Abstract
We consider a type of recurrence equations called “Simple Indexed Recurrences” (SIR) wherein ordinary recurrences of the form X[i]=op i (X[i−1], X[i]) (i = 1...n) are extended to X[g(i)]=op i (X[f(i)],X[g(i)]), such that op i is an associative binary operation, f,g: {1...n} ↿ {1...m} and g is distinct. This extends our capabilties for parallelizing loops of the form: for i=1 to n ×[i]=op i (X[i−1],X[i])} to the form: for i=1 to n & X[g(i)]=op i (X[f(i)],X[g(i)]) }. An efficient solution is presented for the special case where we know how to compute the inverse of op i operator. The algorithm requires O(log n) steps with O(n/ log n) processors. Furthermore, we present a practical and a more improved version of the non-optimal algorithm for SIR presented in [1] which uses repeated iterations of pointer jumping. A sequence of experiments was performed to test the effect of synchronous and asynchronous message-passing executions of the algorithm for p < < n processors. This algorithm computes the final values of X[] in n/p · log p steps and n · log p work, with p processors. The experiments show that pointer jumping requires O(n) work in most practical cases of SIR loops, thus forming a more practical solution.
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References
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© 1998 Springer-Verlag Berlin Heidelberg
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Ben-Asher, Y., Haber, G. (1998). Parallel solutions of Simple Indexed Recurrence equations. In: Pritchard, D., Reeve, J. (eds) Euro-Par’98 Parallel Processing. Euro-Par 1998. Lecture Notes in Computer Science, vol 1470. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0057950
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DOI: https://doi.org/10.1007/BFb0057950
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