We consider the problem of embedding and reconfiguring binary tree structures in faulty hypercubes. We assume that the number of faulty nodes is at most (n-2), where n is the number of dimensions of the hypercube; we further assume that the location of faulty nodes are known. Our embedding 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. Using this approach, two distributed schemes are provided for embedding and reconfiguration in faulty hypercubes. We extend the free dimension concept to degree of occupancy and use this to develop a distributed scheme for reconfiguration of binary tree in faulty hypercubes with up to [3n/2] node faults.

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.>