Parallel solutions of simple indexed recurrence equations

We define a new type of recurrence equations called "Simple Indexed Recurre nces" (SIR). In this type of equations. ordinary recurrences are generalize d to X[g(i)] = op(i) (X[f(i)],X[g(i)]), where f,g : {1... n}bar right arrow {1...m}, op(i)(x, y) is a binary associative operator and g is distinct. i. e., For All (i) not equal j g(i) not equal g(j). This enables us to model c ertain sequential loops as a sequence of SIR equations. A parallel algorith m that solves a set of SIR equations will, in fact, parallelize sequential loops of the above type. Such a parallel SIR algorithm must be efficient en ough to compete with the O(n) work complexity of the original loop. We show why efficient parallel algorithms for the related problems of List Ranking and Tree Contraction. which require O(n.) work, cannot be applied to solvi ng SIR. We instead use repeated iterations of pointer jumping to compute th e final values of X[] in n/p . log p steps and n . logp work, with I, proce ssors. A sequence of experiments was performed to test the effect of synchr onous and asynchronous executions on the actual performance of the algorith m. These experiments show that pointer jumping requires O(n) work in most p ractical cases of SIR loops. An efficient solution is given for the special case where we know how to co mpute the inverse of op(i), and finally, useful applications of SIR to the well-known Livermore Loops benchmark are presented.