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Problem Statement

Given an array of integers, the task is to find the maximum XOR value that can be obtained by performing XOR operations on a suffix of the array. You can choose any starting index and calculate the XOR of all elements from that index to the end of the array. This process can be repeated multiple times with the goal of maximizing the XOR value.

Solution Overview

The key to solving this problem efficiently lies in utilizing a Trie data structure to store the binary representations of the XOR prefixes of the array. A Trie allows us to efficiently search for the binary number that, when XORed with the current prefix, yields the maximum result.

Algorithm Steps

1. Trie Structure: Define a Trie node structure that can store two children (for binary digits 0 and 1) and a value representing the number formed by the path to that node.
2. Insert Function: Implement a function to insert the binary representation of a number into the Trie. For each bit of the number, create a new path if it doesn't already exist.
3. Find Maximum XOR Function: Implement a function that traverses the Trie to find the binary number that, when XORed with the given number, gives the maximum result. This involves choosing the opposite bit at each step of the traversal to maximize the XOR value.
4. Maximum XOR Calculation: Initialize the Trie and insert a 0 (to represent the XOR of an empty prefix). Then, for each element in the array, calculate the current prefix XOR and insert it into the Trie. Use the find function to determine the maximum XOR value possible with the current prefix XOR. Update the maximum result accordingly.
5. Return the Maximum Result: After processing all elements in the array, the maximum stored result is the answer.

Time Complexity Analysis

The time complexity of this algorithm is O(N), where N is the length of the array. This efficiency is achieved because each number is processed in linear time relative to its bit length (which is constant), and the Trie operations (insert and search) are performed in O(1) time for each bit of the number.

Conclusion

This approach combines bit manipulation with a Trie data structure to solve the problem of finding the maximum XOR suffix in linear time. It demonstrates the power of combining algorithmic techniques to tackle complex problems efficiently.

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