# Minimum insertions to make XOR of an Array equal to half of its sum

Given an array of positive integers, the task is to find the minimum number of insertions to be done in the array, to make the XOR of the array equal to half of its sum, i.e. 2 * Xor of all elements = Sum of all elements

Examples:

Input: arr[] = {1 2 3 4 5}
Output: 1 16
Explanation:
In the modified array {1 2 3 4 5 1 16},
Sum = 1 + 2 + 3 + 4 + 5 + 1 + 16 = 32
Xor = 1 ^ 2 ^ 3 ^ 4 ^ 5 ^ 1 ^ 16 = 16
And, 2 * 16 == 32
Thus, the condition 2 * Xor of all elements = Sum of all elements is satisfied.

Input: 7 11 3 25 51 32 9 29
Output: 17 184
Explanation:
In the modified array { 7 11 3 25 51 32 9 29 17 184}
Sum = 7 + 11 + 3 + 25 + 51 + 32 + 9 + 29 + 17 + 184 = 368
Xor = 7 ^ 11 ^ 3 ^ 25 ^ 51 ^ 32 ^ 9 ^ 29 ^ 17 ^ 184 = 184
And, 2 * 184 == 368
Thus, the condition 2 * Xor of all elements = Sum of all elements is satisfied.

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:
To solve the problem, we need to focus on the two basic properties of XOR:

• A xor A = 0
• A xor 0 = A

We need to follow the steps below to solve the problem:

• Calculate the Sum of all array elements(S) and the Xor of all elements (X). If S == 2*X, no change in array is required. Print -1 for this case.
• Otherwise, do the following:
1. If X = 0, just insert S into the array. Now, the XOR is S, and the sum is 2S.
2. Otherwise, Add X to the array to make the new Xor of the array equal to 0. Then, insert S+X in the array. Now, the Sum is 2(S+X) and the Xor is S+X

Below is the implementation of the above approach.

## C++

 `// C++ Program to make XOR of ` `// of all array elements equal ` `// to half of its sum ` `// by minimum insertions ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Function to make XOR of the ` `// array equal to half of its sum ` `int` `make_xor_half(vector<``int``>& arr) ` `{ ` `    ``int` `sum = 0, xr = 0; ` ` `  `    ``// Calculate the sum and ` `    ``// Xor of all the elements ` `    ``for` `(``int` `a : arr) { ` `        ``sum += a; ` `        ``xr ^= a; ` `    ``} ` ` `  `    ``// If the required condition ` `    ``// satisfies already, return ` `    ``// the original array ` `    ``if` `(2 * xr == sum) ` `        ``return` `-1; ` ` `  `    ``// If Xor is already zero, ` `    ``// Insert sum ` `    ``if` `(xr == 0) { ` `        ``arr.push_back(sum); ` `        ``return` `1; ` `    ``} ` ` `  `    ``// Otherwise, insert xr ` `    ``// and insert sum + xr ` `    ``arr.push_back(xr); ` `    ``arr.push_back(sum + xr); ` `    ``return` `2; ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` ` `  `    ``int` `N = 7; ` `    ``vector<``int``> nums ` `        ``= { 3, 4, 7, 1, 2, 5, 6 }; ` ` `  `    ``int` `count = make_xor_half(nums); ` ` `  `    ``if` `(count == -1) ` `        ``cout << ``"-1"` `<< endl; ` `    ``else` `if` `(count == 1) ` `        ``cout << nums[N] << endl; ` `    ``else` `        ``cout << nums[N] << ``" "` `             ``<< nums[N + 1] << endl; ` ` `  `    ``return` `0; ` `} `

## Python3

 `# Python3 program to make XOR of  ` `# of all array elements equal to   ` `# half of its sum by minimum   ` `# insertions  ` ` `  `# Function to make XOR of the  ` `# array equal to half of its sum  ` `def` `make_xor_half(arr):  ` ` `  `    ``sum` `=` `0``; xr ``=` `0``;  ` ` `  `    ``# Calculate the sum and  ` `    ``# Xor of all the elements  ` `    ``for` `a ``in` `arr: ` `        ``sum` `+``=` `a;  ` `        ``xr ^``=` `a;  ` ` `  `    ``# If the required condition  ` `    ``# satisfies already, return  ` `    ``# the original array  ` `    ``if` `(``2` `*` `xr ``=``=` `sum``): ` `        ``return` `-``1``;  ` ` `  `    ``# If Xor is already zero,  ` `    ``# Insert sum  ` `    ``if` `(xr ``=``=` `0``): ` `        ``arr.append(``sum``);  ` `        ``return` `1``;  ` ` `  `    ``# Otherwise, insert xr  ` `    ``# and insert sum + xr  ` `    ``arr.append(xr);  ` `    ``arr.append(``sum` `+` `xr);  ` `    ``return` `2``;  ` ` `  `# Driver code  ` `if` `__name__ ``=``=` `"__main__"``:  ` ` `  `    ``N ``=` `7``;  ` `    ``nums ``=` `[ ``3``, ``4``, ``7``, ``1``, ``2``, ``5``, ``6` `];  ` `    ``count ``=` `make_xor_half(nums);  ` ` `  `    ``if` `(count ``=``=` `-``1``): ` `        ``print``(``"-1"``); ` `         `  `    ``elif` `(count ``=``=` `1``): ` `        ``print``(nums[N]); ` `         `  `    ``else``: ` `        ``print``(nums[N], nums[N ``+` `1``]);  ` ` `  `# This code is contributed by AnkitRai01 `

Output:

```28
```

Time Complexity: O(N) where N is the size of the array. My Personal Notes arrow_drop_up Check out this Author's contributed articles.

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Improved By : AnkitRai01