生成伪随机数的乘法同余法

乘法同余法(Lehmer Method)是一种线性同余生成器，用于生成特定范围内的伪随机数。该方法可以定义为：

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

 

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.

 

方法：

• 选择种子值 ( X ₀ )、模参数 ( m ) 和乘数项 ( a )。
• 初始化所需数量的随机数以生成(例如，整数变量noOfRandomNums )。
• 定义存储以保留大小为noOfRandomNums的生成的随机数(此处考虑向量)。
• 用种子值初始化向量的^第0^个索引。
• 对于其余的索引，遵循乘法同余法来生成随机数。

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

最后，返回随机数。
下面是上述方法的实现：

C++

// C++ implementation of the
// above approach
#include 
using namespace std;
 
// Function to generate random numbers
void multiplicativeCongruentialMethod(
    int Xo, int m, int a,
    vector& 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 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

 
// 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);
       
    // Prlet the generated random numbers
    for(let i = 0; i < noOfRandomNums; i++)
    {
        document.write(randomNums[i] + " ");
    }
 

输出：

3 6 12 9 3 6 12 9 3 6

时间复杂度： O(N)，其中 N 是我们需要生成的随机数总数。
辅助空间：O(1)
伪的字面量意思是false 。这些随机数被称为伪数，因为使用了一些已知的算术程序来生成。甚至生成的序列也形成了一种模式，因此生成的数字似乎是随机的，但可能不是真正的随机。