XOR of all even numbers from a given range

Given two integers L and R, the task is to calculate Bitwise XOR of all even numbers in the range [L, R].

Examples:

Example: 
Input: L = 10, R = 20 
Output: 30 
Explanation: 
Bitwise XOR = 10 ^ 12 ^ 14 ^ 16 ^ 18 ^ 20 = 30 
Therefore, the required output is 30.

Example: 
Input: L = 15, R = 23 
Output: 0 
Explanation: 
Bitwise XOR = 16 ^ 18 ^ 20 ^ 22 = 0 
Therefore, the required output is 0.

Naive Approach:The simplest approach to solve the problem is to traverse all even numbers in the range [L, R] and print the Bitwise XOR of all the even numbers. 

Time Complexity: O(R - L)
Auxiliary Space: O(1)

Efficient Approach: The above approach can be optimized based on the following observations:

If N is an even number: 
2 ^ 4 ... ^ (N) = 2 * (1 ^ 2 ^ ... ^ (N / 2))

If N is an odd number: 
2 ^ 4 ... ^ (N) = 2 * (1 ^ 2 ^ ... ^ ((N - 1) / 2))

Follow the steps below to solve the problem:

• Find the bitwise XOR of all the numbers from the range [1, (R) / 2] and store it in a variable, say xor_r.
• Find the bitwise XOR of all the numbers from the range [1, (L - 1) / 2] and store it in a variable, say xor_l.
• Finally, print the value of xor_r ^ xor_l.

Below is the implementation of the above approach:

// C++ Implementation of the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to calculate XOR of
// numbers in the range [1, n]
int bitwiseXorRange(int n)
{

    // If n is divisible by 4
    if (n % 4 == 0)
        return n;

    // If n mod 4 is equal to 1
    if (n % 4 == 1)
        return 1;

    // If n mod 4 is equal to 2
    if (n % 4 == 2)
        return n + 1;

    return 0;
}

// Function to find XOR of even
// numbers in the range [l, r]
int evenXorRange(int l, int r)
{

    // Stores XOR of even numbers
    // in the range [1, l - 1]
    int xor_l;

    // Stores XOR of even numbers
    // in the range [1, r]
    int xor_r;

    // Update xor_r
    xor_r
        = 2 * bitwiseXorRange(r / 2);

    // Update xor_l
    xor_l
        = 2 * bitwiseXorRange((l - 1) / 2);

    return xor_l ^ xor_r;
}

// Driver Code
int main()
{
    int l = 10;
    int r = 20;
    cout << evenXorRange(l, r);

    return 0;
}

// Java Implementation of the above approach
class GFG 
{
    
  // Function to calculate XOR of
  // numbers in the range [1, n]
  static int bitwiseXorRange(int n)
  {

    // If n is divisible by 4
    if (n % 4 == 0)
      return n;

    // If n mod 4 is equal to 1
    if (n % 4 == 1)
      return 1;

    // If n mod 4 is equal to 2
    if (n % 4 == 2)
      return n + 1;

    return 0;
  }

  // Function to find XOR of even
  // numbers in the range [l, r]
  static int evenXorRange(int l, int r)
  {

    // Stores XOR of even numbers
    // in the range [1, l - 1]
    int xor_l;

    // Stores XOR of even numbers
    // in the range [1, r]
    int xor_r;

    // Update xor_r
    xor_r
      = 2 * bitwiseXorRange(r / 2);

    // Update xor_l
    xor_l
      = 2 * bitwiseXorRange((l - 1) / 2);

    return xor_l ^ xor_r;
  }

  // Driver Code
  public static void main (String[] args)
  {
    int l = 10;
    int r = 20;
    System.out.print(evenXorRange(l, r));    
  }
}

// This code is contributed by AnkThon

# Python3 implementation of the above approach

# Function to calculate XOR of
# numbers in the range [1, n]
def bitwiseXorRange(n):

    # If n is divisible by 4
    if (n % 4 == 0):
        return n

    # If n mod 4 is equal to 1
    if (n % 4 == 1):
        return 1

    # If n mod 4 is equal to 2
    if (n % 4 == 2):
        return n + 1

    return 0

# Function to find XOR of even
# numbers in the range [l, r]
def evenXorRange(l, r):

    # Stores XOR of even numbers
    # in the range [1, l - 1]
    #xor_l

    # Stores XOR of even numbers
    # in the range [1, r]
    #xor_r

    # Update xor_r
    xor_r = 2 * bitwiseXorRange(r // 2)

    # Update xor_l
    xor_l = 2 * bitwiseXorRange((l - 1) // 2)

    return xor_l ^ xor_r

# Driver Code
if __name__ == '__main__':
    
    l = 10
    r = 20
    
    print(evenXorRange(l, r))

# This code is contributed by mohit kumar 29

// C# Implementation of the above approach
using System;
class GFG {
    
  // Function to calculate XOR of
  // numbers in the range [1, n]
  static int bitwiseXorRange(int n)
  {
 
    // If n is divisible by 4
    if (n % 4 == 0)
      return n;
 
    // If n mod 4 is equal to 1
    if (n % 4 == 1)
      return 1;
 
    // If n mod 4 is equal to 2
    if (n % 4 == 2)
      return n + 1;
 
    return 0;
  }
 
  // Function to find XOR of even
  // numbers in the range [l, r]
  static int evenXorRange(int l, int r)
  {
 
    // Stores XOR of even numbers
    // in the range [1, l - 1]
    int xor_l;
 
    // Stores XOR of even numbers
    // in the range [1, r]
    int xor_r;
 
    // Update xor_r
    xor_r
      = 2 * bitwiseXorRange(r / 2);
 
    // Update xor_l
    xor_l
      = 2 * bitwiseXorRange((l - 1) / 2);
 
    return xor_l ^ xor_r;
  }
  
  // Driver code
  static void Main() 
  {
    int l = 10;
    int r = 20;
    Console.Write(evenXorRange(l, r));    
  }
}

// This code is contributed by divyeshrabadiya07

<script>
// JavaScript Implementation of the above approach

// Function to calculate XOR of
// numbers in the range [1, n]
function bitwiseXorRange(n)
{

    // If n is divisible by 4
    if (n % 4 == 0)
        return n;

    // If n mod 4 is equal to 1
    if (n % 4 == 1)
        return 1;

    // If n mod 4 is equal to 2
    if (n % 4 == 2)
        return n + 1;
    return 0;
}

// Function to find XOR of even
// numbers in the range [l, r]
function evenXorRange(l, r)
{

    // Stores XOR of even numbers
    // in the range [1, l - 1]
    let xor_l;

    // Stores XOR of even numbers
    // in the range [1, r]
    let xor_r;

    // Update xor_r
    xor_r
        = 2 * bitwiseXorRange(Math.floor(r / 2));

    // Update xor_l
    xor_l
        = 2 * bitwiseXorRange(Math.floor((l - 1) / 2));

    return xor_l ^ xor_r;
}

// Driver Code

    let l = 10;
    let r = 20;
    document.write(evenXorRange(l, r));

// This code is contributed by Surbhi Tyagi.

</script>

30

Time Complexity: O(1).
Auxiliary Space: O(1)