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Global Commutative and Associative Reduction Operations in Faulty SIMD Hypercubes
April 1996 (vol. 45 no. 4)
pp. 495-498

Abstract—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 n-dimensional SIMD hypercube, Qn, with up to 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 at most n− 1 faults, we develop an ordering d1, d2, ..., dn 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 of this ordering, namely d1, d2, ... dk for any 2 ≤kn, then each k-subcube in the partition contains at most k− 1 faults. We use this result to develop algorithms for global sum. These algorithms use 3n− 2, n + 3 log n + 3 log log n, and n + log n + 4 log log n + O(log log log n) time steps, respectively.

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Index Terms:
Hypercubes, fault tolerance, global sum, algorithms, dimension ordering.
C.s. Raghavendra, M.a. Sridhar, "Global Commutative and Associative Reduction Operations in Faulty SIMD Hypercubes," IEEE Transactions on Computers, vol. 45, no. 4, pp. 495-498, April 1996, doi:10.1109/12.494109
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