XOR of all subarray XORs | Set 1

Last Updated : 19 Sep, 2023

Given an array of integers, we need to get the total XOR of all subarray XORs where subarray XOR can be obtained by XORing all elements of it.

Examples :

Input : arr[] = [3, 5, 2, 4, 6]
Output : 7
Total XOR of all subarray XORs is,
(3) ^ (5) ^ (2) ^ (4) ^ (6)
(3^5) ^ (5^2) ^ (2^4) ^ (4^6)
(3^5^2) ^ (5^2^4) ^ (2^4^6)
(3^5^2^4) ^ (5^2^4^6) ^
(3^5^2^4^6) = 7     

Input : arr[] = {1, 2, 3}
Output : 2

Input : arr[] = {1, 2, 3, 4}
Output : 0

A simple solution is to generate all subarrays and compute XOR of all of them. Below is the implementation of the above idea :

Implementation:

C++

// C++ program to get total xor of all subarray xors
#include <bits/stdc++.h>
using namespace std;

// Returns XOR of all subarray xors
int getTotalXorOfSubarrayXors(int arr[], int N)
{
    //  initialize result by 0 as (a xor 0 = a)
    int res = 0;

    // select the starting element
    for (int i=0; i<N; i++)

        // select the eNding element
        for (int j=i; j<N; j++)

            // Do XOR of elements in current subarray
            for (int k=i; k<=j; k++)
                res = res ^ arr[k];

    return res;
}

// Driver code to test above methods
int main()
{
    int arr[] = {3, 5, 2, 4, 6};
    int N = sizeof(arr) / sizeof(arr[0]);

    cout << getTotalXorOfSubarrayXors(arr, N);
    return 0;
}

Java

// java program to get total XOR 
// of all subarray xors
public class GFG {
        
    // Returns XOR of all subarray xors
    static int getTotalXorOfSubarrayXors(
                          int arr[], int N)
    {
        
        // initialize result by 
        // 0 as (a xor 0 = a)
        int res = 0;
        
        // select the starting element
        for (int i = 0; i < N; i++)
        
            // select the eNding element
            for (int j = i; j < N; j++)
        
            // Do XOR of elements
            // in current subarray
            for (int k = i; k <= j; k++)
                res = res ^ arr[k];
    
        return res;
    }
    
    // Driver code
    public static void main(String args[])
    {
        int arr[] = {3, 5, 2, 4, 6};
        int N = arr.length;
        
        System.out.println(
            getTotalXorOfSubarrayXors(arr, N));
    }
}

// This code is contributed by Sam007.

Python 3

# python program to get total xor 
# of all subarray xors

# Returns XOR of all subarray xors
def getTotalXorOfSubarrayXors(arr, N):
    
    # initialize result by 0 as
    # (a xor 0 = a)
    res = 0

    # select the starting element
    for i in range(0, N):
        
        # select the eNding element
        for j in range(i, N):
            
            # Do XOR of elements in
            # current subarray
            for k in range(i, j + 1):
                res = res ^ arr[k]
            
    return res

# Driver code to test above methods
arr = [3, 5, 2, 4, 6]
N = len(arr)

print(getTotalXorOfSubarrayXors(arr, N))

# This code is contributed by Sam007.

// C# program to get total XOR 
// of all subarray xors
using System;

class GFG {

// Returns XOR of all subarray xors
static int getTotalXorOfSubarrayXors(int []arr, 
                                     int N)
{
    
// initialize result by 
// 0 as (a xor 0 = a)
int res = 0;

// select the starting element
for (int i = 0; i < N; i++)

    // select the eNding element
    for (int j = i; j < N; j++)

        // Do XOR of elements
        // in current subarray
        for (int k = i; k <= j; k++)
            res = res ^ arr[k];

return res;
}

// Driver Code
static void Main()
{
    int []arr = {3, 5, 2, 4, 6};
    int N = arr.Length;
    Console.Write(getTotalXorOfSubarrayXors(arr, N));
}
}

// This code is contributed by Sam007

PHP

<?php
// PHP program to get total 
// xor of all subarray xors

// Returns XOR of all subarray xors
function getTotalXorOfSubarrayXors($arr, $N)
{
    
    // initialize result by
    // 0 as (a xor 0 = a)
    $res = 0;

    // select the starting element
    for($i = 0; $i < $N; $i++)

        // select the eNding element
        for($j = $i; $j < $N; $j++)

            // Do XOR of elements in
            // current subarray
            for($k = $i; $k <= $j; $k++)
                $res = $res ^ $arr[$k];

    return $res;
}

    // Driver code
    $arr = array(3, 5, 2, 4, 6);
    $N = sizeof($arr);
    echo getTotalXorOfSubarrayXors($arr, $N);

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

JavaScript

<script>

// JavaScript program for the above approach

    // Returns XOR of all subarray xors
    function getTotalXorOfSubarrayXors(
                          arr, N)
    {
        
        // initialize result by 
        // 0 as (a xor 0 = a)
        let res = 0;
        
        // select the starting element
        for (let i = 0; i < N; i++)
        
            // select the eNding element
            for (let j = i; j < N; j++)
        
            // Do XOR of elements
            // in current subarray
            for (let k = i; k <= j; k++)
                res = res ^ arr[k];
    
        return res;
    }
    
// Driver Code

    // Both a[] and b[] must be of same size.
    let arr = [3, 5, 2, 4, 6];
    let N = arr.length;
        
    document.write(getTotalXorOfSubarrayXors(arr, N));

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

Output

Time Complexity: O(N³)

Auxiliary Space: O(1)

An efficient solution is based on the idea to enumerate all subarrays, we can count the frequency of each element that occurred totally in all subarrays, if the frequency of an element is odd then it will be included in the final result otherwise not.

As in above example, 
3 occurred 5 times,
5 occurred 8 times,
2 occurred 9 times,
4 occurred 8 times,
6 occurred 5 times
So our final result will be xor of all elements which occurred odd number of times
i.e. 3^2^6 = 7

From above occurrence pattern we can observe that number at i-th index will have 
(i + 1) * (N - i) frequency.

So we can iterate over all elements once and calculate their frequencies and if it is odd then we can include that in our final result by XORing it with the result.
Total time complexity of the solution will be O(N)

Implementation:

C++

// C++ program to get total 
// xor of all subarray xors
#include <bits/stdc++.h>
using namespace std;

// Returns XOR of all subarray xors
int getTotalXorOfSubarrayXors(int arr[], 
                              int N)
{
    // initialize result by 0
    // as (a XOR 0 = a)
    int res = 0;

    // loop over all elements once
    for (int i = 0; i < N; i++)
    {
        // get the frequency of
        // current element
        int freq = (i + 1) * (N - i);

        // Uncomment below line to print 
        // the frequency of arr[i]
        // cout << arr[i] << " " << freq << endl;

        // if frequency is odd, then
        // include it in the result
        if (freq % 2 == 1)
            res = res ^ arr[i];
    }

    // return the result
    return res;
}

// Driver Code
int main()
{
    int arr[] = {3, 5, 2, 4, 6};
    int N = sizeof(arr) / sizeof(arr[0]);

    cout << getTotalXorOfSubarrayXors(arr, N);
    return 0;
}

Java

// java program to get total xor
// of all subarray xors
import java.io.*;

public class GFG {
    
    // Returns XOR of all subarray
    // xors
    static int getTotalXorOfSubarrayXors(
                          int arr[], int N)
    {
        
        // initialize result by 0 
        // as (a XOR 0 = a)
        int res = 0;
    
        // loop over all elements once
        for (int i = 0; i < N; i++)
        {
            // get the frequency of
            // current element
            int freq = (i + 1) * (N - i);
    
            // Uncomment below line to print 
            // the frequency of arr[i]
            
            // if frequency is odd, then 
            // include it in the result
            if (freq % 2 == 1)
                res = res ^ arr[i];
        }
    
        // return the result
        return res;
    }
    
    public static void main(String[] args)
    {

        int arr[] = {3, 5, 2, 4, 6};
        int N = arr.length;
        System.out.println(
            getTotalXorOfSubarrayXors(arr, N));
    }
}

// This code is contributed by Sam007.

Python 3

# Python3 program to get total 
# xor of all subarray xors

# Returns XOR of all
# subarray xors
def getTotalXorOfSubarrayXors(arr, N):

    # initialize result by 0
    # as (a XOR 0 = a)
    res = 0

    # loop over all elements once
    for i in range(0, N):
    
        # get the frequency of
        # current element
        freq = (i + 1) * (N - i)

        # Uncomment below line to print 
        # the frequency of arr[i]

        # if frequency is odd, then
        # include it in the result
        if (freq % 2 == 1):
            res = res ^ arr[i]
    
    # return the result
    return res

# Driver Code
arr = [3, 5, 2, 4, 6]
N = len(arr) 
print(getTotalXorOfSubarrayXors(arr, N))
    
# This code is contributed
# by Smitha

// C# program to get total xor
// of all subarray xors
using System;

class GFG
{
// Returns XOR of all subarray xors
static int getTotalXorOfSubarrayXors(int []arr, 
                                     int N)
{
    // initialize result by 0 
    // as (a XOR 0 = a)
    int res = 0;

    // loop over all elements once
    for (int i = 0; i < N; i++)
    {
        // get the frequency of
        // current element
        int freq = (i + 1) * (N - i);

        // Uncomment below line to print 
        // the frequency of arr[i]
        
        // if frequency is odd, then 
        // include it in the result
        if (freq % 2 == 1)
            res = res ^ arr[i];
    }

    // return the result
    return res;
}
    
    // Driver Code
    public static void Main()
    {
    int []arr = {3, 5, 2, 4, 6};
    int N = arr.Length;

    Console.Write(getTotalXorOfSubarrayXors(arr, N));
    }
}
    

// This code is contributed by Sam007

PHP

<?php
// PHP program to get total 
// xor of all subarray xors

// Returns XOR of all subarray xors
function getTotalXorOfSubarrayXors($arr, 
                                   $N)
{
    
    // initialize result by 0
    // as (a XOR 0 = a)
    $res = 0;

    // loop over all elements once
    for ($i = 0; $i < $N; $i++)
    {
        
        // get the frequency of
        // current element
        $freq = ($i + 1) * ($N - $i);

        // if frequency is odd, then
        // include it in the result
        if ($freq % 2 == 1)
            $res = $res ^ $arr[$i];
    }

    // return the result
    return $res;
}

    // Driver Code
    $arr = array(3, 5, 2, 4, 6);
    $N = count($arr);

    echo getTotalXorOfSubarrayXors($arr, $N);

// This code is contributed by anuj_67.
?>

JavaScript

<script>

// Javascript program to get total 
// xor of all subarray xors

// Returns XOR of all subarray xors
function getTotalXorOfSubarrayXors(arr, N)
{
    // initialize result by 0
    // as (a XOR 0 = a)
    let res = 0;

    // loop over all elements once
    for (let i = 0; i < N; i++)
    {
        // get the frequency of
        // current element
        let freq = (i + 1) * (N - i);

        // Uncomment below line to print 
        // the frequency of arr[i]
        // cout << arr[i] << " " << freq << endl;

        // if frequency is odd, then
        // include it in the result
        if (freq % 2 == 1)
            res = res ^ arr[i];
    }

    // return the result
    return res;
}

// Driver Code
    let arr = [3, 5, 2, 4, 6];
    let N = arr.length;

    document.write(getTotalXorOfSubarrayXors(arr, N));

</script>

Output

Time Complexity : O(N)

Auxiliary Space: O(1)

Sum of XOR of all subarrays

Utkarsh Trivedi

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XOR of all subarray XORs | Set 1

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