Shuffle a given array using Fisher–Yates shuffle Algorithm

Difficulty Level : Medium
Last Updated : 21 Apr, 2021

Given an array, write a program to generate a random permutation of array elements. This question is also asked as “shuffle a deck of cards” or “randomize a given array”. Here shuffle means that every permutation of array element should equally likely.

shuffle-array

Let the given array be arr[]. A simple solution is to create an auxiliary array temp[] which is initially a copy of arr[]. Randomly select an element from temp[], copy the randomly selected element to arr[0] and remove the selected element from temp[]. Repeat the same process n times and keep copying elements to arr[1], arr[2], … . The time complexity of this solution will be O(n^2).
Fisher–Yates shuffle Algorithm works in O(n) time complexity. The assumption here is, we are given a function rand() that generates random number in O(1) time.
The idea is to start from the last element, swap it with a randomly selected element from the whole array (including last). Now consider the array from 0 to n-2 (size reduced by 1), and repeat the process till we hit the first element.
Following is the detailed algorithm

To shuffle an array a of n elements (indices 0..n-1):
  for i from n - 1 downto 1 do
       j = random integer with 0 <= j <= i
       exchange a[j] and a[i]

Following is implementation of this algorithm.

C++

// C++ Program to shuffle a given array
#include<bits/stdc++.h>
#include <stdlib.h>
#include <time.h>
using namespace std;
 
// A utility function to swap to integers
void swap (int *a, int *b)
{
    int temp = *a;
    *a = *b;
    *b = temp;
}
 
// A utility function to print an array
void printArray (int arr[], int n)
{
    for (int i = 0; i < n; i++)
        cout << arr[i] << " ";
    cout << "\n";
}
 
// A function to generate a random
// permutation of arr[]
void randomize (int arr[], int n)
{
    // Use a different seed value so that
    // we don't get same result each time
    // we run this program
    srand (time(NULL));
 
    // Start from the last element and swap
    // one by one. We don't need to run for
    // the first element that's why i > 0
    for (int i = n - 1; i > 0; i--)
    {
        // Pick a random index from 0 to i
        int j = rand() % (i + 1);
 
        // Swap arr[i] with the element
        // at random index
        swap(&arr[i], &arr[j]);
    }
}
 
// Driver Code
int main()
{
    int arr[] = {1, 2, 3, 4, 5, 6, 7, 8};
    int n = sizeof(arr) / sizeof(arr[0]);
    randomize (arr, n);
    printArray(arr, n);
 
    return 0;
}
 
// This code is contributed by
// rathbhupendra

C

// C Program to shuffle a given array
 
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
 
// A utility function to swap to integers
void swap (int *a, int *b)
{
    int temp = *a;
    *a = *b;
    *b = temp;
}
 
// A utility function to print an array
void printArray (int arr[], int n)
{
    for (int i = 0; i < n; i++)
        printf("%d ", arr[i]);
    printf("\n");
}
 
// A function to generate a random permutation of arr[]
void randomize ( int arr[], int n )
{
    // Use a different seed value so that we don't get same
    // result each time we run this program
    srand ( time(NULL) );
 
    // Start from the last element and swap one by one. We don't
    // need to run for the first element that's why i > 0
    for (int i = n-1; i > 0; i--)
    {
        // Pick a random index from 0 to i
        int j = rand() % (i+1);
 
        // Swap arr[i] with the element at random index
        swap(&arr[i], &arr[j]);
    }
}
 
// Driver program to test above function.
int main()
{
    int arr[] = {1, 2, 3, 4, 5, 6, 7, 8};
    int n = sizeof(arr)/ sizeof(arr[0]);
    randomize (arr, n);
    printArray(arr, n);
 
    return 0;
}

Java

// Java Program to shuffle a given array
import java.util.Random;
import java.util.Arrays;
public class ShuffleRand
{
    // A Function to generate a random permutation of arr[]
    static void randomize( int arr[], int n)
    {
        // Creating a object for Random class
        Random r = new Random();
         
        // Start from the last element and swap one by one. We don't
        // need to run for the first element that's why i > 0
        for (int i = n-1; i > 0; i--) {
             
            // Pick a random index from 0 to i
            int j = r.nextInt(i+1);
             
            // Swap arr[i] with the element at random index
            int temp = arr[i];
            arr[i] = arr[j];
            arr[j] = temp;
        }
        // Prints the random array
        System.out.println(Arrays.toString(arr));
    }
     
    // Driver Program to test above function
    public static void main(String[] args)
    {
         
         int[] arr = {1, 2, 3, 4, 5, 6, 7, 8};
         int n = arr.length;
         randomize (arr, n);
    }
}
// This code is contributed by Sumit Ghosh

Python

# Python Program to shuffle a given array
import random
 
# A function to generate a random permutation of arr[]
def randomize (arr, n):
    # Start from the last element and swap one by one. We don't
    # need to run for the first element that's why i > 0
    for i in range(n-1,0,-1):
        # Pick a random index from 0 to i
        j = random.randint(0,i+1)
 
        # Swap arr[i] with the element at random index
        arr[i],arr[j] = arr[j],arr[i]
    return arr
 
# Driver program to test above function.
arr = [1, 2, 3, 4, 5, 6, 7, 8]
n = len(arr)
print(randomize(arr, n))
 
 
# This code is contributed by Pratik Chhajer

C#

// C# Code for Number of digits
// in the product of two numbers
using System;
 
class GFG
{
// A Function to generate a
// random permutation of arr[]
    static void randomize(int []arr, int n)
    {
        // Creating a object
        // for Random class
        Random r = new Random();
         
        // Start from the last element and
        // swap one by one. We don't need to
        // run for the first element
        // that's why i > 0
        for (int i = n - 1; i > 0; i--)
        {
             
            // Pick a random index
            // from 0 to i
            int j = r.Next(0, i+1);
             
            // Swap arr[i] with the
            // element at random index
            int temp = arr[i];
            arr[i] = arr[j];
            arr[j] = temp;
        }
        // Prints the random array
        for (int i = 0; i < n; i++)
        Console.Write(arr[i] + " ");
    }
     
     
// Driver Code
static void Main()
{
    int[] arr = {1, 2, 3, 4,
                 5, 6, 7, 8};
    int n = arr.Length;
    randomize (arr, n);
}
}
 
// This code is contributed by Sam007

PHP

<?php
// PHP Program to shuffle
// a given array
 
// A function to generate
// a random permutation of arr[]
function randomize ($arr, $n)
{
    // Start from the last element
    // and swap one by one. We
    // don't need to run for the
    // first element that's why i > 0
    for($i = $n - 1; $i >= 0; $i--)
    {
        // Pick a random index
        // from 0 to i
        $j = rand(0, $i+1);
 
        // Swap arr[i] with the
        // element at random index
        $tmp = $arr[$i];
        $arr[$i] = $arr[$j];
        $arr[$j] = $tmp;
    }
    for($i = 0; $i < $n; $i++)
    echo $arr[$i]." ";
}
 
// Driver Code
$arr = array(1, 2, 3, 4,
             5, 6, 7, 8);
$n = count($arr);
randomize($arr, $n);
 
// This code is contributed by mits
?>

Javascript

<script>
// JavaScript Program to shuffle a given array
 
// A function to print an array
let printArray = (arr, n)=>
{
    ans = '';
    for (let i = 0; i < n; i++)
    {
        ans += arr[i] + " ";
    }
    console.log(ans);
}
 
// A function to generate a random
// permutation of arr
let randomize = (arr, n) =>
{
 
    // Start from the last element and swap
    // one by one. We don't need to run for
    // the first element that's why i > 0
    for (let i = n - 1; i > 0; i--)
    {
     
        // Pick a random index from 0 to i
        let j = (Math.floor(Math.random())*10000000000) % (i + 1);
 
        // Swap arr[i] with the element
        // at random index
        let temp = arr[i];
        arr[i] = arr[j];
        arr[j] = temp;
    }
}
 
// Driver Code
let arr = [1, 2, 3, 4, 5, 6, 7, 8];
let n = arr.length;
randomize (arr, n);
printArray(arr, n);
 
// This code is contributed by rohitsingh07052.
</script>

Output :

7 8 4 6 3 1 2 5

The above function assumes that rand() generates a random number.
Time Complexity: O(n), assuming that the function rand() takes O(1) time.
How does this work?
The probability that ith element (including the last one) goes to last position is 1/n, because we randomly pick an element in first iteration.
The probability that ith element goes to second last position can be proved to be 1/n by dividing it in two cases.
Case 1: i = n-1 (index of last element):
The probability of last element going to second last position is = (probability that last element doesn’t stay at its original position) x (probability that the index picked in previous step is picked again so that the last element is swapped)
So the probability = ((n-1)/n) x (1/(n-1)) = 1/n
Case 2: 0 < i < n-1 (index of non-last):
The probability of ith element going to second position = (probability that ith element is not picked in previous iteration) x (probability that ith element is picked in this iteration)
So the probability = ((n-1)/n) x (1/(n-1)) = 1/n
We can easily generalize above proof for any other position.

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