Find two non-repeating numbers in an array in O(n) time and O(1) space

Objective:  Given an array of integers that has all the repeating numbers (twice) but two numbers that are non-repeating. Write an algorithm to find out those two numbers.

Example

int [] arrA = {4,5,4,5,3,2,9,3,9,8};
Output: 2 and 8

Approaches:

This problem is similar to the problem “Find two repeating elements in an array”. There could be multiple solutions like sort the array {O(nlogn)} or use hash map{Time complexity: O(n) , space complexity:  O(n)}

In this article, we will discuss a solution that has time complexity: O(n) and space complexity: O(1), constant extra space.

Use XOR: time complexity: O(n) and space complexity: O(1)

• Let’s say non-repeating elements are X, Y.
• A XOR A = 0
• XOR all the elements of the array. This will cancel all the repeated elements.
• The result will be X XOR Y since only X and Y are not repeating.
• 1 XOR 1 =  0 and 1 XOR 0 = 1 with this logic in the result of X XOR Y if any kth bit is set to 1  implies either kth bit is 1 either in X or in Y not in both.
• Use the above step to divide all the ele­ments in the array into 2 groups, one group which has the ele­ments for which the kth bit is set to 1 and a sec­ond group which has the ele­ments for which the kth bit is 0.
• Let’s have that kth bit as the rightmost set bit (Read how to find right most set bit)
• Now we can claim that these two groups are respon­si­ble for pro­duce X and Y.
• Group –1: XOR all the ele­ments whose kth bit is 1 will pro­duce either X or Y.
• Group –2: XOR all the ele­ments whose kth bit is 0 will pro­duce either X or Y.
• See the image below

Run This Code

Code:

Output:

Non Repeating Elements are: 2 and 8