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Multiplicative Congruence method for generating Pseudo Random Numbers

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  • Difficulty Level : Medium
  • Last Updated : 08 Feb, 2022
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Multiplicative Congruential Method (Lehmer Method) is a type of linear congruential generator for generating pseudorandom numbers in a specific range. This method can be defined as: 

    \[X_{i + 1} = aX_{i} \hspace{0.2cm} mod \hspace{0.2cm} m\]

 

where, 
X, the sequence of pseudo-random numbers
m ( > 0), the modulus
a (0, m), the multiplier
X0 [0, m), initial value of the sequence – termed as seed

 

m, a, and X0 should be chosen appropriately to get a period almost equal to m.

 

 

Approach: 

  • Choose the seed value ( X0 ), modulus parameter ( m ), and multiplier term ( a ).
  • Initialize the required amount of random numbers to generate (say, an integer variable noOfRandomNums).
  • Define storage to keep the generated random numbers (here, the vector is considered) of size noOfRandomNums.
  • Initialize the 0th index of the vector with the seed value.
  • For the rest of the indexes follow the Multiplicative Congruential Method to generate the random numbers.

randomNums[i] = (randomNums[i – 1] * a) % m 

Finally, return the random numbers.
Below is the implementation of the above approach:

C++




// C++ implementation of the
// above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to generate random numbers
void multiplicativeCongruentialMethod(
    int Xo, int m, int a,
    vector<int>& randomNums,
    int noOfRandomNums)
{
 
    // Initialize the seed state
    randomNums[0] = Xo;
 
    // Traverse to generate required
    // numbers of random numbers
    for (int i = 1; i < noOfRandomNums; i++) {
 
        // Follow the multiplicative
        // congruential method
        randomNums[i]
            = (randomNums[i - 1] * a) % m;
    }
}
 
// Driver Code
int main()
{
    int Xo = 3; // seed value
    int m = 15; // modulus parameter
    int a = 7; // multiplier term
 
    // Number of Random numbers
    // to be generated
    int noOfRandomNums = 10;
 
    // To store random numbers
    vector<int> randomNums(noOfRandomNums);
 
    // Function Call
    multiplicativeCongruentialMethod(
        Xo, m, a, randomNums,
        noOfRandomNums);
 
    // Print the generated random numbers
    for (int i = 0; i < noOfRandomNums; i++) {
        cout << randomNums[i] << " ";
    }
    return 0;
}


Java




// Java implementation of the above approach
import java.util.*;
 
class GFG{
 
// Function to generate random numbers
static void multiplicativeCongruentialMethod(
    int Xo, int m, int a,
    int[] randomNums,
    int noOfRandomNums)
{
     
    // Initialize the seed state
    randomNums[0] = Xo;
     
    // Traverse to generate required
    // numbers of random numbers
    for(int i = 1; i < noOfRandomNums; i++)
    {
         
        // Follow the multiplicative
        // congruential method
        randomNums[i] = (randomNums[i - 1] * a) % m;
    }
}
 
// Driver code
public static void main(String[] args)
{
     
    // Seed value
    int Xo = 3;
     
    // Modulus parameter
    int m = 15;
     
    // Multiplier term
    int a = 7;
     
    // Number of Random numbers
    // to be generated
    int noOfRandomNums = 10;
     
    // To store random numbers
    int[] randomNums = new int[noOfRandomNums];
     
    // Function Call
    multiplicativeCongruentialMethod(Xo, m, a,
                                     randomNums,
                                     noOfRandomNums);
     
    // Print the generated random numbers
    for(int i = 0; i < noOfRandomNums; i++)
    {
        System.out.print(randomNums[i] + " ");
    }
}
}
 
// This code is contributed by offbeat


Python3




# Python3 implementation of the
# above approach
 
# Function to generate random numbers
def multiplicativeCongruentialMethod(Xo, m, a,
                                     randomNums,
                                     noOfRandomNums):
 
    # Initialize the seed state
    randomNums[0] = Xo
 
    # Traverse to generate required
    # numbers of random numbers
    for i in range(1, noOfRandomNums):
         
        # Follow the linear congruential method
        randomNums[i] = (randomNums[i - 1] * a) % m
 
# Driver Code
if __name__ == '__main__':
     
    # Seed value
    Xo = 3
     
    # Modulus parameter
    m = 15
     
    # Multiplier term
    a = 7
 
    # Number of Random numbers
    # to be generated
    noOfRandomNums = 10
 
    # To store random numbers
    randomNums = [0] * (noOfRandomNums)
 
    # Function Call
    multiplicativeCongruentialMethod(Xo, m, a,
                                     randomNums,
                                     noOfRandomNums)
 
    # Print the generated random numbers
    for i in randomNums:
        print(i, end = " ")
 
# This code is contributed by mohit kumar 29


C#




// C# implementation of the above approach
using System;
 
class GFG{
 
// Function to generate random numbers
static void multiplicativeCongruentialMethod(
    int Xo, int m, int a,
    int[] randomNums,
    int noOfRandomNums)
{
     
    // Initialize the seed state
    randomNums[0] = Xo;
     
    // Traverse to generate required
    // numbers of random numbers
    for(int i = 1; i < noOfRandomNums; i++)
    {
         
        // Follow the multiplicative
        // congruential method
        randomNums[i] = (randomNums[i - 1] * a) % m;
    }
}
 
// Driver code
public static void Main(String[] args)
{
     
    // Seed value
    int Xo = 3;
     
    // Modulus parameter
    int m = 15;
     
    // Multiplier term
    int a = 7;
     
    // Number of Random numbers
    // to be generated
    int noOfRandomNums = 10;
     
    // To store random numbers
    int[] randomNums = new int[noOfRandomNums];
     
    // Function call
    multiplicativeCongruentialMethod(Xo, m, a,
                                     randomNums,
                                     noOfRandomNums);
     
    // Print the generated random numbers
    for(int i = 0; i < noOfRandomNums; i++)
    {
        Console.Write(randomNums[i] + " ");
    }
}
}
 
// This code is contributed by sapnasingh4991


Javascript




<script>
 
// Javascript program to implement
// the above approach
 
// Function to generate random numbers
function multiplicativeCongruentialMethod(
    Xo, m, a,
    randomNums, noOfRandomNums)
{
       
    // Initialize the seed state
    randomNums[0] = Xo;
       
    // Traverse to generate required
    // numbers of random numbers
    for(let i = 1; i < noOfRandomNums; i++)
    {
           
        // Follow the multiplicative
        // congruential method
        randomNums[i] = (randomNums[i - 1] * a) % m;
    }
}
 
    // Driver Code
           
    // Seed value
    let Xo = 3;
       
    // Modulus parameter
    let m = 15;
       
    // Multiplier term
    let a = 7;
       
    // Number of Random numbers
    // to be generated
    let noOfRandomNums = 10;
       
    // To store random numbers
    let randomNums = new Array(noOfRandomNums).fill(0);
       
    // Function Call
    multiplicativeCongruentialMethod(Xo, m, a,
                                     randomNums,
                                     noOfRandomNums);
       
    // Print the generated random numbers
    for(let i = 0; i < noOfRandomNums; i++)
    {
        document.write(randomNums[i] + " ");
    }
 
</script>


Output: 

3 6 12 9 3 6 12 9 3 6

 

Time Complexity: O(N), where N is the total number of random numbers we need to generate.
Auxiliary Space: O(1)
The literal meaning of pseudo is false. These random numbers are called pseudo because some known arithmetic procedure is utilized to generate. Even the generated sequence forms a pattern hence the generated number seems to be random but may not be truly random.
 


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