Maximum XOR Score Subarray Queries

Problem Description

You are given an array nums of n integers, and a 2D integer array queries of size q, where queries[i] = [l_i, r_i]. For each query, you must find the maximum XOR score of any subarray of nums[l_i..r_i]. The XOR score of an array a is found by repeatedly applying the operation of simultaneously replacing a[i] with a[i] XOR a[i + 1] for all indices i except the last one, until only one element remains. Return an array answer of size q where answer[i] is the answer to query i.

Key Insights

• The XOR score can be computed through the XOR operation which combines elements in a specific manner.
• A brute force approach would involve calculating the XOR score for all possible subarrays for each query, leading to high time complexity.
• Efficient calculation can be achieved using dynamic programming and/or segment trees to store and retrieve maximum XOR scores efficiently.

Space and Time Complexity

Time Complexity: O(n^2) for preprocessing (to calculate all subarray XORs) and O(1) for each query if using a precomputed approach. Overall O(n^2 + q) where q is the number of queries.

Space Complexity: O(n^2) for storing XOR scores of all subarrays.

Solution

To solve the problem, we can use a dynamic programming approach to preprocess the maximum XOR scores for all possible subarrays. We then store these scores in a 2D array where the entry at dp[i][j] contains the maximum XOR score for the subarray nums[i..j]. For each query, we simply read the precomputed maximum score from this array.

1. Dynamic Programming Table Construction:

   • Create a 2D array dp of size n x n.
   • Iterate through all possible starting indices of subarrays.
   • For each starting index, compute the XOR score for all ending indices and store the maximum in dp.

2. Answer Queries:

   • For each query [l, r], retrieve the precomputed maximum XOR score from dp[l][r].

Code Solutions