Count ordered pairs of positive numbers such that their sum is S and XOR is K

Given a sum and a number . The task is to count all possible ordered pairs (a, b) of positive numbers such that the two positive integers a and b have a sum of S and a bitwise-XOR of K.

Examples:  

Input : S = 9, K = 5
Output : 4
The ordered pairs are  (2, 7), (3, 6), (6, 3), (7, 2)

Input : S = 2, K = 2
Output : 0
There are no such ordered pair.

Approach: For any two integers a and b

Sum S = a + b can be written as S = (a b) + (a & b)*2
Where a b is the bitwise XOR and a & b is bitwise AND of the two number a and b respectively.  

This is because is non-carrying binary addition. Thus we can write a & b = (S-K)/2 where S=(a + b) and K = (a b).
If (S-K) is odd or (S-K) less than 0, then there is no such ordered pair.

Now, for each bit, a&b {0, 1} and (a b){0, 1}. 

• If, (a b) = 0 then a_i = b_i, so we have one possibility: a_i = b_i = (a_i & b_i).
• If, (a b) = 1 then we must have (a_i & b_i) = 0 (otherwise the output is 0), and we have two choices: either (a_i = 1 and b_i = 0) or (a_i = 0 and b_i = 1).

Where a_i is the i-th bit in a and b_i is the i-th bit in b.
Thus, the answer is 2, where is the number of set bits in K. 
We will subtract 2 if S and K are equal because a and b must be positive(>0).
Below is the implementation of the above approach:

C++

// C++ program to count ordered pairs of
// positive numbers such that their
// sum is S and XOR is K

#include <bits/stdc++.h>
using namespace std;

// Function to count ordered pairs of
// positive numbers such that their
// sum is S and XOR is K
int countPairs(int s, int K)
{
    // Check if no such pair exists
    if (K > s || (s - K) % 2) {
        return 0;
    }

    if ((s - K) / 2 & K) {
        return 0;
    }

    // Calculate set bits in K
    int setBits = __builtin_popcount(K);

    // Calculate pairs
    int pairsCount = pow(2, setBits);

    // If s = k, subtract 2 from result
    if (s == K)
        pairsCount -= 2;

    return pairsCount;
}

// Driver code
int main()
{
    int s = 9, K = 5;

    cout << countPairs(s, K);

    return 0;
}

Java

// Java program to count ordered pairs of
// positive numbers such that their
// sum is S and XOR is K

class GFG {

// Function to count ordered pairs of
// positive numbers such that their
// sum is S and XOR is K
    static int countPairs(int s, int K) {
        // Check if no such pair exists
        if (K > s || (s - K) % 2 ==1) {
            return 0;
        }

        if ((s - K) / 2 == 1 & K == 1) {
            return 0;
        }

        // Calculate set bits in K
        int setBits = __builtin_popcount(K);

        // Calculate pairs
        int pairsCount = (int) Math.pow(2, setBits);

        // If s = k, subtract 2 from result
        if (s == K) {
            pairsCount -= 2;
        }

        return pairsCount;
    }

    static int __builtin_popcount(int n) {
        /* Function to get no of set
    bits in binary representation
    of positive integer n */

        int count = 0;
        while (n > 0) {
            count += n & 1;
            n >>= 1;
        }
        return count;
    }

// Driver program to test above function
    public static void main(String[] args) {
        int s = 9, K = 5;
        System.out.println(countPairs(s, K));

    }

}

Python3

# Python3 program to count ordered pairs of
# positive numbers such that their
# sum is S and XOR is K

# Function to count ordered pairs of
# positive numbers such that their
# sum is S and XOR is K
def countPairs(s,K):
    if(K>s or (s-K)%2==1):
        return 0

    # Calculate set bits in k
    setBits=(str(bin(K))[2:]).count("1")

    # Calculate pairs
    pairsCount = pow(2,setBits)

    # If s = k, subtract 2 from result
    if(s==K):
        pairsCount-=2

    return pairsCount

# Driver code
if __name__=='__main__':
    s,K=9,5
    print(countPairs(s,K))

# This code is contributed by
# Indrajit Sinha.

C#

// C# program to count ordered pairs
// of positive numbers such that their
// sum is S and XOR is K
using System;

class GFG
{

// Function to count ordered pairs of
// positive numbers such that their
// sum is S and XOR is K
static int countPairs(int s, int K)
{
    // Check if no such pair exists
    if (K > s || (s - K) % 2 ==1)
    {
        return 0;
    }

    if ((s - K) / 2 == 1 & K == 1)
    {
        return 0;
    }

    // Calculate set bits in K
    int setBits = __builtin_popcount(K);

    // Calculate pairs
    int pairsCount = (int) Math.Pow(2, setBits);

    // If s = k, subtract 2 from result
    if (s == K)
    {
        pairsCount -= 2;
    }

    return pairsCount;
}

static int __builtin_popcount(int n)
{
    /* Function to get no of set
    bits in binary representation
    of positive integer n */
    int count = 0;
    while (n > 0)
    {
        count += n & 1;
        n >>= 1;
    }
    return count;
}

// Driver Code
public static void Main()
{
    int s = 9, K = 5;
    Console.Write(countPairs(s, K));
}
}

// This code is contributed
// by Rajput-Ji

PHP

<?php
// PHP program to count ordered
// pairs of positive numbers such
// that their sum is S and XOR is K

// Function to count ordered pairs of
// positive numbers such that their
// sum is S and XOR is K
function countPairs($s, $K)
{
    // Check if no such pair exists
    if ($K > $s || ($s - $K) % 2 == 1)
    {
        return 0;
    }

    if (($s - $K) / 2 == 1 & $K == 1)
    {
        return 0;
    }

    // Calculate set bits in K
    $setBits = __builtin_popcount($K);

    // Calculate pairs
    $pairsCount = (int)pow(2, $setBits);

    // If s = k, subtract 2 from result
    if ($s == $K)
    {
        $pairsCount -= 2;
    }

    return $pairsCount;
}

function __builtin_popcount($n)
{
    /* Function to get no of set
    bits in binary representation
    of positive integer n */
    $count = 0;
    while ($n > 0)
    {
        $count += $n & 1;
        $n >>= 1;
    }
    return $count;
}

// Driver Code
$s = 9; $K = 5;
echo countPairs($s, $K) . "\n";

// This code is contributed
// by Akanksha Rai

Javascript

<script>

// Javascript program to count ordered pairs of
// positive numbers such that their
// sum is S and XOR is K

// Function to count ordered pairs of
// positive numbers such that their
// sum is S and XOR is K
function countPairs(s, K)
{
    // Check if no such pair exists
    if (K > s || (s - K) % 2) {
        return 0;
    }

    if (parseInt((s - K) / 2) & K) {
        return 0;
    }

    // Calculate set bits in K
    let setBits = __builtin_popcount(K);

    // Calculate pairs
    let pairsCount = Math.pow(2, setBits);

    // If s = k, subtract 2 from result
    if (s == K)
        pairsCount -= 2;

    return pairsCount;
}

function __builtin_popcount(n)
{
    /* Function to get no of set
    bits in binary representation
    of positive integer n */
    let count = 0;
    while (n > 0)
    {
        count += n & 1;
        n >>= 1;
    }
    return count;
}

// Driver code
    let s = 9, K = 5;

    document.write(countPairs(s, K));

</script>

Output:

4

Time Complexity: O(log(K)), Auxiliary Space: O (1)