You are given an integer array nums and a positive integer k.
The value of a sequence seq of size 2 * x is defined as:
(seq[0] OR seq[1] OR ... OR seq[x - 1]) XOR (seq[x] OR seq[x + 1] OR ... OR seq[2 * x - 1]).Return the maximum value of any subsequence of nums having size 2 * k.
Example 1:
Input: nums = [2,6,7], k = 1
Output: 5
Explanation:
The subsequence [2, 7] has the maximum value of 2 XOR 7 = 5.
Example 2:
Input: nums = [4,2,5,6,7], k = 2
Output: 2
Explanation:
The subsequence [4, 5, 6, 7] has the maximum value of (4 OR 5) XOR (6 OR 7) = 2.
Constraints:
2 <= nums.length <= 4001 <= nums[i] < 271 <= k <= nums.length / 2The key challenge in #3287 Find the Maximum Sequence Value of Array is selecting elements from the array to maximize a value derived from bitwise operations. Since brute‑forcing all subsequences is infeasible, an efficient strategy combines dynamic programming with bit manipulation.
The common idea is to compute possible bitwise OR values obtainable from selecting a fixed number of elements from the prefix and suffix of the array. Using DP, we track all achievable OR states for choosing j elements up to a position. A similar DP is computed from the right side. Once these states are known, we try combining prefix and suffix results and evaluate the sequence value using a XOR operation.
This reduces the search space significantly because the number of distinct OR values is limited by the bit width of the numbers. Efficient state storage (such as sets or bitsets) allows fast merging and evaluation. Overall, this technique avoids exponential enumeration while still exploring all meaningful bitwise combinations.
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Dynamic Programming with Bitwise OR state tracking | O(n * k * M) | O(n * k * M) |
| Combining prefix and suffix OR states to maximize XOR | O(M^2) | O(M) |
Ashish Pratap Singh
Use these hints if you're stuck. Try solving on your own first.
Find all the possible <code>OR</code> till each <code>i</code> with <code>k</code> elements backward and forward.
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Sets or boolean DP tables are commonly used to store reachable OR values for each subsequence length. Since the number of distinct OR values is limited by the number of bits in the integers, these structures remain efficient for state tracking.
Hard problems involving dynamic programming and bit manipulation patterns are common in FAANG-style interviews. While the exact problem may not always appear, the techniques used here—state compression and bitwise optimization—are frequently tested.
The optimal approach uses dynamic programming to track all achievable bitwise OR values for subsequences of a given size from both the prefix and suffix of the array. These states are then combined using XOR to compute the maximum sequence value efficiently without enumerating all subsequences.
Bit manipulation is crucial because the sequence value depends on bitwise operations such as OR and XOR. By tracking OR states, the algorithm reduces the number of possible values to a manageable set, making dynamic programming feasible.