Number of unique pairs whose bitwise XOR and OR are same

Given an integer N, the task is to find the number of pairs (say {a, b})from 1 to N (both inclusive) that satisfies the condition: 

• a and b are distinct elements. 
• The values of a | b and a ^ b are equal and 
• a( a | b ) = b( a ^ b ) – ( a | b ) also holds true.

Examples:

Input: N=5
Output:1
Explanation: If we consider a=3 and b=4, then a | b = 7 and a ^ b=7. Thus it forms a valid pair

Input: N=8
Output: 3 
The possible pairs are  {1, 2}, {3, 4} and {7, 8}

Intuition:

Let us consider two numbers a and b. As we know, the sum of these two values can be written as:

1. a + b = ( a | b ) + ( a & b )
2. a + b = ( a ^ b )  + 2*( a & b)

So equating 1 and 2 we get:
a | b + (a & b) = a^b + 2 * (a & b) or
a | b = a ^ b + (a & b)  (equation 3)

Since it is said that (a ^ b) = (a | b), from third condition, we can derive:
a * ( a | b ) = b * ( a ^ b ) – ( a | b ) or
a * (a | b) = b * (a | b) – (a | b) or 
a = b – 1 i.e. b – a = 1

Since (a ^ b) and (a | b) are same, so from the 3rd equation we can derive that (a & b) = 0.

So the question finally boils down to figure out the adjacent pair of elements whose bitwise AND is 0.

 Naive Approach using Linear Iteration:

The basic idea to solve this problem is to store count of unique adjacent pairs by iterating over the array using a loop from 1 to N.

• Initialize a counter variable (say count) to store the count of unique pairs.
• Start traversing using from i = 1 to N-1:
  • Find the bitwise AND of i and i+1 and if it is 0 increment the count.
• Return count as the required answer.

Below is the implementation of the above approach.

C++

// C++ code to implement the approach

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

// Function to count the number of
// unique pairs that satisfy the conditions
int countPairs(int n)
{
    // stores the count of unique pairs
    int count = 0;

    // Traverse from i = 1 to N-1
    for (int i = 1; i < n; i++) {

        int ans = (i & (i + 1));

        // If AND value is 0,
        // increase count by 1
        if (ans == 0)
            count++;
    }

    return count;
}

// Driver code
int main()
{
    // First test case
    int N = 8;
    cout << "For N = 8: " << countPairs(N) << "\n";

    // Second test case
    N = 1;
    cout << "For N = 1: " << countPairs(N) << "\n";

    // Third test case
    N = 2;
    cout << "For N = 2: " << countPairs(N);

    return 0;
}

Java

// Java code to implement the approach

import java.io.*;

class GFG {

  // Function to count the number of unique pairs that
  // satisfy the conditions
  static int countPairs(int n)
  {
    // stores the count of unique pairs
    int count = 0;

    // Traverse from i=1 to N-1
    for (int i = 1; i < n; i++) {
      int ans = (i & (i + 1));

      if (ans == 0) {
        // If AND value is 0, increase count by 1
        count++;
      }
    }

    return count;
  }

  public static void main(String[] args)
  {
    // First test case
    int N = 8;
    System.out.println("For N = 8: " + countPairs(N));

    // Second test case
    N = 1;
    System.out.println("For N = 1: " + countPairs(N));

    // Third test case
    N = 2;
    System.out.println("For N = 2: " + countPairs(N));
  }
}

// This code is contributed by lokesh.

Python3

# python code to implement the approach

# Function to count the number of
# unique pairs that satisfy the conditions
def countPairs(n):

    # stores the count of unique pairs
    count = 0

    # Traverse from i = 1 to N-1
    for i in range(1, n):
        ans = (i & (i + 1))

        # If AND value is 0,
        # increase count by 1
        if ans == 0:
            count = count+1

    return count


# Driver code
if __name__ == "__main__":

    # First test case
    N = 8
    print("For N = 8: ", countPairs(N))

    # Second test case
    N = 1
    print("For N = 1: ", countPairs(N))

    # Third test case
    N = 2
    print("For N = 2: ", countPairs(N))

    # This code is contributed by garg28harsh.

C#

// C# code to implement the approach
using System;
public class GFG {

    // Function to count the number of unique pairs that
    // satisfy the conditions
    static int countPairs(int n)
    {
        // stores the count of unique pairs
        int count = 0;

        // Traverse from i=1 to N-1
        for (int i = 1; i < n; i++) {
            int ans = (i & (i + 1));

            if (ans == 0) {
                // If AND value is 0, increase count by 1
                count++;
            }
        }

        return count;
    }

    static public void Main()
    {

        // Code
        // First test case
        int N = 8;
        Console.WriteLine("For N = 8: " + countPairs(N));

        // Second test case
        N = 1;
        Console.WriteLine("For N = 1: " + countPairs(N));

        // Third test case
        N = 2;
        Console.WriteLine("For N = 2: " + countPairs(N));
    }
}

// This code is contributed by lokeshmvs21.

Javascript

// JavaScript code to implement the approach

// Function to count the number of
// unique pairs that satisfy the conditions
const countPairs = (n) => {
    // stores the count of unique pairs
    let count = 0;

    // Traverse from i = 1 to N-1
    for (let i = 1; i < n; i++) {

        let ans = (i & (i + 1));

        // If AND value is 0,
        // increase count by 1
        if (ans == 0)
            count++;
    }

    return count;
}

// Driver code

// First test case
let N = 8;
console.log(`For N = 8: ${countPairs(N)}<br/>`);

// Second test case
N = 1;
console.log(`For N = 1: ${countPairs(N)}<br/>`);

// Third test case
N = 2;
console.log(`For N = 2: ${countPairs(N)}`);

// This code is contributed by rakeshsahni

Output

For N = 8: 3
For N = 1: 0
For N = 2: 1

Time Complexity: O(N)
Auxiliary Space: O(N)

Efficient Approach:

Since we know that  both the numbers differ by 1 hence by the property of the AND operation of any power of 2 with a number  that is one less than that gives 0. Hence (2ⁿ) & (2ⁿ-1)==0, So we can directly count the number of pairs of form 2ⁿ and 2ⁿ-1.

Below is the implementation of the above idea.

C++

// C++ code to implement the approach

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

// Function to count the number of
// unique pairs whose
int countPairs(int n)
{
    // Get the maximum power of
    // 2 closest to n
    int max_power_of_2 = log2(n);

    return max_power_of_2;
}

// Driver code
int main()
{
    // First test case
    int N = 8;
    cout << "For N = 8: " << countPairs(N) << "\n";

    // Second test case
    N = 1;
    cout << "For N = 1: " << countPairs(N) << "\n";

    // Third test case
    N = 2;
    cout << "For N = 2: " << countPairs(N);

    return 0;
}

Java

// Java code to implement the approach
import java.util.*;

class GFG {

// Function to count the number of
// unique pairs whose
static int countPairs(int n)
{

    // Get the maximum power of
    // 2 closest to n
    int max_power_of_2 = (int)(Math.log(n) / Math.log(2));

    return max_power_of_2;
}


// Driver Code
public static void main (String[] args) {

    // First test case
    int N = 8;
    System.out.println( "For N = 8: " + countPairs(N));

    // Second test case
    N = 1;
    System.out.println( "For N = 1: " + countPairs(N));

    // Third test case
    N = 2;
    System.out.println( "For N = 2: " + countPairs(N));
}
}

// This code is contributed by sanjoy_62.

Python3

# Python3 code to implement the approach
import math

# Function to count the number of
# unique pairs whose
def countPairs(n):

    # Get the maximum power of
    # 2 closest to n
    max_power_of_2 = int(math.log2(n))

    return max_power_of_2

# Driver code
# First test case
N = 8
print("For N = 8: ",countPairs(N))

# Second test case
N = 1
print("For N = 1: ",countPairs(N))

# Third test case
N = 2
print("For N = 2: ",countPairs(N))

# This code is contributed by akashish__

C#

// C# code to implement the approach
using System;

class GFG {

  // Function to count the number of
  // unique pairs whose
  static int countPairs(int n)
  {

    // Get the maximum power of
    // 2 closest to n
    int max_power_of_2
      = (int)(Math.Log(n) / Math.Log(2));

    return max_power_of_2;
  }

  // Driver Code
  public static void Main(string[] args)
  {

    // First test case
    int N = 8;
    Console.WriteLine("For N = 8: " + countPairs(N));

    // Second test case
    N = 1;
    Console.WriteLine("For N = 1: " + countPairs(N));

    // Third test case
    N = 2;
    Console.WriteLine("For N = 2: " + countPairs(N));
  }
}

// This code is contributed by karandeep1234

Javascript

// Javascript code to implement the approach

// Function to count the number of
// unique pairs whose
function countPairs(n)
{
    // Get the maximum power of
    // 2 closest to n
    let max_power_of_2 = Math.round(Math.log2(n));

    return max_power_of_2;
}

// Driver code
    // First test case
    let N = 8;
    console.log( "For N = 8: " +countPairs(N));

    // Second test case
    N = 1;
    console.log( "For N = 1: " +countPairs(N));

    // Third test case
    N = 2;
    console.log( "For N = 2: " +countPairs(N));

   // This code is contributed by garg28harsh.

Output

For N = 8: 3
For N = 1: 0
For N = 2: 1

Time Complexity: O(log N) since it calculates in log (N) time.
Space Complexity: O(1)