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++ 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 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); } 

// Java implementation of the approach import java.util.*; class GFG {  // Function to find estimated  // value of PI using Monte  // Carlo algorithm  static 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;  // Initializes random points  // with a seed  Random rand = new Random();  // Loop through each iteration  for (int i = 0; i < N; i++) {  // Finds random X co-ordinate  x = rand.nextDouble();  // Finds random Y co-ordinate  y = rand.nextDouble();  // 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 of pi  System.out.println("Final Estimation of Pi = "  + pi);  }  // Driver Code  public static void main(String[] args)  {  // Input  int N = 100000;  int K = 8;  // Function call  monteCarlo(N, K);  } } // This code is contributed by phasing17 

# Python3 program for the above approach import random import time # Function to find estimated # value of PI using Monte # Carlo algorithm def monteCarlo(N, K): # Stores X and Y coordinates # of a random point x = 0 y = 0 # Stores distance of a random # point from origin d = 0 # Stores number of points # lying inside circle pCircle = 0 # Stores number of points # lying inside square pSquare = 0 # Initializes random points # with a seed random.seed(time.time()) for i in range(N): # Finds random X co-ordinate x = random.random() # Finds random X co-ordinate y = random.random() # 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 += 1 # Increment pSquare by 1 pSquare += 1 # Stores the estimated value of PI pi = 4.0 * (pCircle / pSquare) # Prints the value in pi print("Final Estimation of Pi = ", pi) # Driver Code # Input N = 100000 K = 8 # Function call monteCarlo(N, K) # This code is contributed by phasing17. 

// C# equivalent of the above code using System; namespace MonteCarloPi {  class GFG   {    // Function to find estimated  // value of PI using Monte  // Carlo algorithm  static 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;  // Initializes random points  // with a seed  Random rand = new Random();  // Loop through each iteration  for (int i = 0; i < N; i++) {  // Finds random X co-ordinate  x = rand.NextDouble();  // Finds random Y co-ordinate  y = rand.NextDouble();  // 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 of pi  Console.WriteLine("Final Estimation of Pi = " + pi);  }  // Driver Code  static void Main(string[] args)  {  // Input  int N = 100000;  int K = 8;  // Function call  monteCarlo(N, K);  }  } } // This code is contributed by phasing17 

// JS program for the above approach // Function to find estimated value of PI using Monte Carlo algorithm function monteCarlo(N, K) {  // Stores X and Y coordinates of a random point  let x = 0;  let y = 0;      // Stores distance of a random point from origin  let d = 0;    // Stores number of points lying inside circle  let pCircle = 0;    // Stores number of points lying inside square  let pSquare = 0;    let pi;    for (let i = 0; i < N; i++) {  // Finds random X co-ordinate  x = Math.random();    // Finds random Y co-ordinate  y = Math.random();    // 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  pi = 4.0 * (pCircle / pSquare);  }    // Prints the value of pi  console.log("Final Estimation of Pi = " + pi); } // Driver Code // Input const N = 100000; const K = 8; // Function call monteCarlo(N, K); // This code is contributed by phasing17. 

Final Estimation of Pi = 3.146600

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