[GeeksForGeeks] Shuffle a given array

Given an array, write a program to generate a random permuation of array elements. This question is also asked as "shuffle a deck of cards" or "randomize a given array".

 

Solution 1.   O(n^2) runtime, O(n) space

1. create an auxillary array list temp which is initially a copy of the given array arr[].

2. randomly select an element from temp, copy it to arr[0] and remove it from temp.

3. repeat the same process n times and keep copying elements to arr[1], arr[2].....

 

Solution 2. Fisher-Yates shuffle algorithm, O(n) runtime, O(1) space

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.

 

Proof of correctness

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.

 

import java.util.Random;

public class Solution {
    public void randomizeArray(int[] arr) {
        if(arr == null || arr.length <= 1){
            return;
        }
        Random rand = new Random();
        for(int i = arr.length - 1; i >= 1; i--){
            int j = rand.nextInt(i + 1);
            int temp = arr[i];
            arr[i] = arr[j];
            arr[j] = temp;
        }
    }
}