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Finding the Maximum Subarray XOR in a Given Array: A Comprehensive Tutorial

This tutorial will guide you through the process of finding the maximum XOR (exclusive OR) value among all possible subarrays in a given array. We'll explore different approaches, starting with a naive solution and progressing to an efficient solution using a Trie data structure. We'll also provide detailed code examples in Python and C++.

*Understanding the Problem:*

Given an array of integers, `arr`, our goal is to find a contiguous subarray (a sequence of consecutive elements) such that the XOR of all elements within that subarray is maximized. The XOR operation is a bitwise operation that returns 1 if the corresponding bits are different and 0 if they are the same.

*Example:*

Let `arr = [8, 1, 2, 15, 10, 5]`

Possible subarrays and their XOR values:

`[8]`: 8
`[1]`: 1
`[2]`: 2
`[15]`: 15
`[10]`: 10
`[5]`: 5
`[8, 1]`: 9
`[1, 2]`: 3
`[2, 15]`: 13
`[15, 10]`: 5
`[10, 5]`: 15
`[8, 1, 2]`: 11
`[1, 2, 15]`: 14
`[2, 15, 10]`: 7
`[15, 10, 5]`: 0
`[8, 1, 2, 15]`: 10
`[1, 2, 15, 10]`: 6
`[2, 15, 10, 5]`: 5
`[8, 1, 2, 15, 10]`: 12
`[1, 2, 15, 10, 5]`: 1
`[8, 1, 2, 15, 10, 5]`: 13

In this example, the maximum XOR value is 15, achieved by the subarrays `[15]` and `[10, 5]`.

*1. Naive Approach (Brute Force):*

The simplest approach is to iterate through all possible subarrays and calculate their XOR values. We keep track of the maximum XOR encountered so far.

*Algorithm:*

1. Initialize `max_xor` to 0.
2. Iterate through all possible starting indices `i` from 0 to `n-1` (where `n` is the length of the array).
3. For each starting index `i`, iterate through all possible ending indices `j` from `i` to `n-1`.
4. Calculate the XOR of the subarray `arr[i...j]`.
5. Update `max_xor` with the maximum value between `max_xor` and the calculated XOR.
6. Return `max_xor`.

*Python Code:*



**C++ Code ...

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