Find element which maximize the XOR value for N given update query

Given arrays Arr[] and Queries[] of size N and an integer K such that (0 ≤ Arr[i], Queries[i] < 2^K), the task is to form an array, where each element represents an integer such that XOR of all elements of array and element is maximum by updating Arr[i] with Queries[i] at each step (0 ≤ i ≤ N-1).

Examples:

Input: N = 4, Arr[] = {2, 3, 4, 7}, Queries[] = {1, 0, 3, 4}, K = 4 
Output: {14, 13, 10, 9}
Explanation: The queries are answered as follows:

• 1st query: Arr[] = {2, 3, 4, 7}, After Updating Arr[] = {1, 3, 4, 7} then Choosing X = 14 since 1 XOR 3 XOR 4 XOR 7 XOR 14 = 15 which is the maximum.
• 2nd query: Arr[] = {1, 3, 4, 7}, After Updating Arr[] = {1, 0, 4, 7} then Choosing X = 13 since 1 XOR 0 XOR 4 XOR 7 XOR 13 = 15 which is the maximum.
• 3rd query: Arr[] = {1, 0, 4, 7}, After Updating Arr[[ = {1, 0, 3, 7} then Choosing X = 10 since 1 XOR 0 XOR 3 XOR 7 XOR 10 = 15 which is the maximum.
• 1st query: Arr[] = {1, 0, 3, 7}, After Updating Arr[] = {1, 0, 3, 4 ] then Choosing X = 9 since 1 XOR 0 XOR 3 XOR 4 XOR 9 = 15 which is the maximum.

Input: N = 3, Arr[] = {0, 1, 3},  Queries[] = {2, 2, 2}, K = 3
Output: {7, 4, 5}
Explanation: The queries are answered as follows :

• 1st query: Arr[] = {0, 1, 3}, After Updating Arr[] = {2, 1, 3} then Choosing X = 7 since 2 XOR 1 XOR 3 XOR 7 = 7 which is the maximum .
• 2nd query: Arr[] = {2, 1, 3}, After Updating Arr[] = {2, 2, 3} then Choosing X = 4 since 2 XOR 2 XOR 3 XOR 4 = 7 which is the maximum .
• 3rd query: Arr[] = {2, 2, 3}, After Updating Arr[] = {2, 2, 2} then Choosing X = 5 since 2 XOR 2 XOR 2 XOR 5 = 7 which is the maximum.

Approach:  This problem can be solved with the following idea:

Calculate xor of the array Arr[] and let it be Q, We know the xor property that a xor b = c then a xor c = b or b xor c = a . We need xor to be maximum for every query so rhs will always be (2K – 1) then the equation becomes Q ^ X = (2K – 1) or X = Q ^ (2K – 1) .

Follow the steps to implement the idea:

• Declare a vector Answer.
• Initialize preXor with 0.
• Run a for loop from i : 0 to N and compute the Xor of the array and store it in preXor.
• Initialize rhs with (1 << K) – 1. 
• Run a loop from i : 0 to N and perform the following steps:
  • update preXor = ( preXor ^ Arr[i] )
  • update preXor = ( preXor ^ Queries[i] ) . 
  • Store ( rhs ^ preXor ) in the Answer array.

Below is the implementation of the above algorithm :

C++

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

// Function performing calculation
vector<int> MaxXor(int N, vector<int>& Arr,
                   vector<int>& Queries, int K)
{

    vector<int> answer;
    int preXor = 0;

    // Precomputing the Xor of the array
    for (int i = 0; i < N; i++) {

        preXor = (preXor ^ Arr[i]);
    }

    // Right hand side variable which have
    // to be maximum
    int rhs = (1 << K) - 1;

    for (int i = 0; i < N; i++) {

        // Updating preXor for each update
        // query and the result in
        // answer array
        preXor = (preXor ^ Arr[i]);

        preXor = (preXor ^ Queries[i]);
        answer.push_back(rhs ^ preXor);
    }

    // Returning the answer array
    return answer;
}

// Function for printing the array
void print(int N, vector<int>& answer)
{

    for (int i = 0; i < N; i++) {
        cout << answer[i] << " ";
    }
}

// Driver code
int main()
{

    int N = 4;
    vector<int> Arr = { 2, 3, 4, 7 };
    vector<int> Queries = { 1, 0, 3, 4 };
    int K = 4;

    // Function call
    vector<int> answer = MaxXor(N, Arr, Queries, K);

    // Function call to print
    print(N, answer);

    return 0;
}

Java

// Java code for the above approach
import java.util.*;

public class Main {
    // Function performing calculation
    public static List<Integer> MaxXor(int N, List<Integer> Arr,
    List<Integer> Queries, int K) {
        List<Integer> answer = new ArrayList<>(); // list to store the result
        int preXor = 0; // variable to store the Xor of the array

        // Precomputing the Xor of the array
        for (int i = 0; i < N; i++) {
            preXor = (preXor ^ Arr.get(i));
        }
// Right hand side variable which have to be maximum
        int rhs = (1 << K) - 1;

        for (int i = 0; i < N; i++) {
      // Updating preXor for each update query and the result in answer array
            preXor = (preXor ^ Arr.get(i));
            preXor = (preXor ^ Queries.get(i));
            answer.add(rhs ^ preXor);
        }

        // Returning the answer list
        return answer;
    }

    // Function for printing the result
    public static void print(int N, List<Integer> answer) {
        for (int i = 0; i < N; i++) {
            System.out.print(answer.get(i) + " ");
        }
    }

    public static void main(String[] args) {
        int N = 4;
        List<Integer> Arr = Arrays.asList(2, 3, 4, 7);
        List<Integer> Queries = Arrays.asList(1, 0, 3, 4);
        int K = 4;

        List<Integer> answer = MaxXor(N, Arr, Queries, K);

        print(N, answer);
    }
}

Python3

#Python code for the above approach
from typing import List

# Function performing calculation
def MaxXor(N: int, Arr: List[int], Queries: List[int], K: int) -> List[int]:
    answer = []
    preXor = 0

    # Precomputing the Xor of the array
    for i in range(N):
        preXor = (preXor ^ Arr[i])

    # Right hand side variable which have
    # to be maximum
    rhs = (1 << K) - 1

    for i in range(N):
        # Updating preXor for each update
        # query and the result in
        # answer array
        preXor = (preXor ^ Arr[i])
        preXor = (preXor ^ Queries[i])
        answer.append(rhs ^ preXor)

    # Returning the answer array
    return answer

# Function for printing the array
def print_ans(N: int, answer: List[int]):
    for i in range(N):
        print(answer[i], end = " ")
    print()

# Driver code
if __name__ == "__main__":
    N = 4
    Arr = [2, 3, 4, 7]
    Queries = [1, 0, 3, 4]
    K = 4

    # Function call
    answer = MaxXor(N, Arr, Queries, K)

    # Function call to print
    print_ans(N, answer)

    # This code is contributed by lokehpotta20.

C#

// C# code for the above approach

using System;
using System.Linq;
using System.Collections.Generic;

class GFG
{

    // Function performing calculation
    static List<int> MaxXor(int N, List<int> Arr, List<int> Queries, int K)
    {

        List<int> answer=new List<int>();
        int preXor = 0;

        // Precomputing the Xor of the array
        for (int i = 0; i < N; i++) {

            preXor = (preXor ^ Arr[i]);
        }

        // Right hand side variable which have
        // to be maximum
        int rhs = (1 << K) - 1;

        for (int i = 0; i < N; i++) {

            // Updating preXor for each update
            // query and the result in
            // answer array
            preXor = (preXor ^ Arr[i]);

            preXor = (preXor ^ Queries[i]);
            answer.Add(rhs ^ preXor);
        }

        // Returning the answer array
        return answer;
    }

    // Function for printing the array
    static void print(int N, List<int> answer)
    {

        for (int i = 0; i < N; i++) {
            Console.Write(answer[i] + " ");
        }
    }

    // Driver code
    static public void Main()
    {

        int N = 4;
        List<int> Arr = new List<int>{ 2, 3, 4, 7 };
        List<int> Queries = new List<int>{ 1, 0, 3, 4 };
        int K = 4;

        // Function call
        List<int> answer = MaxXor(N, Arr, Queries, K);

        // Function call to print
        print(N, answer);

    }
}

Javascript

// Javascript code for the above approach

// Function performing calculation
function MaxXor(N, Arr, Queries, K)
{

    let answer=[];
    let preXor = 0;

    // Precomputing the Xor of the array
    for (let i = 0; i < N; i++) {

        preXor = (preXor ^ Arr[i]);
    }

    // Right hand side variable which have
    // to be maximum
    let rhs = (1 << K) - 1;

    for (let i = 0; i < N; i++) {

        // Updating preXor for each update
        // query and the result in
        // answer array
        preXor = (preXor ^ Arr[i]);

        preXor = (preXor ^ Queries[i]);
        answer.push(rhs ^ preXor);
    }

    // Returning the answer array
    return answer;
}

// Function for printing the array
function print(N,  answer)
{

    for (let i = 0; i < N; i++) {
        console.log(answer[i] + " ");
    }
}

// Driver code
let N = 4;
let Arr = [ 2, 3, 4, 7 ];
let Queries = [ 1, 0, 3, 4 ];
let K = 4;

// Function call
let answer = MaxXor(N, Arr, Queries, K);

// Function call to print
print(N, answer);

Output

14 13 10 9 

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