Shuffle a given array

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”.

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 C++ implementation of this algorithm.

// 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; }

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.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.