Find the maximum subarray XOR in a given array

Given an array of integers. find the maximum XOR subarray value in given array. Expected time complexity O(n).

Examples:

Input: arr[] = {1, 2, 3, 4}
Output: 7
The subarray {3, 4} has maximum XOR value

Input: arr[] = {8, 1, 2, 12, 7, 6}
Output: 15
The subarray {1, 2, 12} has maximum XOR value

Input: arr[] = {4, 6}
Output: 6
The subarray {6} has maximum XOR value

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

A Simple Solution is to use two loops to find XOR of all subarrays and return the maximum.

C++

// A simple C++ program to find max subarray XOR 
#include<bits/stdc++.h> 
using namespace std; 
  
int maxSubarrayXOR(int arr[], int n) 
{ 
    int ans = INT_MIN;     // Initialize result 
  
    // Pick starting points of subarrays 
    for (int i=0; i<n; i++) 
    { 
        int curr_xor = 0; // to store xor of current subarray 
  
        // Pick ending points of subarrays starting with i 
        for (int j=i; j<n; j++) 
        { 
            curr_xor = curr_xor ^ arr[j]; 
            ans = max(ans, curr_xor); 
        } 
    } 
    return ans; 
} 
  
// Driver program to test above functions 
int main() 
{ 
    int arr[] = {8, 1, 2, 12}; 
    int n = sizeof(arr)/sizeof(arr[0]); 
    cout << "Max subarray XOR is " << maxSubarrayXOR(arr, n); 
    return 0; 
} 

Java

// A simple Java program to find max subarray XOR 
class GFG { 
    static int maxSubarrayXOR(int arr[], int n) 
    { 
        int ans = Integer.MIN_VALUE; // Initialize result 
       
        // Pick starting points of subarrays 
        for (int i=0; i<n; i++) 
        { 
                // to store xor of current subarray    
            int curr_xor = 0;  
       
            // Pick ending points of subarrays starting with i 
            for (int j=i; j<n; j++) 
            { 
                curr_xor = curr_xor ^ arr[j]; 
                ans = Math.max(ans, curr_xor); 
            } 
        } 
        return ans; 
    } 
       
    // Driver program to test above functions 
    public static void main(String args[]) 
    { 
        int arr[] = {8, 1, 2, 12}; 
        int n = arr.length; 
        System.out.println("Max subarray XOR is " +  
                                 maxSubarrayXOR(arr, n)); 
    } 
} 
//This code is contributed by Sumit Ghosh 

Python3

# A simple Python program 
# to find max subarray XOR 
  
def maxSubarrayXOR(arr,n): 
  
    ans = -2147483648     #Initialize result 
   
    # Pick starting points of subarrays 
    for i in range(n): 
          
        # to store xor of current subarray 
        curr_xor = 0 
   
        # Pick ending points of 
        # subarrays starting with i 
        for j in range(i,n): 
          
            curr_xor = curr_xor ^ arr[j] 
            ans = max(ans, curr_xor) 
          
      
    return ans 
  
  
# Driver code 
  
arr = [8, 1, 2, 12] 
n = len(arr) 
  
print("Max subarray XOR is ", 
     maxSubarrayXOR(arr, n)) 
  
# This code is contributed 
# by Anant Agarwal. 

C#

// A simple C# program to find  
// max subarray XOR 
using System; 
  
class GFG 
{ 
      
    // Function to find max subarray 
    static int maxSubarrayXOR(int []arr, int n) 
    { 
        int ans = int.MinValue;  
        // Initialize result 
      
        // Pick starting points of subarrays 
        for (int i = 0; i < n; i++) 
        { 
            // to store xor of current subarray  
            int curr_xor = 0;  
      
            // Pick ending points of  
            // subarrays starting with i 
            for (int j = i; j < n; j++) 
            { 
                curr_xor = curr_xor ^ arr[j]; 
                ans = Math.Max(ans, curr_xor); 
            } 
        } 
        return ans; 
    } 
      
    // Driver code 
    public static void Main() 
    { 
        int []arr = {8, 1, 2, 12}; 
        int n = arr.Length; 
        Console.WriteLine("Max subarray XOR is " +  
                           maxSubarrayXOR(arr, n)); 
    } 
} 
  
// This code is contributed by Sam007. 

PHP

<?php 
// A simple PHP program to 
// find max subarray XOR 
  
function maxSubarrayXOR($arr, $n) 
{ 
      
    // Initialize result 
    $ans = PHP_INT_MIN;  
  
    // Pick starting points 
    // of subarrays 
    for ($i = 0; $i < $n; $i++) 
    { 
        // to store xor of 
        // current subarray 
        $curr_xor = 0;  
  
        // Pick ending points of  
        // subarrays starting with i 
        for ($j = $i; $j < $n; $j++) 
        { 
            $curr_xor = $curr_xor ^ $arr[$j]; 
            $ans = max($ans, $curr_xor); 
        } 
    } 
    return $ans; 
} 
  
    // Driver Code 
    $arr = array(8, 1, 2, 12); 
    $n = count($arr); 
    echo "Max subarray XOR is " 
         , maxSubarrayXOR($arr, $n); 
           
// This code is contributed by anuj_67. 
?> 

Output:

Max subarray XOR is 15

Time Complexity of above solution is O(n²).

An Efficient Solution can solve the above problem in O(n) time under the assumption that integers take fixed number of bits to store. The idea is to use Trie Data Structure. Below is algorithm.

1) Create an empty Trie.  Every node of Trie is going to 
   contain two children, for 0 and 1 value of bit.
2) Initialize pre_xor = 0 and insert into the Trie.
3) Initialize result = minus infinite
4) Traverse the given array and do following for every 
   array element arr[i].
       a) pre_xor  = pre_xor  ^ arr[i]
          pre_xor now contains xor of elements from 
          arr[0] to arr[i].
       b) Query the maximum xor value ending with arr[i] 
          from Trie.
       c) Update result if the value obtained in step 
          4.b is more than current value of result.

How does 4.b work?
We can observe from above algorithm that we build a Trie that contains XOR of all prefixes of given array. To find the maximum XOR subarray ending with arr[i], there may be two cases.
i) The prefix itself has the maximum XOR value ending with arr[i]. For example if i=2 in {8, 2, 1, 12}, then the maximum subarray xor ending with arr[2] is the whole prefix.
ii) We need to remove some prefix (ending at index from 0 to i-1). For example if i=3 in {8, 2, 1, 12}, then the maximum subarray xor ending with arr[3] starts with arr[1] and we need to remove arr[0].

To find the prefix to be removed, we find the entry in Trie that has maximum XOR value with current prefix. If we do XOR of such previous prefix with current prefix, we get the maximum XOR value ending with arr[i].
If there is no prefix to be removed (case i), then we return 0 (that’s why we inserted 0 in Trie).

Below is the implementation of above idea :

C++

// C++ program for a Trie based O(n) solution to find max 
// subarray XOR 
#include<bits/stdc++.h> 
using namespace std; 
  
// Assumed int size 
#define INT_SIZE 32 
  
// A Trie Node 
struct TrieNode 
{ 
    int value;  // Only used in leaf nodes 
    TrieNode *arr[2]; 
}; 
  
// Utility function to create a Trie node 
TrieNode *newNode() 
{ 
    TrieNode *temp = new TrieNode; 
    temp->value = 0; 
    temp->arr[0] = temp->arr[1] = NULL; 
    return temp; 
} 
  
// Inserts pre_xor to trie with given root 
void insert(TrieNode *root, int pre_xor) 
{ 
    TrieNode *temp = root; 
  
    // Start from the msb, insert all bits of 
    // pre_xor into Trie 
    for (int i=INT_SIZE-1; i>=0; i--) 
    { 
        // Find current bit in given prefix 
        bool val = pre_xor & (1<<i); 
  
        // Create a new node if needed 
        if (temp->arr[val] == NULL) 
            temp->arr[val] = newNode(); 
  
        temp = temp->arr[val]; 
    } 
  
    // Store value at leaf node 
    temp->value = pre_xor; 
} 
  
// Finds the maximum XOR ending with last number in 
// prefix XOR 'pre_xor' and returns the XOR of this maximum 
// with pre_xor which is maximum XOR ending with last element 
// of pre_xor. 
int query(TrieNode *root, int pre_xor) 
{ 
    TrieNode *temp = root; 
    for (int i=INT_SIZE-1; i>=0; i--) 
    { 
        // Find current bit in given prefix 
        bool val = pre_xor & (1<<i); 
  
        // Traverse Trie, first look for a 
        // prefix that has opposite bit 
        if (temp->arr[1-val]!=NULL) 
            temp = temp->arr[1-val]; 
  
        // If there is no prefix with opposite 
        // bit, then look for same bit. 
        else if (temp->arr[val] != NULL) 
            temp = temp->arr[val]; 
    } 
    return pre_xor^(temp->value); 
} 
  
// Returns maximum XOR value of a subarray in arr[0..n-1] 
int maxSubarrayXOR(int arr[], int n) 
{ 
    // Create a Trie and insert 0 into it 
    TrieNode *root = newNode(); 
    insert(root, 0); 
  
    // Initialize answer and xor of current prefix 
    int result = INT_MIN, pre_xor =0; 
  
    // Traverse all input array element 
    for (int i=0; i<n; i++) 
    { 
        // update current prefix xor and insert it into Trie 
        pre_xor = pre_xor^arr[i]; 
        insert(root, pre_xor); 
  
        // Query for current prefix xor in Trie and update 
        // result if required 
        result = max(result, query(root, pre_xor)); 
    } 
    return result; 
} 
  
// Driver program to test above functions 
int main() 
{ 
    int arr[] = {8, 1, 2, 12}; 
    int n = sizeof(arr)/sizeof(arr[0]); 
    cout << "Max subarray XOR is " << maxSubarrayXOR(arr, n); 
    return 0; 
} 

Java

// Java program for a Trie based O(n) solution to  
// find max subarray XOR 
class GFG  
{ 
    // Assumed int size 
    static final int INT_SIZE = 32; 
       
    // A Trie Node 
    static class TrieNode 
    { 
        int value;  // Only used in leaf nodes 
        TrieNode[] arr =  new TrieNode[2]; 
        public TrieNode() { 
            value = 0; 
            arr[0] = null; 
            arr[1] = null; 
        } 
    } 
    static TrieNode root; 
      
    // Inserts pre_xor to trie with given root 
    static void insert(int pre_xor) 
    { 
        TrieNode temp = root; 
       
        // Start from the msb, insert all bits of 
        // pre_xor into Trie 
        for (int i=INT_SIZE-1; i>=0; i--) 
        { 
            // Find current bit in given prefix 
            int val = (pre_xor & (1<<i)) >=1 ? 1 : 0; 
       
            // Create a new node if needed 
            if (temp.arr[val] == null) 
                temp.arr[val] = new TrieNode(); 
       
            temp = temp.arr[val]; 
        } 
       
        // Store value at leaf node 
        temp.value = pre_xor; 
    } 
       
    // Finds the maximum XOR ending with last number in 
    // prefix XOR 'pre_xor' and returns the XOR of this  
    // maximum with pre_xor which is maximum XOR ending  
    // with last element of pre_xor. 
    static int query(int pre_xor) 
    { 
        TrieNode temp = root; 
        for (int i=INT_SIZE-1; i>=0; i--) 
        { 
            // Find current bit in given prefix 
            int val = (pre_xor & (1<<i)) >= 1 ? 1 : 0; 
       
            // Traverse Trie, first look for a 
            // prefix that has opposite bit 
            if (temp.arr[1-val] != null) 
                temp = temp.arr[1-val]; 
       
            // If there is no prefix with opposite 
            // bit, then look for same bit. 
            else if (temp.arr[val] != null) 
                temp = temp.arr[val]; 
        } 
        return pre_xor^(temp.value); 
    } 
       
    // Returns maximum XOR value of a subarray in  
        // arr[0..n-1] 
    static int maxSubarrayXOR(int arr[], int n) 
    { 
        // Create a Trie and insert 0 into it 
        root = new TrieNode(); 
        insert(0); 
       
        // Initialize answer and xor of current prefix 
        int result = Integer.MIN_VALUE; 
        int pre_xor =0; 
       
        // Traverse all input array element 
        for (int i=0; i<n; i++) 
        { 
            // update current prefix xor and insert it  
                // into Trie 
            pre_xor = pre_xor^arr[i]; 
            insert(pre_xor); 
       
            // Query for current prefix xor in Trie and  
            // update result if required 
            result = Math.max(result, query(pre_xor)); 
  
        } 
        return result; 
    } 
       
    // Driver program to test above functions 
    public static void main(String args[]) 
    { 
        int arr[] = {8, 1, 2, 12}; 
        int n = arr.length; 
        System.out.println("Max subarray XOR is " +  
                                 maxSubarrayXOR(arr, n)); 
    } 
} 
// This code is contributed by Sumit Ghosh 

Python3

"""Python implementation for a Trie based solution 
to find max subArray XOR"""
  
"""structure of Trie Node"""
class Node: 
  
    def __init__(self, data): 
  
        self.data = data 
        self.left = None # left node for 0 
        self.right = None # right node for 1 
  
""" class for implementing Trie """
  
class Trie: 
  
    def __init__(self): 
  
        self.root = Node(0) 
  
    """insert pre_xor to trie with given root"""
    def insert(self, pre_xor): 
  
        self.temp = self.root 
  
        """start from msb, insert all bits of pre_xor 
        into the Trie"""
        for i in range(31, -1, -1): 
  
            """Find current bit in prefix sum"""
            val = pre_xor & (1<<i) 
  
            if val : 
                """create new node if needed"""
                if not self.temp.right: 
                    self.temp.right = Node(0) 
                self.temp = self.temp.right 
  
            if not val: 
                """create new node if needed"""
                if not self.temp.left: 
                    self.temp.left = Node(0) 
                self.temp = self.temp.left 
  
        """store value at leaf node"""
        self.temp.data = pre_xor 
  
    """find the maximum xor ending with last number 
        in prefix XOR and return the XOR of this"""
    def query(self, xor): 
  
        self.temp = self.root 
  
        for i in range(31, -1, -1): 
  
            """find the current bit in prefix xor"""
            val = xor & (1<<i) 
  
            """Traverse the trie, first look for opposite bit 
                and then look for same bit"""
            if val: 
                if self.temp.left: 
                    self.temp = self.temp.left 
                elif self.temp.right: 
                    self.temp = self.temp.right 
            else: 
                if self.temp.right: 
                    self.temp = self.temp.right 
                elif self.temp.left: 
                    self.temp = self.temp.left 
  
        return xor ^ self.temp.data 
  
    """returns maximum XOR value of subarray"""
    def maxSubArrayXOR(self, n, Arr): 
  
        """insert 0 in the trie"""
        self.insert(0) 
  
        """initialize result and pre_xor"""
        result = -float('inf') 
        pre_xor = 0
  
        """traverse all input array element"""
        for i in range(n): 
  
            """update current prefix xor and insert it into Trie"""
            pre_xor = pre_xor ^ Arr[i] 
            self.insert(pre_xor) 
  
            """Query for current prefix xor in Trie and update result"""
            result = max(result, self.query(pre_xor)) 
  
        return result 
  
"""Driver program to test above functions"""
if __name__ == "__main__": 
  
    Arr = [8, 1, 2, 12] 
    n = len(Arr) 
    trie = Trie() 
    print(trie.maxSubArrayXOR(n, Arr)) 
  
# This code is contributed by chaudhary_19 

C#

using System; 
  
// C# program for a Trie based O(n) solution to   
// find max subarray XOR  
public class GFG 
{ 
    // Assumed int size  
    public const int INT_SIZE = 32; 
  
    // A Trie Node  
    public class TrieNode 
    { 
        public int value; // Only used in leaf nodes 
        public TrieNode[] arr = new TrieNode[2]; 
        public TrieNode() 
        { 
            value = 0; 
            arr[0] = null; 
            arr[1] = null; 
        } 
    } 
    public static TrieNode root; 
  
    // Inserts pre_xor to trie with given root  
    public static void insert(int pre_xor) 
    { 
        TrieNode temp = root; 
  
        // Start from the msb, insert all bits of  
        // pre_xor into Trie  
        for (int i = INT_SIZE-1; i >= 0; i--) 
        { 
            // Find current bit in given prefix  
            int val = (pre_xor & (1 << i)) >= 1 ? 1 : 0; 
  
            // Create a new node if needed  
            if (temp.arr[val] == null) 
            { 
                temp.arr[val] = new TrieNode(); 
            } 
  
            temp = temp.arr[val]; 
        } 
  
        // Store value at leaf node  
        temp.value = pre_xor; 
    } 
  
    // Finds the maximum XOR ending with last number in  
    // prefix XOR 'pre_xor' and returns the XOR of this   
    // maximum with pre_xor which is maximum XOR ending   
    // with last element of pre_xor.  
    public static int query(int pre_xor) 
    { 
        TrieNode temp = root; 
        for (int i = INT_SIZE-1; i >= 0; i--) 
        { 
            // Find current bit in given prefix  
            int val = (pre_xor & (1 << i)) >= 1 ? 1 : 0; 
  
            // Traverse Trie, first look for a  
            // prefix that has opposite bit  
            if (temp.arr[1 - val] != null) 
            { 
                temp = temp.arr[1 - val]; 
            } 
  
            // If there is no prefix with opposite  
            // bit, then look for same bit.  
            else if (temp.arr[val] != null) 
            { 
                temp = temp.arr[val]; 
            } 
        } 
        return pre_xor ^ (temp.value); 
    } 
  
    // Returns maximum XOR value of a subarray in   
        // arr[0..n-1]  
    public static int maxSubarrayXOR(int[] arr, int n) 
    { 
        // Create a Trie and insert 0 into it  
        root = new TrieNode(); 
        insert(0); 
  
        // Initialize answer and xor of current prefix  
        int result = int.MinValue; 
        int pre_xor = 0; 
  
        // Traverse all input array element  
        for (int i = 0; i < n; i++) 
        { 
            // update current prefix xor and insert it   
                // into Trie  
            pre_xor = pre_xor ^ arr[i]; 
            insert(pre_xor); 
  
            // Query for current prefix xor in Trie and   
            // update result if required  
            result = Math.Max(result, query(pre_xor)); 
  
        } 
        return result; 
    } 
  
    // Driver program to test above functions  
    public static void Main(string[] args) 
    { 
        int[] arr = new int[] {8, 1, 2, 12}; 
        int n = arr.Length; 
        Console.WriteLine("Max subarray XOR is " + maxSubarrayXOR(arr, n)); 
    } 
} 
  
  // This code is contributed by Shrikant13 

Output:

Max subarray XOR is 15

Exercise: Extend the above solution so that it also prints starting and ending indexes of subarray with maximum value (Hint: we can add one more field to Trie node to achieve this)

This article is contributed by Romil Punetha. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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My Personal Notes

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

C++

Java

Python3

C#

PHP

C++

Java

Python3

C#

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