#3287

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Find the Maximum Sequence Value of Array

Determine the maximum value of a subsequence in an integer array using state transition dynamic programming and bit operations efficiently.

array dynamic programming bit manipulation

Problem Statement

Given an integer array nums and a positive integer k, find the maximum value obtainable from any subsequence of nums of size 2 * k. Each subsequence's value is computed using bitwise OR and XOR operations on pairs within the subsequence, emphasizing careful state management.

Return the maximum value of such subsequences, ensuring that the selection respects the original array order. Optimize your solution using dynamic programming to handle OR combinations and sequence transitions for each subsequence efficiently.

Examples

Example 1

Input: nums = [2,6,7], k = 1

Output: 5

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

The subsequence [4, 5, 6, 7] has the maximum value of (4 OR 5) XOR (6 OR 7) = 2 .

Constraints

2 <= nums.length <= 400
1 <= nums[i] < 27
1 <= k <= nums.length / 2

Solution Approach

State Transition DP Setup

Define dp[i][j] as the maximum value for the first i elements selecting j pairs. Initialize base states for zero pairs and iterate over array elements. This ensures proper state tracking for sequence combinations.

Bitwise OR and XOR Propagation

For each element, propagate possible OR values backward for k elements and forward for next elements. Compute XOR across pair combinations efficiently to avoid redundant calculations and maintain maximal values per state.

Iterative Maximization

Iteratively update dp states for all i and j by considering taking or skipping the current element. Keep the running maximum of subsequence values, ensuring the final dp table provides the maximum result for 2 * k subsequence length.

Complexity Analysis

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

Time complexity depends on iterating through all elements and possible k-length OR combinations, roughly O(n * k * 2^m) for bit states. Space complexity is O(n * k) for DP storage with optimization possible via rolling arrays.

What Interviewers Usually Probe

They may hint at optimizing bitwise operations rather than naive pair checking.
Expect prompts about state definition for dynamic programming and handling OR/XOR transitions.
Clarifications may focus on array length constraints and avoiding recomputation across subsequences.

Common Pitfalls or Variants

Common pitfalls

Forgetting to consider all OR combinations leading to suboptimal maximum.
Mismanaging DP state transitions causing index out-of-bounds errors.
Assuming XOR distribution is linear without correctly pairing OR segments.

Follow-up variants

Compute maximum subsequence value with additional sum constraints alongside OR/XOR evaluation.
Allow subsequences of variable even lengths instead of strictly 2 * k.
Modify operations to include AND or custom bitwise operations while maintaining state transition DP.

FAQ

What is the best approach for Find the Maximum Sequence Value of Array?

Use state transition dynamic programming with careful tracking of OR values and XOR computations across 2 * k subsequences.

Can this problem be solved without DP?

A brute-force method is possible but infeasible due to exponential combinations; DP ensures efficient state propagation.

How do OR and XOR combine in subsequence evaluation?

Compute OR for each pair or segment, then XOR the results to get the subsequence value, updating DP accordingly.

What are typical interviewer hints?

Expect suggestions on propagating OR states, handling array boundaries, and optimizing repeated XOR calculations.

Does array length affect complexity?

Yes, longer arrays increase DP table size and possible OR states, but rolling arrays can reduce space usage.

terminal

Solution

Solution 1: Dynamic Programming + Prefix and Suffix Decomposition + Enumeration

We consider dividing the sequence into two parts, the first $k$ elements and the last $k$ elements, and calculate all possible XOR values for the prefixes and suffixes.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

class Solution:
    def maxValue(self, nums: List[int], k: int) -> int:
        m = 1 << 7
        n = len(nums)
        f = [[[False] * m for _ in range(k + 2)] for _ in range(n + 1)]
        f[0][0][0] = True
        for i in range(n):
            for j in range(k + 1):
                for x in range(m):
                    f[i + 1][j][x] |= f[i][j][x]
                    f[i + 1][j + 1][x | nums[i]] |= f[i][j][x]

        g = [[[False] * m for _ in range(k + 2)] for _ in range(n + 1)]
        g[n][0][0] = True
        for i in range(n, 0, -1):
            for j in range(k + 1):
                for y in range(m):
                    g[i - 1][j][y] |= g[i][j][y]
                    g[i - 1][j + 1][y | nums[i - 1]] |= g[i][j][y]

        ans = 0
        for i in range(k, n - k + 1):
            for x in range(m):
                if f[i][k][x]:
                    for y in range(m):
                        if g[i][k][y]:
                            ans = max(ans, x ^ y)
        return ans

Continue Practicing

#3276 select cells in grid with maximum score #3376 minimum time to break locks i #3181 maximum total reward using operations ii