Choose X such that (A xor X) + (B xor X) is minimized

Given two integers A and B. The task is to choose an integer X such that (A xor X) + (B xor X) is the minimum possible.

Examples:

Input: A = 2, B = 3 
Output: X = 2, Sum = 1

Input: A = 7, B = 8 
Output: X = 0, Sum = 15 

A simple solution is to generate all possible sum by taking xor of A and B with all possible values of X ≤ min(A, B). To generate all possible sums it would take O(N) time where N = min(A, B). 

An efficient solution is based on the fact that the number X will contain the set bits only at that index where both A and B contain a set bit such that after xor operation with X that bit will be unset. This would take only O(Log N) time.
Other cases: If at a particular index one or both the numbers contain 0 (unset bit) and the number X contains 1 (set bit) then 0 will be set after xor with X in A and B then the sum couldn’t be minimized.

Below is the implementation of the above approach: 

C++

// C++ implementation of the approach
#include <iostream>
using namespace std;

// Function to return the integer X such that
// (A xor X) + (B ^ X) is minimized
int findX(int A, int B)
{
    int j = 0, x = 0;

    // While either A or B is non-zero
    while (A || B) {

        // Position at which both A and B
        // have a set bit
        if ((A & 1) && (B & 1)) {

            // Inserting a set bit in x
            x += (1 << j);
        }

        // Right shifting both numbers to
        // traverse all the bits
        A >>= 1;
        B >>= 1;
        j += 1;
    }
    return x;
}

// Driver code
int main()
{
    int A = 2, B = 3;
    int X = findX(A, B);

    cout << "X = " << X << ", Sum = " << (A ^ X) + (B ^ X);

    return 0;
}

Java

// Java implementation of the approach
class GFG {

    // Function to return the integer X such that
    // (A xor X) + (B ^ X) is minimized
    static int findX(int A, int B)
    {
        int j = 0, x = 0;

        // While either A or B is non-zero
        while (A != 0 || B != 0) {

            // Position at which both A and B
            // have a set bit
            if ((A % 2 == 1) && (B % 2 == 1)) {

                // Inserting a set bit in x
                x += (1 << j);
            }

            // Right shifting both numbers to
            // traverse all the bits
            A >>= 1;
            B >>= 1;
            j += 1;
        }
        return x;
    }

    // Driver code
    public static void main(String[] args)
    {
        int A = 2, B = 3;
        int X = findX(A, B);

        System.out.println(
            "X = " + X + ", Sum = " + ((A ^ X) + (B ^ X)));
    }
}

// This code has been contributed by 29AjayKumar

Python3

# Python 3 implementation of the approach

# Function to return the integer X such that
# (A xor X) + (B ^ X) is minimized


def findX(A, B):
    j = 0
    x = 0

    # While either A or B is non-zero
    while (A or B):

        # Position at which both A and B
        # have a set bit
        if ((A & 1) and (B & 1)):

            # Inserting a set bit in x
            x += (1 << j)

        # Right shifting both numbers to
        # traverse all the bits
        A >>= 1
        B >>= 1
        j += 1
    return x


# Driver code
if __name__ == '__main__':
    A = 2
    B = 3
    X = findX(A, B)

    print("X =", X, ", Sum =", (A ^ X) + (B ^ X))

# This code is contributed by
# Surendra_Gangwar

C#

// C# implementation of the approach
using System;

class GFG {

    // Function to return the integer X such that
    // (A xor X) + (B ^ X) is minimized
    static int findX(int A, int B)
    {
        int j = 0, x = 0;

        // While either A or B is non-zero
        while (A != 0 || B != 0) {

            // Position at which both A and B
            // have a set bit
            if ((A % 2 == 1) && (B % 2 == 1)) {

                // Inserting a set bit in x
                x += (1 << j);
            }

            // Right shifting both numbers to
            // traverse all the bits
            A >>= 1;
            B >>= 1;
            j += 1;
        }
        return x;
    }

    // Driver code
    public static void Main(String[] args)
    {
        int A = 2, B = 3;
        int X = findX(A, B);

        Console.WriteLine(
            "X = " + X + ", Sum = " + ((A ^ X) + (B ^ X)));
    }
}

// This code has been contributed by 29AjayKumar

PHP

<?php
// PHP implementation of the approach

// Function to return the integer X such that
// (A xor X) + (B ^ X) is minimized
function findX($A, $B)
{
    $j = 0;
    $x = 0;

    // While either A or B is non-zero
    while ($A || $B) {

        // Position at which both A and B
        // have a set bit
        if (($A & 1) && ($B & 1))
        {

            // Inserting a set bit in x
            $x += (1 << $j);
        }

        // Right shifting both numbers to
        // traverse all the bits
        $A >>= 1;
        $B >>= 1;
        $j += 1;
    }
    return $x;
}

// Driver code
    $A = 2;
    $B = 3;
    $X = findX($A, $B);

    echo "X = " , $X , ", Sum = ",
        ($A ^ $X) + ($B ^ $X);

// This code is contributed by ajit.
?>

Javascript

<script>
    // Javascript implementation of the approach

    // Function to return the integer X such that
    // (A xor X) + (B ^ X) is minimized
    function findX(A, B)
    {
        let j = 0, x = 0;

        // While either A or B is non-zero
        while (A != 0 || B != 0) {

            // Position at which both A and B
            // have a set bit
            if ((A % 2 == 1) && (B % 2 == 1)) {

                // Inserting a set bit in x
                x += (1 << j);
            }

            // Right shifting both numbers to
            // traverse all the bits
            A >>= 1;
            B >>= 1;
            j += 1;
        }
        return x;
    }

    let A = 2, B = 3;
    let X = findX(A, B);

    document.write("X = " + X + ", Sum = " + ((A ^ X) + (B ^ X)));

        // This code is contributed by suresh07.
</script>

Output

X = 2, Sum = 1

Time Complexity: O(log(max(A, B)))

Auxiliary Space: O(1)

Most Efficient Approach:

Using the idea that X will contain only the set bits as A and B,  X = A & B. On replacing X, the above equation becomes (A ^ (A & B)) + (B ^ (A & B)) which further equates to A^B. 

Proof:
Given (A ^ X) + (B ^ X)
Taking X = (A & B), we have 
(A ^ (A & B)) + (B ^ (A & B))
   (using x ^ y = x'y + y'x )
= (A'(A & B) + A(A & B)') + (B'(A & B) + B(A & B)')
   (using (x & y)' = x' + y')
= (A'(A & B) + A(A' + B')) + (B'(A & B) + B(A' + B'))
  (A'(A & B) = A'A & A'B = 0, B'(A & B) 
   = B'A & B'B = 0)
= (A(A' + B')) + (B(A' + B'))                               
= (AA' + AB') + (BA' + BB')
  (using xx' = x'x = 0)                                   
= (AB') + (BA')                                             
= (A ^ B)

Click here to know more about Boolean Properties.

Below is the implementation of the above approach: 

C++

// c++ implementation of above approach
#include <iostream>
using namespace std;

// finding X
int findX(int A, int B) {
  return A & B;
}

// finding Sum
int findSum(int A, int B) {
  return A ^ B;
}

// Driver code
int main()
{
    int A = 2, B = 3;
    cout << "X = " << findX(A, B)
         << ", Sum = " << findSum(A, B);
    return 0;
}

// This code is contributed by yashbeersingh42

Java

// Java implementation of above approach
import java.io.*;

class GFG
{
    // finding X
    public static int findX(int A, int B)
    {
      return A & B;
    }

    // finding Sum
    public static int findSum(int A, int B)
    {
        return A ^ B;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int A = 2, B = 3;
        System.out.print("X = " + findX(A, B)
                         + ", Sum = " + findSum(A, B));
    }
}
// This code is contributed by yashbeersingh42

Python3

# Python3 implementation of above approach

# finding X
def findX(A, B):
    return A & B

# finding Sum
def findSum(A, B):
    return A ^ B

# Driver code
A, B = 2, 3
print("X =", findX(A, B) , ", Sum =" , findSum(A, B))

# This code is contributed by divyeshrabadiya07

C#

// C# implementation of above approach
using System;

class GFG{

// Finding X
public static int findX(int A, int B)
{
    return A & B;
}

// Finding Sum
public static int findSum(int A, int B)
{
    return A ^ B;
}

// Driver Code
public static void Main(String[] args)
{
    int A = 2, B = 3;

    Console.Write("X = " + findX(A, B) +
                  ", Sum = " + findSum(A, B));
}
}

// This code is contributed by Princi Singh

Javascript

<script>

    // Javascript implementation of the approach

    // finding X
    function findX(A, B) {
      return A & B;
    }

    // finding Sum
    function findSum(A, B) {
      return A ^ B;
    }

    let A = 2, B = 3;
    document.write("X = " + findX(A, B) + ", Sum = " + findSum(A, B));

</script>

Output

X = 2, Sum = 1

Time Complexity: O(1)

Auxiliary Space: O(1)