LeetCode #3287 — HARD

Find the Maximum Sequence Value of Array

Break down a hard problem into reliable checkpoints, edge-case handling, and complexity trade-offs.

Solve on LeetCode

The Problem

Problem Statement

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

Step 01

Brute Force Baseline

Problem summary: 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.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Dynamic Programming · Bit Manipulation

Example 1

[2,6,7]
1

Example 2

[4,2,5,6,7]
2

Core Insight

What unlocks the optimal approach

Find all the possible <code>OR</code> till each <code>i</code> with <code>k</code> elements backward and forward.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Largest constraint values

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #3287: Find the Maximum Sequence Value of Array
class Solution {
    public int maxValue(int[] nums, int k) {
        int m = 1 << 7;
        int n = nums.length;
        boolean[][][] f = new boolean[n + 1][k + 2][m];
        f[0][0][0] = true;

        for (int i = 0; i < n; i++) {
            for (int j = 0; j <= k; j++) {
                for (int x = 0; x < m; x++) {
                    if (f[i][j][x]) {
                        f[i + 1][j][x] = true;
                        f[i + 1][j + 1][x | nums[i]] = true;
                    }
                }
            }
        }

        boolean[][][] g = new boolean[n + 1][k + 2][m];
        g[n][0][0] = true;

        for (int i = n; i > 0; i--) {
            for (int j = 0; j <= k; j++) {
                for (int y = 0; y < m; y++) {
                    if (g[i][j][y]) {
                        g[i - 1][j][y] = true;
                        g[i - 1][j + 1][y | nums[i - 1]] = true;
                    }
                }
            }
        }

        int ans = 0;

        for (int i = k; i <= n - k; i++) {
            for (int x = 0; x < m; x++) {
                if (f[i][k][x]) {
                    for (int y = 0; y < m; y++) {
                        if (g[i][k][y]) {
                            ans = Math.max(ans, x ^ y);
                        }
                    }
                }
            }
        }

        return ans;
    }
}

// Accepted solution for LeetCode #3287: Find the Maximum Sequence Value of Array
func maxValue(nums []int, k int) int {
	m := 1 << 7
	n := len(nums)

	f := make([][][]bool, n+1)
	for i := range f {
		f[i] = make([][]bool, k+2)
		for j := range f[i] {
			f[i][j] = make([]bool, m)
		}
	}
	f[0][0][0] = true

	for i := 0; i < n; i++ {
		for j := 0; j <= k; j++ {
			for x := 0; x < m; x++ {
				if f[i][j][x] {
					f[i+1][j][x] = true
					f[i+1][j+1][x|nums[i]] = true
				}
			}
		}
	}

	g := make([][][]bool, n+1)
	for i := range g {
		g[i] = make([][]bool, k+2)
		for j := range g[i] {
			g[i][j] = make([]bool, m)
		}
	}
	g[n][0][0] = true

	for i := n; i > 0; i-- {
		for j := 0; j <= k; j++ {
			for y := 0; y < m; y++ {
				if g[i][j][y] {
					g[i-1][j][y] = true
					g[i-1][j+1][y|nums[i-1]] = true
				}
			}
		}
	}

	ans := 0

	for i := k; i <= n-k; i++ {
		for x := 0; x < m; x++ {
			if f[i][k][x] {
				for y := 0; y < m; y++ {
					if g[i][k][y] {
						ans = max(ans, x^y)
					}
				}
			}
		}
	}

	return ans
}

# Accepted solution for LeetCode #3287: Find the Maximum Sequence Value of Array
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

// Accepted solution for LeetCode #3287: Find the Maximum Sequence Value of Array
/**
 * [3287] Find the Maximum Sequence Value of Array
 */
pub struct Solution {}

// submission codes start here
use std::collections::HashSet;

impl Solution {
    pub fn max_value(nums: Vec<i32>, k: i32) -> i32 {
        let k = k as usize;
        let length = nums.len();
        let search = |array: &Vec<i32>| {
            let mut dp = Vec::with_capacity(array.len());
            let mut prev = vec![HashSet::new(); k + 1];
            prev[0].insert(0);

            for i in 0..array.len() {
                for j in (0..=(i + 1).min(k - 1)).rev() {
                    // 使用split_at_mut以通过借用检查
                    let (first, second) = prev.split_at_mut(j + 1);
                    for x in first[j].iter().map(|x| *x | array[i]) {
                        second[0].insert(x);
                    }
                }

                dp.push(prev[k].clone());
            }

            dp
        };

        let a = search(&nums);
        let reversed_nums: Vec<i32> = nums.iter().rev().map(|x| x.clone()).collect();
        let b = search(&reversed_nums);

        let mut result = 0;

        for i in k - 1..length - k {
            for &a_val in a[i].iter() {
                for &b_val in b[length - i - 2].iter() {
                    result = result.max(a_val ^ b_val);
                }
            }
        }

        result
    }
}

// submission codes end

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_3287() {
        assert_eq!(5, Solution::max_value(vec![2, 6, 7], 1));
        assert_eq!(2, Solution::max_value(vec![4, 2, 5, 6, 7], 2));
    }
}

// Accepted solution for LeetCode #3287: Find the Maximum Sequence Value of Array
function maxValue(nums: number[], k: number): number {
    const m = 1 << 7;
    const n = nums.length;

    const f: boolean[][][] = Array.from({ length: n + 1 }, () =>
        Array.from({ length: k + 2 }, () => Array(m).fill(false)),
    );
    f[0][0][0] = true;

    for (let i = 0; i < n; i++) {
        for (let j = 0; j <= k; j++) {
            for (let x = 0; x < m; x++) {
                if (f[i][j][x]) {
                    f[i + 1][j][x] = true;
                    f[i + 1][j + 1][x | nums[i]] = true;
                }
            }
        }
    }

    const g: boolean[][][] = Array.from({ length: n + 1 }, () =>
        Array.from({ length: k + 2 }, () => Array(m).fill(false)),
    );
    g[n][0][0] = true;

    for (let i = n; i > 0; i--) {
        for (let j = 0; j <= k; j++) {
            for (let y = 0; y < m; y++) {
                if (g[i][j][y]) {
                    g[i - 1][j][y] = true;
                    g[i - 1][j + 1][y | nums[i - 1]] = true;
                }
            }
        }
    }

    let ans = 0;

    for (let i = k; i <= n - k; i++) {
        for (let x = 0; x < m; x++) {
            if (f[i][k][x]) {
                for (let y = 0; y < m; y++) {
                    if (g[i][k][y]) {
                        ans = Math.max(ans, x ^ y);
                    }
                }
            }
        }
    }

    return ans;
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n × m × k)

Space

O(n × m × k)

Approach Breakdown

RECURSIVE

O(2ⁿ) time

O(n) space

Pure recursion explores every possible choice at each step. With two choices per state (take or skip), the decision tree has 2ⁿ leaves. The recursion stack uses O(n) space. Many subproblems are recomputed exponentially many times.

DYNAMIC PROGRAMMING

O(n × m) time

O(n × m) space

Each cell in the DP table is computed exactly once from previously solved subproblems. The table dimensions determine both time and space. Look for the state variables — each unique combination of state values is one cell. Often a rolling array can reduce space by one dimension.

Shortcut: Count your DP state dimensions → that’s your time. Can you drop one? That’s your space optimization.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

State misses one required dimension

Wrong move: An incomplete state merges distinct subproblems and caches incorrect answers.

Usually fails on: Correctness breaks on cases that differ only in hidden state.

Fix: Define state so each unique subproblem maps to one DP cell.