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<p><b>Abstract</b>—We consider the problem of computing a global commutative and associative operation, also known as semi-group operation, (such as addition and multiplication) on a faulty hypercube. In particular, we study the problem of performing such an operation in an <it>n</it>-dimensional SIMD hypercube, <it>Q</it><sub><it>n</it></sub>, with up to <it>n</it>− 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.</p><p>Given a set of at most <it>n</it>− 1 faults, we develop an ordering <it>d</it><sub>1</sub>, <it>d</it><sub>2</sub>, ..., <it>d</it><sub><it>n</it></sub> of <it>n</it> dimensions, depending on where the faults are located. An important and useful property of this dimension ordering is the following: if the <it>n</it>-cube is partitioned into <it>k</it>-subcubes using the first <it>k</it> dimensions of this ordering, namely <it>d</it><sub>1</sub>, <it>d</it><sub>2</sub>, ... <it>d</it><sub><it>k</it></sub> for any 2 ≤<it>k</it>≤<it>n</it>, then each <it>k</it>-subcube in the partition contains at most <it>k</it>− 1 faults. We use this result to develop algorithms for global sum. These algorithms use 3<it>n</it>− 2, <it>n</it> + 3 log <it>n</it> + 3 log log <it>n</it>, and <it>n</it> + log <it>n</it> + 4 log log <it>n</it> + <it>O</it>(log log log <it>n</it>) time steps, respectively.</p>
Hypercubes, fault tolerance, global sum, algorithms, dimension ordering.

C. Raghavendra and M. Sridhar, "Global Commutative and Associative Reduction Operations in Faulty SIMD Hypercubes," in IEEE Transactions on Computers, vol. 45, no. , pp. 495-498, 1996.
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