerance in Hypercubes Shobana Balakrishnan, F¨ usun ¨ Ozg¨ uner, and Baback A. Izadi Department of Electrical Engineering, The Ohio State University, Columbus, OH 43210, USA Abstract: This paper describes different schemes for tolerating faults in hypercube multi- processors. A study of hypercube algorithms reveals that in many cases, the computations that require local communication are mapped onto topologies such as meshes or rings and the hyper- cube topology is used for global data communication. Therefore, a faulty hypercube needs to be reconfigured to perform both local and global communication as required by the algorithm, ef- fectively and with minimal performance degradation. Two general approaches can be identified. The first approach looks into ways of utilizing the healthy processors and links of a hypercube with faulty nodes/links, for embedding topologies such as lower dimensional hypercubes, rings, meshes and trees for performing communication. The second approach makes use of hardware redundancy in the form of spare nodes and/or links and usually requires modifications in the communication hardware. Augmented hypercubes and spare allocation schemes are described. Keywords: Hypercube, fault tolerance, embeddings, global communication, spare alloca- tion, reconfiguration 1 Introduction The area of reconfigurability in the presence of faults is becoming increasingly important with the emergence of massively parallel architectures. Therefore, fault tolerance is an important issue that needs to be addressed in the design of node architectures, communication hardware and software design as well as parallel algorithm development. A –dimensional hypercube multicomputer consists of processors (nodes) interconnected as a Boolean -cube, with each processor having only local memory. Inter–processor communication is done by explicit message passing. Each processor in a –cube can be represented by a d–tuple where , and a subcube in a -cube can be represented by a d–tuple Coordinate values “0” and “1” can be referred to as bound coordinates and “x” as free.A –dimensional subcube ( –subcube) in a –cube is represented by a d–tuple with k bound coordinates