Reconfiguring a hypercube in the presence of faults

We consider the computational power of a hypercube containing a potentially large number of randomly located faulty components. In particular, we describe algorithms for embedding an N/2-node hypercube in an N-node hypercube with faulty processors. Provided that the processors of the N-node hypercube are faulty with probability p < 1/2, and that the faults are independently distributed, we show that with high probability, adjacent cells in the N/2-node hypercube are mapped to functioning cells at distance 3 or less apart in the N-node hypercube. The algorithm is deterministic, easy to implement and runs in Ο(log N) steps using only local control. We also describe ways to produce embeddings which allow for low delay simulations, as well as ways to use a faulty hypercube to efficiently simulate a completely functioning hypercube of the same size.