Hello Everyone,

Add each number once and multiply the sum by 3, we will get thrice the sum of each element of the array. Store it as thrice_sum. Subtract the sum of the whole array from the thrice_sum and divide the result by 2. The number we get is the required number (which appears once in the array).

Array [] : [a, a, a, b, b, b, c, c, c, d]

Mathematical Equation = ( 3*(a+b+c+d) – (a + a + a + b + b + b + c + c + c + d) ) / 2

In more simple words: ( 3*(sum_of_array_without_duplicates) – (sum_of_array) ) / 2

let arr[] = {12, 1, 12, 3, 12, 1, 1, 2, 3, 3}

Required no = ( 3*(sum_of_array_without_duplicates) - (sum_of_array) ) / 2

= ( 3*(12 + 1 + 3 + 2) - (12 + 1 + 12 + 3 + 12 + 1 + 1 + 2 + 3 + 3))/2

= ( 3* 18 - 50) / 2

= (54 - 50) / 2

= 2 (required answer)

As we know that set does not contain any duplicate element,

But, std::set is commonly implemented as a red-black binary search tree. Insertion on this data structure has a worst-case of O(log(n)) complexity, as the tree is kept balanced.

Given an array where every element occurs three times, except one element which occurs only once. Find the element that occurs once. Expected time complexity is O(n) and O(1) extra space.

**Examples:**

Input:arr[] = {12, 1, 12, 3, 12, 1, 1, 2, 3, 3}

Output:2

In the given array all element appear three times except 2 which appears once.

Input:arr[] = {10, 20, 10, 30, 10, 30, 30}

Output:20

In the given array all element appear three times except 20 which appears once.

We can use sorting to do it in O(nLogn) time. We can also use hashing, it has the worst case time complexity of O(n), but requires extra space.

The idea is to use bitwise operators for a solution that is O(n) time and uses O(1) extra space. The solution is not easy like other XOR based solutions, because all elements appear odd number of times here. Run a loop for all elements in array. At the end of every iteration, maintain following two values.

ones: The bits that have appeared 1st time or 4th time or 7th time … etc.

twos: The bits that have appeared 2nd time or 5th time or 8th time … etc.

Finally, we return the value of ‘ones’

How to maintain the values of ‘ones’ and ‘twos’?

‘ones’ and ‘twos’ are initialized as 0. For every new element in array, find out the common set bits in the new element and previous value of ‘ones’. These common set bits are actually the bits that should be added to ‘twos’. So do bitwise OR of the common set bits with ‘twos’. ‘twos’ also gets some extra bits that appear third time. These extra bits are removed later.

Update ‘ones’ by doing XOR of new element with previous value of ‘ones’. There may be some bits which appear 3rd time. These extra bits are also removed later.

Both ‘ones’ and ‘twos’ contain those extra bits which appear 3rd time. Remove these extra bits by finding out common set bits in ‘ones’ and ‘twos’.

We can sum the bits in same positions for all the numbers and take modulo with 3. The bits for which sum is not multiple of 3, are the bits of number with single occurrence.

Let us consider the example array {5, 5, 5, 8}. The 101, 101, 101, 1000

Sum of first bits%3 = (1 + 1 + 1 + 0)%3 = 0;

Sum of second bits%3 = (0 + 0 + 0 + 0)%3 = 0;

Sum of third bits%3 = (1 + 1 + 1 + 0)%3 = 0;

Sum of fourth bits%3 = (1)%3 = 1;

Hence number which appears once is 1000

**Note:** this approach wont work for negative numbers

Below is the implementation of above approach:

`// C++ program to find the element`

`// that occur only once`

`#include <bits/stdc++.h>`

`using`

`namespace`

`std;`

`// function which find number`

`int`

`singleNumber(`

`int`

`a[], `

`int`

`n)`

`{`

` `

`unordered_set<`

`int`

`> s(a, a + n);`

` `

`int`

`arr_sum = accumulate(a, a + n, 0); `

`// sum of array`

` `

`int`

`set_sum = accumulate(s.begin(), s.end(), 0); `

`// sum of set`

` `

`// applying the formula.`

` `

`return`

`(3 * set_sum - arr_sum) / 2;`

`}`

`// driver function`

`int`

`main()`

`{`

` `

`int`

`a[] = { 12, 1, 12, 3, 12, 1, 1, 2, 3, 2, 2, 3, 7 };`

` `

`int`

`n = `

`sizeof`

`(a) / `

`sizeof`

`(a[0]);`

` `

`cout << `

`"The element with single occurrence is "`

`<< singleNumber(a, n);`

`}`

**Output**

The element with single occurrence is 7

**Time Complexity:** O(Nlog(N))

**Auxiliary Space:** O(N)