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POParallelization of a quadratic algorithm
 

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    <p>Let's say I have a vector (array, list, whatever...) V of N elements, let's say V<sub>0</sub> to V<sub>(N-1)</sub>. For each element V<sub>i</sub>, a function f(V<sub>i</sub>,V<sub>j</sub>) needs to be computed for every index j (including the case i=j). The function is symmetric so that once f(V<sub>i</sub>, V<sub>j</sub>) is computed, there is no need to recompute f(V<sub>j</sub>,V<sub>i</sub>). Then we have N(N+1)/2 total evaluations of the function, making this a O(N<sup>2</sup>) algorithm. Let's assume the time it takes to compute f is relatively long but consistent.</p> <p>Now, I want to parallelize the execution of the algorithm. I need to determine a schedule for (some number M of) worker threads so that two threads do not use the same part of memory (i.e. the same element) at the same time. For example, f(V<sub>1</sub>,V<sub>2</sub>) could be evaluated parallel to f(V<sub>3</sub>,V<sub>4</sub>), but not parallel to f(V<sub>2</sub>,V<sub>3</sub>). The workflow is divided into steps such that for each step, every worker thread performs one evaluation of f. The threads are then synchronized, after which they proceed to the next step and so on.</p> <p>The question is, how do I determine (preferably optimally) the schedule for each thread as a series of index pairs (i,j) so that the complete problem is solved (i.e. each index pair visited exactly once, considering the symmetry)? While a direct answer would of course be nice, I'd also appreciate a pointer to an algorithm or even to relevant websites/literature.</p>
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    1. COWhy f(V1,V2) could not be evaluated parallel to f(V2,V3)? Does this algorithm change the vector in-place?
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    2. CONot as such, but the function f(Vi,Vj) writes to another vector of the same length as V, at both indices i and j, and those writes need to be protected. Maybe I should have said that explicitly.
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    3. COWhat's `f()` doing with this second vector? If an adjustment to this can be made to allow overlapping pairs to be handled simultaneously then the problem of allocating work becomes *much* easier. For example, is this second vector maybe used for recording results? If so, could `f()` instead write to `[i][j]` (and maybe `[j][i]`) of a 2D vector - then once all parallel work is complete you'd just need to merge this 2D vector down to the required 1D version.
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