Parallel Solutions of Simple Index Recurrence Equations
نویسندگان
چکیده
We deene a new type of recurrence equation called \Simple Indexed Recurrences" (SIR). In this type of equation, ordinary recurrences of the form Xi] = opi(Xi ? 1]; Xi]) i = 1 : : : n are generalized to Xg(i)] = opi(Xf(i)];Xg(i)]), where f; g : f1 : : : ng 7 ! f1 : : : mg and g is distinct. This enables us to model sequential loops of the form for i = 1 to n Xg(i)] = opi(Xf(i)];Xg(i)]); as a sequence of SIR equations. A parallel algorithm that solves a set of SIR equations will in fact parallelize sequential loops of the above type. Such a parallel SIR algorithm must be eecient enough to compete with the O(n) work of the original loop. We show why eecient parallel algorithms of related problems of List Ranking and Tree Contraction , which require O(n) work, cannot be applied to SIR, as is. We use instead, repeated iterations of pointer jumping to compute the nal values of X] in n p logp steps and nlogp work, with p processors. A sequence of experiments was performed to test the eeect of synchronous and asynchronous executions on the actual performance of the algorithm. These experiments show that pointer jumping requires O(n) work in most practical cases of SIR loops. Finally, useful applications for SIR to the well-known Livermore Loops benchmark are presented.
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