# Shuffle a given array using Fisher–Yates shuffle Algorithm

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.

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` |

## Python3

`# Python Program to shuffle a given array` `from` `random ` `import` `randint` `# 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 ` `=` `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 inclusive` ` ` `let j = Math.floor(Math.random() * (i + 1));` ` ` `// Swap arr[i] with the element` ` ` `// at random index` ` ` `[arr[i], arr[j]] = [arr[j], arr[i]];` ` ` `}` `}` `// 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.

**Auxiliary Space: **O(1)**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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