Sum of XOR of All Pairs in an Array in C++

In this problem, we are given an array arr[] of size n. Our task is to create a program to find the sum of XOR of sum of all pairs in an array.

Let’s see an example to understand the problem, 

Input: arr[5, 7, 9]

Output: 22

Explanation: 

(5+5) ^ (5+7) ^ (5+9) ^ (7+5) ^ (7+7) ^ (7+9) ^ (9+5) ^ (9+7) ^ (9+9) = 22

A simple solution to the problem is by using a nested loop. And creating all possible pairs from the array. And calculate the XOR of the sum of each pair.

Algorithm: 

Initialise XorSum = 0

Step 1: iterate from 0 to n. Follow:

Step 1.1: iterate from 0 to n. Follow

Step 1.1.1: Update XorSum i.e. XorSum = XorSum ^ (arr[i]+arr[j]).

Step 2: return sum

Program to illustrate the working of our code

Example

Live Demo

#include <iostream>
using namespace std;

int findSumXORPair(int arr[], int n) {

   int XorSum = 0;
   for (int i = 0; i < n; i++)
      for (int j = 0; j < n; j++)
       XorSum = XorSum^(arr[i]+arr[j]);
   return XorSum;
}

int main() {

   int arr[] = {5, 7, 9};
   int n = sizeof(arr)/sizeof(arr[0]);
   cout<<"XOR of sum of all pairs in an array is "<<findSumXORPair(arr, n);
   return 0;
}

Output

XOR of sum of all pairs in an array is 22

This solution is not efficient as its time complexity is of the order n².

An efficient solution is using the properties of XOR. To solve the problem, we will calculate the XOR of all elements of the array and then multiply it by two.

Algorithm: 

Initialize XorSum = 0

Step 1: Iterate from 0 to n.

Step 1.1: update XorSum i.e. XorSum = XorSum ^ arr[i]

Step 2: double the sum and return it.

Program to illustrate the working of our solution,

Example

Live Demo

#include <iostream>
using namespace std;

int findSumXORPair(int arr[], int n) {

   int XorSum = 0;
   for (int i = 0; i < n; i++)
    XorSum = XorSum^arr[i];
   XorSum = 2*XorSum;
   return XorSum;
}

int main() {

int arr[] = {5, 7, 9};
   int n = sizeof(arr)/sizeof(arr[0]);
   cout<<"XOR of sum of all pairs in an array is "<<findSumXORPair(arr, n);
   return 0;
}

Output

XOR of sum of all pairs in an array is 22