Global semigroup operations in faulty SIMD hypercubes

The authors consider the problem of computing a global semigroup operation (such as addition and multiplication) on a faulty hypercube. In particular, they study the problem of performing such an operation in an n-dimensional SIMD hypercube Q_n, with upto n-1 node and/or link faults. In an SIMD hypercube, during a communication step, nodes can exchange information with their neighbors only across a specific dimension. Given a set of most n-1 faults they develop an ordering d₁, d₂,. . .,d_n of n dimensions, depending on where the faults are located. An important and useful property of this dimension ordering is the following: if the n-cube is partitioned into k-subcubes using the first k dimensions f this ordering, namely d₁,d₂. . .d_k for any 1⩽k⩽n, then each k-subcube in the partition contains at most k-1 faults. They use this result to develop algorithms for global sum. This ordering can be obtained in the presence of node as well as link faults. They also consider larger fault size, and show how to extend the dimension ordering theorem to handle up to (₂ⁿ) faults. Using this result, it seems possible to obtain even more fault-tolerant algorithms for the semigroup operation problem