Bitwise XOR of a submatrix of a matrix generated from a given array

Given an array arr[] of length N, , a matrix of dimensions N * N was defined on the array arr[] where M_{i, j} = arr_i& arr_j. Given four integers X, Y, S and T, the task is to find the Bitwise XOR of all the elements of the submatrix from top-left (X, Y) to bottom-right (S, T).

Examples:

Input: N = 3, A[] = {2, 3, 4}, (X, Y)=(0, 1), (S, T)=(2, 2)
Output: 5
Explanation:
Matrix defined on A is
{{(2&2), (2&3), (2&4)},
{(3&2), (3&3), (3&4)},
{(4&2), (4&3), (4&4)}}
Finally, the matrix will be:
{{2, 2, 0},
{2, 3, 0},
{0, 0, 4}}
XOR value= (2^0)^(3^0)^(0^4) = 5
Input: N=3, A[]={1, 2, 3}, (X, Y)=(0, 1), (S, T)=(1, 2)
Output: 1

Naive approach: The simplest approach is to generate the matrix M from the given array and calculate the Bitwise XOR of all the elements present in the given submatrix of M.

Time Complexity: O(N²)
Auxiliary Space: O(N²)

Efficient Approach: The idea is to use the following distributive property of the ‘XOR’ and ‘AND’ operations:

(A & B) ^ (A & C) = A & (B ^ C)

Therefore, the final XOR of the sub-matrix from top-left (X, Y) to bottom-right (S, T) can be calculated from the following equation:

Final XOR
= (XOR of row X)^(XOR of row X+1)^. . . . ^(XOR of row S)
= (A_X& (A_Y^. . . .^ A_T)) ^ ….
. . . ^(A_S& (A_Y^. . . .^A_T))
= (A_Y^. . . .^A_T)&(A_X^. . .^A_S)

Iterate over the array from indices Y to T and compute the XOR of the elements.
Traverse the array from indices X to S and compute the XOR of the elements.
Finally, compute the Bitwise AND of computed XOR’s, which is equal to the Bitwise XOR of the submatrix from (X, Y) to (S, T)

Below is the implementation of the above approach:

C++

// C++ program of the
// above approach
 
#include <iostream>
using namespace std;
 
// Function to find the submatrix
// XOR of the given matrix
int submatrix_xor(int* A, int N,
                  int X, int Y,
                  int S, int T)
{
    int left_xor = 0, i, right_xor = 0;
 
    // Calculating left xor
    // i.e A[Y]^A[Y+1]^. . .^A[T]
    for (i = Y; i <= T; i++) {
        left_xor ^= A[i];
    }
 
    // Calculating right xor
    // i.e A[X]^A[X+1]^. . .^A[S]
    for (i = X; i <= S; i++) {
        right_xor ^= A[i];
    }
 
    // Bitwise AND of left_xor and
    // right_xor gives required result
    return left_xor & right_xor;
}
 
// Driver Code
int main()
{
    int A[3] = { 2, 3, 4 }, X = 0,
        Y = 1, S = 2, T = 2, N = 3;
 
    // Printing xor of submatrix
    cout << submatrix_xor(A, N, X,
                          Y, S, T);
    return 0;
}

Java

// Java program of the
// above approach
import java.io.*;
 
class GFG{
  
// Function to find the submatrix
// XOR of the given matrix
static int submatrix_xor(int[] A, int N,
                         int X, int Y,
                         int S, int T)
{
    int left_xor = 0, i, right_xor = 0;
  
    // Calculating left xor
    // i.e A[Y]^A[Y+1]^. . .^A[T]
    for(i = Y; i <= T; i++)
    {
        left_xor ^= A[i];
    }
  
    // Calculating right xor
    // i.e A[X]^A[X+1]^. . .^A[S]
    for(i = X; i <= S; i++)
    {
        right_xor ^= A[i];
    }
  
    // Bitwise AND of left_xor and
    // right_xor gives required result
    return left_xor & right_xor;
}
  
// Driver Code
public static void main (String[] args)
{
    int[] A = { 2, 3, 4 };
    int X = 0, Y = 1, S = 2,
        T = 2, N = 3;
  
    // Printing xor of submatrix
    System.out.print(submatrix_xor(A, N, X,
                                   Y, S, T));
}
}
 
// This code is contributed by code_hunt

Python3

# Python3 program of the
# above approach
 
# Function to find the submatrix
# XOR of the given matrix
def submatrix_xor(A, N, X, Y, S, T):
     
    left_xor = 0
    i = 0
    right_xor = 0
 
    # Calculating left xor
    # i.e A[Y]^A[Y+1]^. . .^A[T]
    for i in range(Y, T + 1):
        left_xor ^= A[i]
 
    # Calculating right xor
    # i.e A[X]^A[X+1]^. . .^A[S]
    for i in range(X, S + 1):
        right_xor ^= A[i]
 
    # Bitwise AND of left_xor and
    # right_xor gives required result
    return left_xor & right_xor
 
# Driver Code
if __name__ == '__main__':
     
    A = [ 2, 3, 4 ]
    X = 0
    Y = 1
    S = 2
    T = 2
    N = 3
 
    # Printing xor of submatrix
    print(submatrix_xor(A, N, X, Y, S, T))
 
# This code is contributed by mohit kumar 29

C#

// C# program for the above approach 
using System;
 
class GFG{
  
// Function to find the submatrix
// XOR of the given matrix
static int submatrix_xor(int[] A, int N,
                         int X, int Y,
                         int S, int T)
{
    int left_xor = 0, i, right_xor = 0;
  
    // Calculating left xor
    // i.e A[Y]^A[Y+1]^. . .^A[T]
    for(i = Y; i <= T; i++)
    {
        left_xor ^= A[i];
    }
  
    // Calculating right xor
    // i.e A[X]^A[X+1]^. . .^A[S]
    for(i = X; i <= S; i++)
    {
        right_xor ^= A[i];
    }
  
    // Bitwise AND of left_xor and
    // right_xor gives required result
    return left_xor & right_xor;
}
  
// Driver Code
public static void Main ()
{
    int[] A = { 2, 3, 4 };
    int X = 0, Y = 1, S = 2,
        T = 2, N = 3;
  
    // Printing xor of submatrix
    Console.Write(submatrix_xor(A, N, X,
                                Y, S, T));
}
}
 
// This code is contributed by code_hunt

Javascript

<script>
 
// JavaScript program to implement
// the above approach
 
// Function to find the submatrix
// XOR of the given matrix
function submatrix_xor(A, N, X, Y, S, T)
{
    let left_xor = 0, i, right_xor = 0;
   
    // Calculating left xor
    // i.e A[Y]^A[Y+1]^. . .^A[T]
    for(i = Y; i <= T; i++)
    {
        left_xor ^= A[i];
    }
   
    // Calculating right xor
    // i.e A[X]^A[X+1]^. . .^A[S]
    for(i = X; i <= S; i++)
    {
        right_xor ^= A[i];
    }
   
    // Bitwise AND of left_xor and
    // right_xor gives required result
    return left_xor & right_xor;
}
 
// Driver code
    let A = [ 2, 3, 4 ];
    let X = 0, Y = 1, S = 2,
        T = 2, N = 3;
   
    // Printing xor of submatrix
    document.write(submatrix_xor(A, N, X,
                                   Y, S, T));
 
// This code is contributed by target_2.
</script>