Count minimum bits to flip such that XOR of A and B equal to C

1.5

Given a sequence of three binary sequences A, B and C of N bits. Count the minimum bits required to flip in A and B such that XOR of A and B is equal to C. For example:-

Input: N = 3
       A = 110
       B = 101
       C = 001
Output: 1
We only need to flip the bit of 2nd position
of either A or B, such that A ^ B = C i.e., 
100 ^ 101 = 001

A Naive approach is to generate all possible combination of bits in A and B and then XORing them to Check whether it is equal to C or not. Time complexity of this approach grows exponentially so it would not be better for large value of N.

An Efficient approach is to use concept of XOR.

XOR Truth Table
  Input    Output
 X     Y     Z
 0     0  -  0
 0     1  -  1
 1     0  -  1
 1     1  -  0

If we generalize, we will find that at any position of A and B, we just only need to flip ith (0 to N-1) position of either A or B otherwise we will not able to achieve minimum no of Bits.
So at any position of i (0 to N-1) you will encounter two type of situation i.e., either A[i] == B[i] or A[i] != B[i]. Let’s discuss it one by one.

  • If A[i] == B[i] then XOR of these bits will be 0, two cases arise in C[]: C[i]==0 or C[i]==1.
    If C[i] == 0, then no need to flip the bit but if C[i] == 1 then we have to flip the bit either in A[i] or B[i] so that 1^0 == 1 or 0^1 == 1.
  • If A[i] != B[i] then XOR of these Bits gives a 1, In C two cases again arise i.e., either C[i] == 0 or C[i] == 1.
    Therefore if C[i] == 1, then we need not to flip the bit but if C[i] == 0, then we need to flip the bit either in A[i] or B[i] so that 0^0==0 or 1^1==0

C++

// C++ code to count the Minimum bits in A and B
#include<bits/stdc++.h>
using namespace std;

int totalFlips(char *A, char *B, char *C, int N)
{
    int count = 0;
    for (int i=0; i < N; ++i)
    {
        // If both A[i] and B[i] are equal
        if (A[i] == B[i] && C[i] == '1')
            ++count;

        // If Both A and B are unequal
        else if (A[i] != B[i] && C[i] == '0')
            ++count;
    }
    return count;
}

//Driver Code
int main()
{
    //N represent total count of Bits
    int N = 5;
    char a[] = "10100";
    char b[] = "00010";
    char c[] = "10011";

    cout << totalFlips(a, b, c, N);

    return 0;
}

Python

# Python code to find minimum bits to be flip
def totalFlips(A, B, C, N):

    count = 0
    for i in range(N):
        
        # If both A[i] and B[i] are equal
        if A[i] == B[i] and C[i] == '1':
            count=count+1

        # if A[i] and B[i] are unequal
        elif A[i] != B[i] and C[i] == '0':
            count=count+1
    return count


# Driver Code
# N represent total count of Bits
N = 5
a = "10100"
b = "00010"
c = "10011"
print(totalFlips(a, b, c, N))
Output:
2

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

This article is contributed by Shubham Bansal. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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