Estimating the value of Pi using Monte Carlo | Parallel Computing Method

Given two integers N and K representing number of trials and number of total threads in parallel processing. The task is to find the estimated value of PI using the Monte Carlo algorithm using the Open Multi-processing (OpenMP) technique of parallelizing sections of the program.

Examples:

Input: N = 100000, K = 8 
Output: Final Estimation of Pi = 3.146600

Input: N = 10, K = 8
Output: Final Estimation of Pi = 3.24

Input: N = 100, K = 8
Output: Final Estimation of Pi = 3.0916

Approach: The above given problem Estimating the value of Pi using Monte Carlo is already been solved using standard algorithm. Here the idea is to use parallel computing using OpenMp to solve the problem. Follow the steps below to solve the problem:

• Initialize 3 variables say x, y, and d to store the X and Y co-ordinates of a random point and the square of the distance of the random point from origin.
• Initialize 2 variables say pCircle and pSquare with values 0 to store the points lying inside circle of radius 0.5 and square of side length 1.
• Now starts the parallel processing with OpenMp together with reduction() of the following section:
  • Iterate over the range [0, N] and find x and y in each iteration using srand48() and drand48() then find the square of distance of point (x, y) from origin and then if the distance is less than or equal to 1 then increment pCircle by 1.
  • In each iteration of the above step, increment the count of pSquare by 1.
• Finally, after the above step calculate the value of estimated pi as below and then print the obtained value.
  •  Pi = 4.0 * ((double)pCircle / (double)(pSquare))

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <iostream>
using namespace std;

// Function to find estimated
// value of PI using Monte
// Carlo algorithm
void monteCarlo(int N, int K)
{

    // Stores X and Y coordinates
    // of a random point
    double x, y;

    // Stores distance of a random
    // point from origin
    double d;

    // Stores number of points
    // lying inside circle
    int pCircle = 0;

    // Stores number of points
    // lying inside square
    int pSquare = 0;
    int i = 0;

// Parallel calculation of random
// points lying inside a circle
#pragma omp parallel firstprivate(x, y, d, i) reduction(+ : pCircle, pSquare) num_threads(K)
    {

        // Initializes random points
        // with a seed
        srand48((int)time(NULL));

        for (i = 0; i < N; i++)
        {

            // Finds random X co-ordinate
            x = (double)drand48();

            // Finds random X co-ordinate
            y = (double)drand48();

            // Finds the square of distance
            // of point (x, y) from origin
            d = ((x * x) + (y * y));

            // If d is less than or
            // equal to 1
            if (d <= 1)
            {

                // Increment pCircle by 1
                pCircle++;
            }

            // Increment pSquare by 1
            pSquare++;
        }
    }

    // Stores the estimated value of PI
    double pi = 4.0 * ((double)pCircle / (double)(pSquare));

    // Prints the value in pi
    cout << "Final Estimation of Pi = "<< pi;
}

// Driver Code
int main()
{

    // Input
    int N = 100000;
    int K = 8;

    // Function call
    monteCarlo(N, K);
}

// This code is contributed by shivanisinghss2110

C

// C program for the above approach

#include <omp.h>
#include <stdio.h>
#include <stdlib.h>
#include <time.h>

// Function to find estimated
// value of PI using Monte
// Carlo algorithm
void monteCarlo(int N, int K)
{
    // Stores X and Y coordinates
    // of a random point
    double x, y;
    // Stores distance of a random
    // point from origin
    double d;

    // Stores number of points
    // lying inside circle
    int pCircle = 0;

    // Stores number of points
    // lying inside square
    int pSquare = 0;

    int i = 0;

// Parallel calculation of random
// points lying inside a circle
#pragma omp parallel firstprivate(x, y, d, i) reduction(+ : pCircle, pSquare) num_threads(K)
    {
        // Initializes random points
        // with a seed
        srand48((int)time(NULL));

        for (i = 0; i < N; i++) {
            // Finds random X co-ordinate
            x = (double)drand48();

            // Finds random X co-ordinate
            y = (double)drand48();

            // Finds the square of distance
            // of point (x, y) from origin
            d = ((x * x) + (y * y));

            // If d is less than or
            // equal to 1
            if (d <= 1) {
                // Increment pCircle by 1
                pCircle++;
            }
            // Increment pSquare by 1
            pSquare++;
        }
    }
    // Stores the estimated value of PI
    double pi = 4.0 * ((double)pCircle / (double)(pSquare));

    // Prints the value in pi
    printf("Final Estimation of Pi = %f\n", pi);
}

// Driver Code
int main()
{
    // Input
    int N = 100000;
    int K = 8;
    // Function call
    monteCarlo(N, K);
}

Output

Final Estimation of Pi = 3.146600

Output of above C program

Time Complexity: O(N*K)
Auxiliary Space: O(1)