Number of ways to erase exactly one element in the Binary Array to make XOR zero

Given a binary array of 0’s and 1’s, the task is to find the number of ways to erase exactly one element from this array to make XOR zero.

Examples:

Input: arr = {1, 1, 1, 1, 1 }
Output: 5 
You can erase any of the given 5 1's,
it will make the XOR of the rest equal to zero.

Input: arr = {1, 0, 0, 1, 0 }
Output: 3 
Since the XOR of array is already 0,
You can erase any of the given 3 0's
so that the XOR remains unaffected.

Approach: Since we know that, to make XOR of binary elements be 0, the number of 1’s should be even. Hence this problem can be broken into 4 cases:



  • When the number of 1’s is even and the number of 0’s is also even in the given array: In this scenario, the XOR of the array is already 0. Hence to keep the XOR unaffected, we can remove only the 0’s. Hence the number of ways to erase exactly one element from this array to make XOR zero is the number of 0’s in this array.
  • When the number of 1’s is even and the number of 0’s is odd in the given array: In this scenario, the XOR of the array is already 0. Hence to keep the XOR unaffected, we can remove only the 0’s. Hence the number of ways to erase exactly one element from this array to make XOR zero is the number of 0’s in this array.
  • When the number of 1’s is odd and the number of 0’s is even in the given array: In this scenario, the XOR of the array is 1. Hence to make the XOR 0, we can remove any of the 1’s. Hence the number of ways to erase exactly one element from this array to make XOR zero is the number of 1’s in this array.
  • When the number of 1’s is odd and the number of 0’s is also odd in the given array: In this scenario, the XOR of the array is 1. Hence to make the XOR 0, we can remove any of the 1’s. Hence the number of ways to erase exactly one element from this array to make XOR zero is the number of 1’s in this array.

Below is the implementation of the above approach:

C++

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// C++ program to find the number of ways
// to erase exactly one element
// from this array to make XOR zero
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the number of ways
int no_of_ways(int a[], int n)
{
    int count_0 = 0, count_1 = 0;
  
    // Calculate the number of 1's and 0's
    for (int i = 0; i < n; i++) {
        if (a[i] == 0)
            count_0++;
        else
            count_1++;
    }
  
    // Considering the 4 cases
    if (count_1 % 2 == 0)
        return count_0;
    else
        return count_1;
}
  
// Driver code
int main()
{
    int n = 4;
    int a1[4] = { 1, 1, 0, 0 };
    cout << no_of_ways(a1, n) << endl;
  
    n = 5;
    int a2[5] = { 1, 1, 1, 0, 0 };
    cout << no_of_ways(a2, n) << endl;
  
    n = 5;
    int a3[5] = { 1, 1, 0, 0, 0 };
    cout << no_of_ways(a3, n) << endl;
  
    n = 6;
    int a4[6] = { 1, 1, 1, 0, 0, 0 };
    cout << no_of_ways(a4, n) << endl;
  
    return 0;
}

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Java

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// Java program to find the number of ways 
// to erase exactly one element 
// from this array to make XOR zero 
class GFG
{
      
    // Function to find the number of ways 
    static int no_of_ways(int a[], int n) 
    
        int count_0 = 0, count_1 = 0
      
        // Calculate the number of 1's and 0's 
        for (int i = 0; i < n; i++) 
        
            if (a[i] == 0
                count_0++; 
            else
                count_1++; 
        
      
        // Considering the 4 cases 
        if (count_1 % 2 == 0
            return count_0; 
        else
            return count_1; 
    
      
    // Driver code 
    public static void main (String[] args) 
    
        int n = 4
        int a1[] = { 1, 1, 0, 0 }; 
        System.out.println(no_of_ways(a1, n)); 
      
        n = 5
        int a2[] = { 1, 1, 1, 0, 0 }; 
        System.out.println(no_of_ways(a2, n)); 
      
        n = 5
        int a3[] = { 1, 1, 0, 0, 0 }; 
        System.out.println(no_of_ways(a3, n)); 
      
        n = 6
        int a4[] = { 1, 1, 1, 0, 0, 0 }; 
        System.out.println(no_of_ways(a4, n)); 
    
}
  
// This code is contributed by AnkitRai01

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Python3

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# Python3 program to find the number of ways 
# to erase exactly one element 
# from this array to make XOR zero 
  
# Function to find the number of ways 
def no_of_ways(a, n): 
  
    count_0 = 0
    count_1 = 0
  
    # Calculate the number of 1's and 0's
    for i in range(0, n): 
        if (a[i] == 0): 
            count_0 += 1
        else:
            count_1 += 1
      
    # Considering the 4 cases 
    if (count_1 % 2 == 0): 
        return count_0 
    else:
        return count_1 
  
# Driver code 
if __name__ == '__main__'
    n = 4
    a1 = [ 1, 1, 0, 0
    print(no_of_ways(a1, n)) 
  
    n = 5
    a2 = [ 1, 1, 1, 0, 0
    print(no_of_ways(a2, n)) 
  
    n = 5
    a3 = [ 1, 1, 0, 0, 0
    print(no_of_ways(a3, n)) 
  
    n = 6
    a4 = [ 1, 1, 1, 0, 0, 0
    print(no_of_ways(a4, n)) 
  
# This code is contributed by ashutosh450

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C#

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// C# program to find the number of ways 
// to erase exactly one element 
// from this array to make XOR zero 
using System;
  
class GFG
{
      
    // Function to find the number of ways 
    static int no_of_ways(int []a, int n) 
    
        int count_0 = 0, count_1 = 0; 
      
        // Calculate the number of 1's and 0's 
        for (int i = 0; i < n; i++) 
        
            if (a[i] == 0) 
                count_0++; 
            else
                count_1++; 
        
      
        // Considering the 4 cases 
        if (count_1 % 2 == 0) 
            return count_0; 
        else
            return count_1; 
    
      
    // Driver code 
    public static void Main () 
    
        int n = 4; 
        int [] a1 = { 1, 1, 0, 0 }; 
        Console.WriteLine(no_of_ways(a1, n)); 
      
        n = 5; 
        int [] a2 = { 1, 1, 1, 0, 0 }; 
        Console.WriteLine(no_of_ways(a2, n)); 
      
        n = 5; 
        int [] a3 = { 1, 1, 0, 0, 0 }; 
        Console.WriteLine(no_of_ways(a3, n)); 
      
        n = 6; 
        int [] a4 = { 1, 1, 1, 0, 0, 0 }; 
        Console.WriteLine(no_of_ways(a4, n)); 
    
}
  
// This code is contributed by Mohit kumar

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Output:

2
3
3
3

Time Complexity: O(n)



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