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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.
Citation:
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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