#3277

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximum XOR Score Subarray Queries

Solve the Maximum XOR Score Subarray Queries problem using state transition dynamic programming for optimal subarray computation.

array dynamic programming

Problem Statement

You are given an integer array nums of length n and a list of queries, where each query specifies a subarray from index li to ri. For each query, determine the subarray that produces the maximum XOR score within the specified range.

The XOR score of an array is computed by applying repeated XOR operations over its elements until only one number remains. Return an array of integers representing the maximum XOR scores for each query, highlighting the need for a state transition dynamic programming approach to handle overlapping subarrays efficiently.

Examples

Example 1

Input: nums = [2,8,4,32,16,1], queries = [[0,2],[1,4],[0,5]]

Output: [12,60,60]

In the first query, nums[0..2] has 6 subarrays [2] , [8] , [4] , [2, 8] , [8, 4] , and [2, 8, 4] each with a respective XOR score of 2, 8, 4, 10, 12, and 6. The answer for the query is 12, the largest of all XOR scores. In the second query, the subarray of nums[1..4] with the largest XOR score is nums[1..4] with a score of 60. In the third query, the subarray of nums[0..5] with the largest XOR score is nums[1..4] with a score of 60.

Example 2

Input: nums = [0,7,3,2,8,5,1], queries = [[0,3],[1,5],[2,4],[2,6],[5,6]]

Output: [7,14,11,14,5]

Constraints

1 <= n == nums.length <= 2000
0 <= nums[i] <= 231 - 1
1 <= q == queries.length <= 105
queries[i].length == 2
queries[i] = [li, ri]
0 <= li <= ri <= n - 1

Solution Approach

Precompute Subarray XOR Scores

Build a 2D DP table where dp[i][j] stores the XOR score of nums[i..j]. This allows O(1) retrieval of any subarray's XOR score and avoids redundant recalculation across multiple queries.

Process Queries Using DP States

For each query [li, ri], iterate through all subarrays within the range using precomputed DP values. Track the maximum XOR score by comparing the DP results, applying the state transition dynamic programming principle to ensure efficient updates.

Optimize Time with XOR Properties

Leverage the property that XOR is associative and commutative to reduce computation. By combining prefix XOR with DP, you can quickly calculate scores for subarrays without recalculating each element, which is critical for large input sizes.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity depends on precomputing the DP table, which is O(n^2), and answering q queries using precomputed states is O(1) per subarray access. Space complexity is O(n^2) for the DP table, though prefix XOR optimizations can reduce space to O(n).

What Interviewers Usually Probe

Checks if you precompute subarray XOR scores to save repeated calculations.
Looks for use of dynamic programming to handle overlapping subarray states efficiently.
Wants recognition of XOR properties to optimize computation for multiple queries.

Common Pitfalls or Variants

Common pitfalls

Attempting to compute XOR scores on the fly without DP, leading to TLE for large n.
Misapplying the state transition logic and incorrectly combining subarrays.
Ignoring the associative property of XOR, which can simplify computation.

Follow-up variants

Maximum XOR score for all subarrays without queries, returning a single global max.
Queries asking for minimum XOR score subarrays instead of maximum.
Applying the same DP approach for XOR with other operations like sum or AND.

FAQ

What is the main pattern used in Maximum XOR Score Subarray Queries?

The core pattern is state transition dynamic programming, which precomputes subarray XOR scores and updates efficiently across queries.

Can prefix XOR reduce space complexity in this problem?

Yes, using prefix XOR allows O(n) space while still computing subarray XOR in O(1) time, avoiding a full O(n^2) DP table.

Why can't we compute XOR on the fly for each query?

Computing XOR on the fly leads to redundant calculations and results in TLE for large arrays due to the quadratic number of subarrays.

Does this problem require handling overlapping subarrays?

Yes, state transition dynamic programming handles overlapping subarrays efficiently by reusing previously computed XOR scores.

What is a common mistake when implementing this solution?

A common mistake is ignoring XOR's associative property, leading to incorrect accumulation when combining subarrays or misapplying DP updates.

terminal

Solution

Solution 1: Dynamic Programming

We define $f[i][j]$ to represent the XOR value of $\textit{nums}[i..j]$. According to the problem description, we can derive the state transition equation:

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class Solution:
    def maximumSubarrayXor(
        self, nums: List[int], queries: List[List[int]]
    ) -> List[int]:
        n = len(nums)
        f = [[0] * n for _ in range(n)]
        g = [[0] * n for _ in range(n)]
        for i in range(n - 1, -1, -1):
            f[i][i] = g[i][i] = nums[i]
            for j in range(i + 1, n):
                f[i][j] = f[i][j - 1] ^ f[i + 1][j]
                g[i][j] = max(f[i][j], g[i][j - 1], g[i + 1][j])
        return [g[l][r] for l, r in queries]

Continue Practicing

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