Count subsequences having odd Bitwise XOR values from an array

Given an array A[] of size N, the task is to count the number of subsequences from the given array whose Bitwise XOR value is odd.

Examples:

Input: A[] = {1, 3, 4}
Output: 4
Explanation: Subsequences with odd Bitwise XOR are {1}, {3}, {1, 4}, {3, 4}.

Input: A[] = {2, 8, 6}
Output: 0
Explanation: No such subsequences are present in the array.

Naive Approach: The simplest approach to solve the problem is to generate all the subsequences of the given array and for each subsequence, check if its Bitwise XOR value is odd or not. If found to be odd, then increase the count by one. After checking for all subsequences, print the count obtained.

Time Complexity: O(N * 2^N)
Auxiliary Space: O(1)

Efficient Approach: To optimize the above approach, the idea is based on the observation that a subsequence will have odd Bitwise XOR only if the number of odd elements present in it is odd. This is because odd elements have the least significant bit equal to 1. Therefore, XOR of the least significant bits (1^1^1....) up to an odd number of times sets the least significant bit of the new number formed. Therefore, the new number formed is odd.
Follow the steps below to solve the problem:

1. Store the count of even and odd elements present in the array A[] in even and odd respectively.
2. Traverse the array A[] using the variable i
   • If the value of A[i] is odd, increase the value of odd by 1.
   • Otherwise, increase the value of even by 1.
3. If the value of odd is equal to 0, then print 0 and return.
4. Otherwise,
   • Find the total combinations with an odd number of odd elements.
   • This can be calculated by (^XC₁+^XC₃+...+^XC_(x-(x+1)%2)) * (^MC₀+^MC₁+...+^MC_M) = 2^X-1 * 2^M = 2^X+M-1= 2^N-1.
   • Therefore, Print 2^N-1

Below is the implementation of the above approach:

// C++ program for the above approach
#include<bits/stdc++.h>

using namespace std;

// Function to count the subsequences
// having odd bitwise XOR value
void countSubsequences(vector<int> A)
{
    
    // Stores count of odd elements
    int odd = 0;

    // Stores count of even elements
    int even = 0;

    // Traverse the array A[]
    for(int el : A) 
    {
        
        // If el is odd
        if (el % 2 == 1)
            odd++;
        else
            even++;
    }

    // If count of odd elements is 0
    if (odd == 0)
        cout << (0);
    else
        cout << (1 << (A.size() - 1));
}

// Driver Code
int main()
{
    
    // Given array A[]
    vector<int> A = { 1, 3, 4 };
    
    // Function call to count subsequences
    // having odd bitwise XOR value
    countSubsequences(A);
}

// This code is contributed by mohit kumar 29

// Java program for
// the above approach

import java.io.*;

class GFG {

    // Function to count the subsequences
    // having odd bitwise XOR value
    public static void countSubsequences(int[] A)
    {

        // Stores count of odd elements
        int odd = 0;

        // Stores count of even elements
        int even = 0;

        // Traverse the array A[]
        for (int el : A) {

            // If el is odd
            if (el % 2 == 1)
                odd++;
            else
                even++;
        }

        // If count of odd elements is 0
        if (odd == 0)
            System.out.println(0);

        else
            System.out.println(1 << (A.length - 1));
    }

    // Driver Code
    public static void main(String[] args)
    {
        // Given array A[]
        int[] A = { 1, 3, 4 };

        // Function call to count subsequences
        // having odd bitwise XOR value
        countSubsequences(A);
    }
}

# Python3 program for the above approach

# Function to count the subsequences
# having odd bitwise XOR value
def countSubsequences(A):
    
    # Stores count of odd elements
    odd = 0

    # Stores count of even elements
    even = 0

    # Traverse the array A[]
    for el in A:

        # If el is odd
        if (el % 2 == 1):
            odd += 1
        else:
            even += 1

    # If count of odd elements is 0
    if (odd == 0):
        print(0)
    else:
        print(1 << len(A) - 1)

# Driver Code
if __name__ == "__main__":

    # Given array A[]
    A = [1, 3, 4]

    # Function call to count subsequences
    # having odd bitwise XOR value
    countSubsequences(A)

# This code is contributed by ukasp

// C# program for above approach
using System;

public class GFG 
{
  
  // Function to count the subsequences
  // having odd bitwise XOR value
  public static void countSubsequences(int[] A)
  {

    // Stores count of odd elements
    int odd = 0;

    // Stores count of even elements
    int even = 0;

    // Traverse the array A[]
    foreach (int el in A) {

      // If el is odd
      if (el % 2 == 1)
        odd++;
      else
        even++;
    }

    // If count of odd elements is 0
    if (odd == 0)
      Console.WriteLine(0);

    else
      Console.WriteLine(1 << (A.Length - 1));
  }

  // Driver code
  public static void Main(String[] args)
  {

    // Given array A[]
    int[] A = { 1, 3, 4 };

    // Function call to count subsequences
    // having odd bitwise XOR value
    countSubsequences(A);
  }
}

// This code is contributed by splevel62.

<script>

// Javascript program for the above approach

// Function to count the subsequences
// having odd bitwise XOR value
function countSubsequences( A)
{
    
    // Stores count of odd elements
    var odd = 0;

    // Stores count of even elements
    var even = 0;

    // Traverse the array A[]
    for(var e1 = 0; e1<A.length; e1++)
    {
        // If el is odd
        if (A[e1] % 2 == 1)
            odd++;
        else
            even++;
    }

    // If count of odd elements is 0
    if (odd == 0)
        document.write(0);
    else
        document.write((1 << (A.length - 1)));
}

// Driver Code
// Given array A[]
var A = [ 1, 3, 4 ];

// Function call to count subsequences
// having odd bitwise XOR value
countSubsequences(A);

</script>  

4

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