Free dimensions-an effective approach to achieving fault tolerance in hypercubes

Hypercube network is an attractive structure for parallel processing due to its symmetry and regularity. We use the concept of free dimensions to achieve fault tolerance in hypercubes without requiring additional spare processing nodes; such additional redundancy requires modification of hypercube structure. A free dimension is defined to be a dimension across which both end nodes are not faulty. Given an n-dimensional hypercube, Qn, and a set of f⩽n faulty nodes, we present an efficient algorithm to find free dimensions, and show that at least n-f+1 free dimensions exist. Free dimensions can be used to partition Q_n into subcubes such that each subcube contains at most one fault. Such a partitioning helps in achieving fault tolerance via emulation, embedding, reconfiguration. It also helps in designing efficient routing and broadcasting algorithms in faulty hypercubes