The document describes Monte Carlo methods for estimating pi by randomly generating points in the unit square and counting the proportion that fall within the unit circle. It discusses parallelizing the algorithm using OpenMP and shows that optimized parallelization reduces execution time compared to a naive parallel or sequential implementation. Performance scales sublinearly with the number of processors due to overhead from workload imbalance and thread synchronization.

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auto ps = generate(n, randpair);
auto rs = map([](pair p){return p.x*p.x+p.y*p.y},ps);
auto ios = map([](double r2){return r2 <= 1;}, rs);
double pi = reduce(std::plus<double>(), ios) /size*4;
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70
naïve parallel
60
sequential
Execution Time (s)
50 optimized parallel
40
30
20
10
0
1 2 3 4 5 6 7 8
#processor
On a workstation with two Intel®Xeon® X5550 (8 cores in total,
without hyper-threading), 12GB memory, and Linux 2.6.31.

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