Three Methods for Subcube determination in Faulty Hypercubes

Abstract

For a large hypercube system, the probability of fault occurrence can be high. It is often desired to reconfigure the faulty hypercube that operates in a gracefully degraded manner as to retain as many nonfaulty nodes and links as possible. Therefore the subcube determination problem is essential that the time for executing a parallel algorithm trends to depend on the dimension of the assigned subcube. Here, we present three different methods to determine prime subcubes which are hypercube function method, Q-map method , and synchronized message passing (SMP) method. The basic ideas of these three methods are from the similarities between Boolean algebra and hypercube topology. A faulty hypercube system can be described by a hypercube function. The hypercube function is similar to the logic function of a switching circuit. The nonfaulty subcubes in the system can be obtained from the complement of set of faulty nodes by DeMorgan’s law easily. We propose the Q-map method which improved from the K-map method of logic circuit to provide a easy method for finding subcubes in a small faulty hypercube system. The proposed SMP method is a parallel algorithm, which exhibits polynomial time complexity with respect to the system’s dimension. This method can deal with node failures and link failures.