Dimension Ordering and Broadcast Algorithms in Faulty SIMD Hypercubes

In this paper, the problem of broadcasting in ann-dimensional SIMD hypercube,Qn, with up ton? 1 node faults is studied. In an SIMD hypercube, during a communication step, nodes can exchange information with their neighbors only across a specific dimension. The broadcasting algorithms must work independent of the location of the source node and faulty nodes. In a fault-free hypercube, any source node can broadcast a message to all nodes innsteps, by successive communication along any arbitrary ordering of thendimensions. Given a set of at mostn? 1 faults, an orderingd1,d2, ...,dnofndimensions is developed, depending on where the faults are located. An important and useful property of this dimension ordering is the following: if then-cube is partitioned intok-subcubes using the firstkdimensions of this ordering, namely,d1,d2, ...,dkfor any 2 ?k?n, then eachk-subcube contains at mostk? 1 faults. This result is then used to develop several new algorithms for broadcasting. These algorithms usen+ 3 logn,n+ 2 logn+ 2,n+ logn+O(log logn),n+ logn+ 5, andn+ 12 time steps respectively, and thus improve upon the best known algorithms for this problem. This ordering of dimensions is also demonstrated in the presence of node as well as link faults. In this paper, it is also known how to extend the dimension ordering theorem for handling up to (n2) faults. Using this result, it seems possible to obtain even more fault-tolerant algorithms for the broadcasting problem.

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