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

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.
Another 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 i^th (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;
}

Java

// Java code to count the Minimum bits in A and B
class GFG {

    static int totalFlips(String A, String B,
                                String C, int N)
    {
        int count = 0;

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

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

        return count;
    }

    //driver code
    public static void main (String[] args)
    {
        //N represent total count of Bits
        int N = 5;
        String a = "10100";
        String b = "00010";
        String c = "10011";

        System.out.print(totalFlips(a, b, c, N));
    }
}

// This code is contributed by Anant Agarwal.

Python3

# 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
        else if 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))

C#

// C# code to count the Minimum
// bits flip in A and B
using System;

class GFG {

    static int totalFlips(string A, string B,
                          string 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
    public static void Main()
    {
        // N represent total count of Bits
        int N = 5;
        string a = "10100";
        string b = "00010";
        string c = "10011";

        Console.Write(totalFlips(a, b, c, N));
    }
}

// This code is contributed by Anant Agarwal.

PHP

<?php
// PHP code to count the
// Minimum bits in A and B

function totalFlips($A, $B, $C, $N)
{
    $count = 0;
    for ($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

// N represent total count of Bits
$N = 5;
$a = "10100";
$b = "00010";
$c = "10011";

echo totalFlips($a, $b, $c, $N);

// This code is contributed by nitin mittal.
?>

Javascript

<script>

// Javascript code to count the Minimum bits in A and B

function totalFlips(A, B, C, N)
    {
        let count = 0;
        for (let 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

    // N represent total count of Bits
    let N = 5;
    let a = "10100";
    let b = "00010";
    let c = "10011";

    document.write(totalFlips(a, b, c, N));

</script>

Output

2

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

Efficient Approach:

This approach follows O(log N) time complexity.

C++

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

int totalFlips(string A, string B, string C, int N)
{
    int INTSIZE = 31;
    int ans = 0;
    int i = 0;
    while (N > 0) {
        // Considering only 31 bits
        int a = stoi(A.substr(i * INTSIZE, min(INTSIZE, N)),
                     0, 2);
        int b = stoi(B.substr(i * INTSIZE, min(INTSIZE, N)),
                     0, 2);
        int c = stoi(C.substr(i * INTSIZE, min(INTSIZE, N)),
                     0, 2);
        int Z = a ^ b ^ c;
        // builtin function for
        // counting the number of set bits.
        ans += __builtin_popcount(Z);
        i++;
        N -= 32;
    }

    return ans;
}

// 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;
}

// This code is contributed by Kasina Dheeraj.

Java

// Java code to count the Minimum bits in A and B
class GFG {

    static int totalFlips(String A, String B, String C,
                          int N)
    {

        int INTSIZE = 31;
        int ans = 0;
        int i = 0;
        while (N > 0) {
            // Considering only 31 bits
            int a = Integer.parseInt(
                A.substring(i * INTSIZE,
                            i * INTSIZE
                                + Math.min(INTSIZE, N)),
                2);
            int b = Integer.parseInt(
                B.substring(i * INTSIZE,
                            i * INTSIZE
                                + Math.min(INTSIZE, N)),
                2);
            int c = Integer.parseInt(
                C.substring(i * INTSIZE,
                            i * INTSIZE
                                + Math.min(INTSIZE, N)),
                2);
            int Z = a ^ b ^ c;
            // builtin function for
            // counting the number of set bits.
            ans += Integer.bitCount(Z);
            i++;
            N -= 32;
        }

        return ans;
    }

    // driver code
    public static void main(String[] args)
    {
        // N represent total count of Bits
        int N = 5;
        String a = "10100";
        String b = "00010";
        String c = "10011";

        System.out.print(totalFlips(a, b, c, N));
    }
}

// This code is contributed by Kasina Dheeraj.

Python3

def totalFlips(A, B, C, N):
    INTSIZE = 31
    ans = 0
    i = 0
    while N > 0:
        # Considering only 31 bits
        a = int(A[i * INTSIZE: min(INTSIZE + i * INTSIZE, N)], 2)
        b = int(B[i * INTSIZE: min(INTSIZE + i * INTSIZE, N)], 2)
        c = int(C[i * INTSIZE: min(INTSIZE + i * INTSIZE, N)], 2)
        Z = a ^ b ^ c
        # builtin function for counting the number of set bits.
        ans += bin(Z).count('1')
        i += 1
        N -= 32
    return ans


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

C#

using System;

class Program {
    static int TotalFlips(string A, string B, string C,
                          int N)
    {
        int INTSIZE = 31;
        int ans = 0;
        int i = 0;
        while (N > 0) {
            // Considering only 31 bits
            int a = Convert.ToInt32(
                A.Substring(i * INTSIZE,
                            Math.Min(INTSIZE, N)),
                2);
            int b = Convert.ToInt32(
                B.Substring(i * INTSIZE,
                            Math.Min(INTSIZE, N)),
                2);
            int c = Convert.ToInt32(
                C.Substring(i * INTSIZE,
                            Math.Min(INTSIZE, N)),
                2);
            int Z = a ^ b ^ c;
            // builtin function for
            // counting the number of set bits.
            ans += BitCount(Z);
            i++;
            N -= 32;
        }

        return ans;
    }

    static int BitCount(int i)
    {
        i = i - ((i >> 1) & 0x55555555);
        i = (i & 0x33333333) + ((i >> 2) & 0x33333333);
        return (((i + (i >> 4)) & 0x0F0F0F0F) * 0x01010101)
            >> 24;
    }

    static void Main(string[] args)
    {
        // N represent total count of Bits
        int N = 5;
        string a = "10100";
        string b = "00010";
        string c = "10011";

        Console.WriteLine(TotalFlips(a, b, c, N));
    }
}

Javascript

function TotalFlips(A, B, C, N)
{
    let INTSIZE = 31;
    let ans = 0;
    let i = 0;
    while (N > 0)
    {
        // Considering only 31 bits
        let a = parseInt(A.substring(i * INTSIZE, Math.min(INTSIZE + i * INTSIZE, N)), 2);
        let b = parseInt(B.substring(i * INTSIZE, Math.min(INTSIZE + i * INTSIZE, N)), 2);
        let c = parseInt(C.substring(i * INTSIZE, Math.min(INTSIZE + i * INTSIZE, N)), 2);
        let Z = a ^ b ^ c;
        // builtin function for
        // counting the number of set bits.
        ans += Z.toString(2).split('1').length - 1;
        i++;
        N -= 32;
    }

    return ans;
}

// Driver Code
let N = 5;
let a = "10100";
let b = "00010";
let c = "10011";

console.log(TotalFlips(a, b, c, N));

Output

2

Why this code works?

We observe that bit must be flipped if A[i]^B[i] !=C[i]. So, we can get the number of flips by calculating the number of set bits in a^b^c where a,b,c are integer representations of binary string. But string length may be greater than 32, size of a typical int type. So, the plan is to divide the string into substrings of length 31 ,perform operations and count set bits as mentioned for each substring.

Time Complexity: O(log N) as the while loop runs for log₃₁N times and counting set bits account for at most O(32) for 32-bit and O(64) for 64-bit and for each substring operation O(31).

Space Complexity: O(1) , to be noted that substring operation need O(32) space.

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