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/spl les/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/sub 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.>

Fault tolerance in hypercubes is achieved by exploiting inherent redundancy and executing tasks on faulty hypercubes. The authors consider tasks that require linear chain, ring, mesh, and torus structure, which are quite useful in parallel and pipeline computations. They assume the number of faults is on the order of the number of dimensions of the hypercube. The techniques are based on a key concept called free dimension, which can be used to partition a cube into subcubes such that each subcube contains, at most, one faulty node. Subgraphs are embedded in each subcube and then merged to form the entire graph.>

In an n-dimensional hypercube Qn, with the fault set mod F mod>