Find the MEX of given XOR sequence

Given two integers N and M. The task is to determine the MEX (Minimal Excluded) of the following sequence: N ^ 0, N ^ 1, N ^ 2... N ^ M, where ^ is the Bitwise XOR operator.

Examples:

Input: N = 4, M = 2
Output: 0
Explanation: The sequence will be: 4 ^ 0, 4 ^ 1, 4 ^ 2 = 4, 5, 6. So, the MEX will be 0.

Input: N = 5, M = 7
Output:
Explanation: The sequence will be: 5 ^ 0, 5 ^ 1, 5 ^ 2, 5 ^ 3, 5 ^ 4, 5 ^ 5, 5 ^ 6, 5 ^ 7 = 5, 4, 7, 6, 1, 0, 3, 2. So, the MEX will be 8.

Approach: To solve the problem, follow the below idea:

The problem is to find whether a number K is present in the sequence N ^ 0, N ^ 1, N ^ 2 ... N ^ M. If K is present in the sequence, then there must be an integer X (X <= M), such that N ^ X = K. We can also write N ^ X = K as N ^ K = X. So, now if we want to find whether a number K is present in the sequence N ^ 0, N ^ 1, N ^ 2 ... N ^ M, we can simply check N ^ K <= M.

Now, the problem has been reduced to find the smallest number K such that N ^ K >= M + 1. For simplicity, let's consider P=M + 1. So, we need to find the smallest K such that N ^ K >= P.

We can build P bit by bit starting from the highest bit and moving towards the lowest bit.

• If P_i = 1 and N_i = 1, then set K_i = 0, as N_i ^ 0 >= P_i.
• If Pi = 0 and Ni = 0, then set K_i = 0, as N_i ^ 0 >= P_i.
• If P_i = 0 and N_i = 1, then set K_i = 0, we can break here by setting the remaining bits of K off as no matter what the remaining bits of N are N ^ K > P.
• If P_i = 0 and N_i = 1, then set K_i = 1 as we want to have N_i ^ 1 >= P_i.

Step-by-step algorithm:

• Run a loop from i = 31 to 0 to iterate from Most Significant Bit to Least Significant Bit.
• Start building K bit by bit according to the above conditions.
• Finally return K as the final answer.

Below is the implementation of the approach:

#include <bits/stdc++.h>
using namespace std;

int solve(int N, int M) {
    int P = M + 1;
    int K = 0;
    for (int i = 30; i >= 0; i--) {
        int ithBitN = (N >> i & 1);
        int ithBitP = (P >> i & 1);
        if (ithBitN == ithBitP)
            continue;
        if (ithBitN == 1)
            break;
        if (ithBitP == 1) {
            K |= 1 << i;
        }
    }
    return K;
}

int main()
{
    int N = 5, M = 7;
    cout << solve(N, M) << "\n";
    return 0;
}

import java.util.*;

class Main {
    // Function to find the largest value K such that K XOR M is less than N
    static int solve(int N, int M) {
        int P = M + 1;
        int K = 0;
        // Iterate through each bit position from the most significant bit to the least significant bit
        for (int i = 30; i >= 0; i--) {
            int ithBitN = (N >> i) & 1;
            int ithBitP = (P >> i) & 1;
            // Check if the corresponding bits in N and P are different
            if (ithBitN == ithBitP)
                continue;
            // If the bit in N is 1, break the loop
            if (ithBitN == 1)
                break;
            // If the bit in P is 1, set the corresponding bit in K
            if (ithBitP == 1) {
                K |= 1 << i;
            }
        }
        // Return the final value of K
        return K;
    }

    public static void main(String[] args) {
        // Example input values
        int N = 5, M = 7;
        // Print the result of the solve function
        System.out.println(solve(N, M));
    }
}

// This code is contributed by shivamgupta310570

using System;

class MainClass {
    // Function to find the largest value K such that K XOR M is less than N
    static int Solve(int N, int M) {
        int P = M + 1;
        int K = 0;
        // Iterate through each bit position from the most significant bit to the least significant bit
        for (int i = 30; i >= 0; i--) {
            int ithBitN = (N >> i) & 1;
            int ithBitP = (P >> i) & 1;
            // Check if the corresponding bits in N and P are different
            if (ithBitN == ithBitP)
                continue;
            // If the bit in N is 1, break the loop
            if (ithBitN == 1)
                break;
            // If the bit in P is 1, set the corresponding bit in K
            if (ithBitP == 1) {
                K |= 1 << i;
            }
        }
        // Return the final value of K
        return K;
    }

    public static void Main(string[] args) {
        // Example input values
        int N = 5, M = 7;
        // Print the result of the Solve function
        Console.WriteLine(Solve(N, M));
    }
}

// This code is contributed by shivamgupta310570

// Function to solve the problem
function solve(N, M) {
    // Increment M by 1
    let P = M + 1;
    let K = 0;
    // Loop from 30 to 0
    for (let i = 30; i >= 0; i--) {
        // Get the ith bit of N
        let ithBitN = (N >> i) & 1;
        // Get the ith bit of P
        let ithBitP = (P >> i) & 1;
        // If the ith bits of N and P are the same, continue to the next iteration
        if (ithBitN == ithBitP)
            continue;
        // If the ith bit of N is 1, break the loop
        if (ithBitN == 1)
            break;
        // If the ith bit of P is 1, set the ith bit of K
        if (ithBitP == 1) {
            K |= 1 << i;
        }
    }
    // Return the result
    return K;
}

// Main function
function main() {
    let N = 5, M = 7;
    console.log(solve(N, M));
}

// Call the main function
main();

def solve(N, M):
    # Initialize P with M + 1 and K with 0
    P = M + 1
    K = 0

    # Iterate over each bit position (from 30 to 0)
    for i in range(30, -1, -1):
        # Extract the i-th bit of N and P
        ithBitN = (N >> i) & 1
        ithBitP = (P >> i) & 1

        # If the i-th bits of N and P are equal, continue to the next bit
        if ithBitN == ithBitP:
            continue

        # If the i-th bit of N is 1, break the loop
        if ithBitN == 1:
            break

        # If the i-th bit of P is 1, set the corresponding bit in K
        if ithBitP == 1:
            K |= 1 << i

    return K

def main():
    # Example values for N and M
    N = 5
    M = 7

    # Call the solve function and print the result
    print(solve(N, M))

if __name__ == "__main__":
    # Call the main function if the script is run directly
    main()

8

Time Complexity: O(log (max(N, M))), where N and M are given as input.
Auxiliary Space: O(1)