Number of triplets in array having subarray xor equal

Given an array of integers Arr. The task is to count the number of triplets (i, j, k) such that A_i ^ A_i+1 ^ A_i+2 ^ …. ^ A_j-1 = A_j ^ A_j+1 ^ A_j+2 ^ ….. ^ A_k, and 0 < (i, j, k) < N , where N is the size of the array.

Where ^ is the bitwise xor of two numbers.

Examples:

Input: Arr = {5, 2, 7}
Output: 2

The triplets are (0, 2, 2) since 5 ^ 2 = 7 and (0, 1, 2) since 2 ^ 7 = 5.

Input: Arr = {1, 2, 3, 4, 5}
Output: 5

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Bruteforce Approach: A simple solution is to run three nested loops iterating through all possible values of i, j and k and then another loop to calculate the xor of both first and second subarray.
The time complexity of this solution is O(N⁴).

C++

// A simple C++ program to find Number of triplets 
// in array having subarray xor equal 
#include <bits/stdc++.h> 
using namespace std; 
  
// Function to return the count 
int xor_triplet(int arr[], int n) 
{ 
    // Initialse result 
    int ans = 0; 
  
    // Pick 1st element of the triplet 
    for (int i = 0; i < n; i++) { 
  
        // Pick 2nd element of the triplet 
        for (int j = i + 1; j < n; j++) { 
  
            // Pick 3rd element of the triplet 
            for (int k = j; k < n; k++) { 
  
                int xor1 = 0, xor2 = 0; 
  
                // Taking xor in the 
                // first subarray 
                for (int x = i; x < j; x++) { 
                    xor1 ^= arr[x]; 
                } 
  
                // Taking xor in the 
                // second subarray 
                for (int x = j; x <= k; x++) { 
                    xor2 ^= arr[x]; 
                } 
  
                // If both xor is equal 
                if (xor1 == xor2) { 
                    ans++; 
                } 
            } 
        } 
    } 
    return ans; 
} 
  
// Driver Code 
int main() 
{ 
    int arr[] = { 1, 2, 3, 4, 5 }; 
    int n = sizeof(arr) / sizeof(arr[0]); 
  
    // Function Calling 
    cout << xor_triplet(arr, n); 
    return 0; 
} 

Java

// Java program to find Number of triplets  
// in array having subarray xor equal  
class GFG 
{ 
  
// Function to return the count  
static int xor_triplet(int arr[], int n)  
{  
    // Initialse result  
    int ans = 0;  
  
    // Pick 1st element of the triplet  
    for (int i = 0; i < n; i++)  
    {  
  
        // Pick 2nd element of the triplet  
        for (int j = i + 1; j < n; j++) 
        {  
  
            // Pick 3rd element of the triplet  
            for (int k = j; k < n; k++) 
            {  
                int xor1 = 0, xor2 = 0;  
  
                // Taking xor in the  
                // first subarray  
                for (int x = i; x < j; x++)  
                {  
                    xor1 ^= arr[x];  
                }  
  
                // Taking xor in the  
                // second subarray  
                for (int x = j; x <= k; x++) 
                {  
                    xor2 ^= arr[x];  
                }  
  
                // If both xor is equal  
                if (xor1 == xor2)  
                {  
                    ans++;  
                }  
            }  
        }  
    }  
    return ans;  
}  
  
// Driver Code  
public static void main (String[] args)  
{  
    int arr[] = { 1, 2, 3, 4, 5 };  
    int n = arr.length;  
  
    // Function Calling  
    System.out.println(xor_triplet(arr, n));  
}  
} 
  
// This code is contributed by AnkitRai01 

Python3

# A simple python3 program to find Number of triplets 
# in array having subarray xor equal 
  
# Function to return the count 
def xor_triplet(arr, n): 
      
    # Initialse result 
    ans = 0; 
  
    # Pick 1st element of the triplet 
    for i in range(n): 
  
        # Pick 2nd element of the triplet 
        for j in range(i + 1, n): 
  
            # Pick 3rd element of the triplet 
            for k in range(j, n): 
  
                xor1 = 0; xor2 = 0; 
  
                # Taking xor in the 
                # first subarray 
                for x in range(i, j): 
                    xor1 ^= arr[x]; 
  
                # Taking xor in the 
                # second subarray 
                for x in range(j, k + 1): 
                    xor2 ^= arr[x]; 
  
                # If both xor is equal 
                if (xor1 == xor2): 
                    ans += 1; 
  
    return ans; 
  
# Driver Code 
if __name__ == '__main__': 
    arr = [1, 2, 3, 4, 5]; 
    n = len(arr); 
  
    # Function Calling 
    print(xor_triplet(arr, n)); 
      
# This code is contributed by PrinciRaj1992 

C#

// C# program to find Number of triplets  
// in array having subarray xor equal  
using System; 
      
class GFG 
{ 
  
// Function to return the count  
static int xor_triplet(int []arr, int n)  
{  
    // Initialse result  
    int ans = 0;  
  
    // Pick 1st element of the triplet  
    for (int i = 0; i < n; i++)  
    {  
  
        // Pick 2nd element of the triplet  
        for (int j = i + 1; j < n; j++) 
        {  
  
            // Pick 3rd element of the triplet  
            for (int k = j; k < n; k++) 
            {  
                int xor1 = 0, xor2 = 0;  
  
                // Taking xor in the  
                // first subarray  
                for (int x = i; x < j; x++)  
                {  
                    xor1 ^= arr[x];  
                }  
  
                // Taking xor in the  
                // second subarray  
                for (int x = j; x <= k; x++) 
                {  
                    xor2 ^= arr[x];  
                }  
  
                // If both xor is equal  
                if (xor1 == xor2)  
                {  
                    ans++;  
                }  
            }  
        }  
    }  
    return ans;  
}  
  
// Driver Code  
public static void Main (String[] args)  
{  
    int []arr = { 1, 2, 3, 4, 5 };  
    int n = arr.Length;  
  
    // Function Calling  
    Console.WriteLine(xor_triplet(arr, n));  
}  
} 
  
// This code is contributed by PrinciRaj1992 

Output:

Efficient Approach:

An efficient solution is to use Trie.
One observatoion is that if the subarray from i to k have A_i ^ A_i+1 ^ …. ^ A_k = 0 then we can select any j such that i < j <= k and that will always satisfy our condition.
This observation will be used to calculate the number of triplets.
We will take the cumulative xor of the elements in the array and check that this xor value exists in the Trie or not.
1. If the xor already exists in the Trie then we have encounted a subarray having 0 xor and count all the triplets.
2. Else push the value of xor into the Trie.

Below is the implementation of the above approach:

C++

// C++ trie based program to find the Number of 
// triplets in array having subarray xor equal 
#include <bits/stdc++.h> 
using namespace std; 
  
// maximum number of bits 
// in an integer <= 1e9 
#define lg 31 
  
// Structure of a Trie Node 
struct TrieNode { 
  
    // [0] index is bit 0 
    // and [1] index is bit 1 
    TrieNode* children[2]; 
  
    // Sum of indexes 
    // inserted at at a node 
    int sum_of_indexes; 
  
    // Number of indexes 
    // inserted at a node 
    int number_of_indexes; 
  
    // Constructor to initialize 
    // a newly created node 
    TrieNode() 
    { 
  
        this->children[0] = nullptr; 
        this->children[1] = nullptr; 
        this->sum_of_indexes = 0; 
        this->number_of_indexes = 0; 
    } 
}; 
  
// Function to insert curr_xor 
// into the trie 
void insert(TrieNode* node, 
            int num, 
            int index) 
{ 
  
    // Iterate from the 31st bit 
    // to the 0th bit of curr_xor 
    // number 
    for (int bits = lg; bits >= 0; bits--) { 
  
        // Check if the current 
        // bit is set or not 
        int curr_bit = (num >> bits) & 1; 
  
        // If this node isn't already 
        // present in the trie structure 
        // insert it into the trie. 
        if (node->children[curr_bit] 
            == nullptr) { 
            node->children[curr_bit] 
                = new TrieNode(); 
        } 
  
        node = node->children[curr_bit]; 
    } 
  
    // Increase the sum of 
    // indexes by the current 
    // index value 
    node->sum_of_indexes += index; 
  
    // Increase the number 
    // of indexes by 1 
    node->number_of_indexes++; 
} 
  
// Function to check if curr_xor 
// is present in trie or not 
int query(TrieNode* node, 
          int num, 
          int index) 
{ 
  
    // Iterate from the 31st bit 
    // to the 0th bit of curr_xor number 
    for (int bits = lg; bits >= 0; bits--) { 
  
        // Check if the current bit 
        // is set or not 
        int curr_bit = (num >> bits) & 1; 
  
        // If this node isn't already 
        // present in the trie structure 
        // that means no sub array till 
        // current index has 0 xor so 
        // return 0 
        if (node->children[curr_bit] 
            == nullptr) { 
            return 0; 
        } 
  
        node = node->children[curr_bit]; 
    } 
  
    // Calculate the number of index 
    // inserted at final node 
    int sz = node->number_of_indexes; 
  
    // Calculate the sum of index 
    // inserted at final node 
    int sum = node->sum_of_indexes; 
  
    int ans = (sz * index) - (sum); 
  
    return ans; 
} 
  
// Function to return the count of 
// valid triplets 
int no_of_triplets(int arr[], int n) 
{ 
  
    // To store cumulative xor 
    int curr_xor = 0; 
  
    int number_of_triplets = 0; 
  
    // The root of the trie 
    TrieNode* root = new TrieNode(); 
  
    for (int i = 0; i < n; i++) { 
  
        int x = arr[i]; 
  
        // Insert the curr_xor in the trie 
        insert(root, curr_xor, i); 
  
        // Update the cumulative xor 
        curr_xor ^= x; 
  
        // Check if the cumulative xor 
        // is present in the trie or not 
        // if present then add 
        // (sz * index) - sum 
        number_of_triplets 
            += query(root, curr_xor, i); 
    } 
  
    return number_of_triplets; 
} 
  
// Driver Code 
int main() 
{ 
    // Given array 
    int arr[] = { 5, 2, 7 }; 
    int n = sizeof(arr) / sizeof(arr[0]); 
  
    cout << no_of_triplets(arr, n); 
  
    return 0; 
} 

Java

// Java trie based program to find the Number of 
// triplets in array having subarray xor equal 
class GFG 
{ 
      
// maximum number of bits 
// in an integer <= 1e9 
static int lg = 31; 
  
// Structure of a Trie Node 
static class TrieNode 
{ 
  
    // [0] index is bit 0 
    // and [1] index is bit 1 
    TrieNode children[]; 
  
    // Sum of indexes 
    // inserted at at a node 
    int sum_of_indexes; 
  
    // Number of indexes 
    // inserted at a node 
    int number_of_indexes; 
  
    // Constructor to initialize 
    // a newly created node 
    TrieNode() 
    { 
        children = new TrieNode[2]; 
        this.children[0] = null; 
        this.children[1] = null; 
        this.sum_of_indexes = 0; 
        this.number_of_indexes = 0; 
    } 
}; 
  
// Function to insert curr_xor 
// into the trie 
static void insert(TrieNode node, 
                    int num, 
                    int index) 
{ 
  
    // Iterate from the 31st bit 
    // to the 0th bit of curr_xor 
    // number 
    for (int bits = lg; bits >= 0; bits--)  
    { 
  
        // Check if the current 
        // bit is set or not 
        int curr_bit = (num >> bits) & 1; 
  
        // If this node isn't already 
        // present in the trie structure 
        // insert it into the trie. 
        if (node.children[curr_bit] 
            == null) 
        { 
            node.children[curr_bit] 
                = new TrieNode(); 
        } 
  
        node = node.children[curr_bit]; 
    } 
  
    // Increase the sum of 
    // indexes by the current 
    // index value 
    node.sum_of_indexes += index; 
  
    // Increase the number 
    // of indexes by 1 
    node.number_of_indexes++; 
} 
  
// Function to check if curr_xor 
// is present in trie or not 
static int query(TrieNode node, 
                  int num, 
                  int index) 
{ 
  
    // Iterate from the 31st bit 
    // to the 0th bit of curr_xor number 
    for (int bits = lg; bits >= 0; bits--) 
    { 
  
        // Check if the current bit 
        // is set or not 
        int curr_bit = (num >> bits) & 1; 
  
        // If this node isn't already 
        // present in the trie structure 
        // that means no sub array till 
        // current index has 0 xor so 
        // return 0 
        if (node.children[curr_bit] 
            == null) 
        { 
            return 0; 
        } 
  
        node = node.children[curr_bit]; 
    } 
  
    // Calculate the number of index 
    // inserted at final node 
    int sz = node.number_of_indexes; 
  
    // Calculate the sum of index 
    // inserted at final node 
    int sum = node.sum_of_indexes; 
  
    int ans = (sz * index) - (sum); 
  
    return ans; 
} 
  
// Function to return the count of 
// valid triplets 
static int no_of_triplets(int arr[], int n) 
{ 
  
    // To store cumulative xor 
    int curr_xor = 0; 
  
    int number_of_triplets = 0; 
  
    // The root of the trie 
    TrieNode root = new TrieNode(); 
  
    for (int i = 0; i < n; i++) 
    { 
  
        int x = arr[i]; 
  
        // Insert the curr_xor in the trie 
        insert(root, curr_xor, i); 
  
        // Update the cumulative xor 
        curr_xor ^= x; 
  
        // Check if the cumulative xor 
        // is present in the trie or not 
        // if present then add 
        // (sz * index) - sum 
        number_of_triplets 
            += query(root, curr_xor, i); 
    } 
  
    return number_of_triplets; 
} 
  
// Driver Code 
public static void main(String args[]) 
{ 
    // Given array 
    int arr[] = { 5, 2, 7 }; 
    int n = arr.length; 
  
    System.out.println(no_of_triplets(arr, n)); 
}  
} 
  
// This code is contributed by Arnab Kundu 

Output:

Time complexity: O(31*N)

My Personal Notes

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#

C++

Java

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