Minimum size of the array with MEX as A and XOR of the array elements as B

Given two integers A and B, the task is to find the minimum possible size of the array whose MEX of the array is A and Bitwise XOR of all array elements is B.

Examples:

Input: A = 1, B = 1
Output: 3
Explanation: The array which can satisfy the given condition is {0, 3, 2} which is of minimum possible size i.e, 3.

Input: A = 1, B = 999
Output: 2

Approach: The given problem can be solved by the following observations:

• As the MEX of the array should be A as all the elements over the range [0, A – 1] must be included.
• In order to make the XOR of all array elements as B, a few integers must be added to the array which can be classified into 3 cases. Suppose variable currXOR stores the value of XOR over the range [0, A – 1] which can be calculated using the approach discussed in this article.
  • Case 1 where currXor = B. In this case, no integers are required to be added.
  • Case 2 where currXor^B = A. In this case, since A cannot be added to the array, 2 integers must be added such that their XOR is A.
  • In all the other cases, 1 integer must be added in order to make the XOR of the given array as B.

Therefore, the above problem can be solved by finding the XOR of all numbers from 0 to A-1 and checking for the above-mentioned cases.

Below is the implementation of the above approach:

C++

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

// Function to find the minimum size
// of array with given MEX and XOR
int minimumSizeArr(int A, int B)
{
    int currXor = 0;

    // Find the XOR of values from
    // 0 to A-1
    int reminder = (A - 1) % 4;

    // If A is a multiple of 4
    if (reminder == 0)
        currXor = A - 1;

    // If A % 4 gives remainder 1
    else if (reminder == 1)
        currXor = 1;

    // If A % 4 gives remainder 2
    else if (reminder == 2)
        currXor = A;

    // Initializing array size by A
    int minSize = A;

    // If XOR of all values of array
    // is equal to B
    if (currXor == B)
        return minSize;

    // If the required integer to be
    // added is equal to A
    else if (currXor ^ B == A)
        return minSize + 2;

    // Else any integer can be added
    else
        return minSize + 1;
}

// Driver Code
int main()
{
    int A = 1;
    int B = 999;
    cout << minimumSizeArr(A, B);

    return 0;
}

Java

// Java program for the above approach
public class GFG {

    // Function to find the minimum size
    // of array with given MEX and XOR
    static int minimumSizeArr(int A, int B)
    {
        int currXor = 0;

        // Find the XOR of values from
        // 0 to A-1
        int reminder = (A - 1) % 4;

        // If A is a multiple of 4
        if (reminder == 0)
            currXor = A - 1;

        // If A % 4 gives remainder 1
        else if (reminder == 1)
            currXor = 1;

        // If A % 4 gives remainder 2
        else if (reminder == 2)
            currXor = A;

        // Initializing array size by A
        int minSize = A;

        // If XOR of all values of array
        // is equal to B
        if (currXor == B)
            return minSize;

        // If the required integer to be
        // added is equal to A
        else if ((currXor ^ B) == A)
            return minSize + 2;

        // Else any integer can be added
        else
            return minSize + 1;
    }

    // Driver Code
    public static void main (String[] args) {

        int A = 1;
        int B = 999;
        System.out.println(minimumSizeArr(A, B));
    }
}

// This code is contributed by AnkThon

Python3

# python program for the above approach

# Function to find the minimum size
# of array with given MEX and XOR
def minimumSizeArr(A, B):

    currXor = 0

    # Find the XOR of values from
    # 0 to A-1
    reminder = (A - 1) % 4

    # If A is a multiple of 4
    if (reminder == 0):
        currXor = A - 1

    # If A % 4 gives remainder 1
    elif (reminder == 1):
        currXor = 1

    # If A % 4 gives remainder 2
    elif (reminder == 2):
        currXor = A

    # Initializing array size by A
    minSize = A

    # If XOR of all values of array
    # is equal to B
    if (currXor == B):
        return minSize

    # If the required integer to be
    # added is equal to A
    elif (currXor ^ B == A):
        return minSize + 2

    # Else any integer can be added
    else:
        return minSize + 1

# Driver Code
if __name__ == "__main__":

    A = 1
    B = 999
    print(minimumSizeArr(A, B))

# This code is contributed by rakeshsahni

C#

// C# program for the above approach
using System;
public class GFG {

    // Function to find the minimum size
    // of array with given MEX and XOR
    static int minimumSizeArr(int A, int B)
    {
        int currXor = 0;

        // Find the XOR of values from
        // 0 to A-1
        int reminder = (A - 1) % 4;

        // If A is a multiple of 4
        if (reminder == 0)
            currXor = A - 1;

        // If A % 4 gives remainder 1
        else if (reminder == 1)
            currXor = 1;

        // If A % 4 gives remainder 2
        else if (reminder == 2)
            currXor = A;

        // Initializing array size by A
        int minSize = A;

        // If XOR of all values of array
        // is equal to B
        if (currXor == B)
            return minSize;

        // If the required integer to be
        // added is equal to A
        else if ((currXor ^ B) == A)
            return minSize + 2;

        // Else any integer can be added
        else
            return minSize + 1;
    }

    // Driver Code
    public static void Main () {

        int A = 1;
        int B = 999;
        Console.WriteLine(minimumSizeArr(A, B));
    }
}

// This code is contributed by ihritik

Javascript

<script>
// Javascript program for the above approach

// Function to find the minimum size
// of array with given MEX and XOR
function minimumSizeArr(A, B)
{
  let currXor = 0;

  // Find the XOR of values from
  // 0 to A-1
  let reminder = (A - 1) % 4;

  // If A is a multiple of 4
  if (reminder == 0) currXor = A - 1;

  // If A % 4 gives remainder 1
  else if (reminder == 1) currXor = 1;

  // If A % 4 gives remainder 2
  else if (reminder == 2) currXor = A;

  // Initializing array size by A
  let minSize = A;

  // If XOR of all values of array
  // is equal to B
  if (currXor == B) return minSize;
  // If the required integer to be
  // added is equal to A
  else if (currXor ^ (B == A)) return minSize + 2;
  // Else any integer can be added
  else return minSize + 1;
}

// Driver Code

let A = 1;
let B = 999;
document.write(minimumSizeArr(A, B));

// This code is contributed by saurabh_jaiswal.
</script>

Output

2

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