2407. Longest Increasing Subsequence II

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You are given an integer array nums and an integer k.

Find the longest subsequence of nums that meets the following requirements:

The subsequence is strictly increasing and
The difference between adjacent elements in the subsequence is at most k.

Return the length of the longest subsequence that meets the requirements.

A subsequence is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.

Example 1:

Input: nums = [4,2,1,4,3,4,5,8,15], k = 3
Output: 5
Explanation:
The longest subsequence that meets the requirements is [1,3,4,5,8].
The subsequence has a length of 5, so we return 5.
Note that the subsequence [1,3,4,5,8,15] does not meet the requirements because 15 - 8 = 7 is larger than 3.

Example 2:

Input: nums = [7,4,5,1,8,12,4,7], k = 5
Output: 4
Explanation:
The longest subsequence that meets the requirements is [4,5,8,12].
The subsequence has a length of 4, so we return 4.

Example 3:

Input: nums = [1,5], k = 1
Output: 1
Explanation:
The longest subsequence that meets the requirements is [1].
The subsequence has a length of 1, so we return 1.

Constraints:

1 <= nums.length <= 10⁵
1 <= nums[i], k <= 10⁵

ArrayDivide and ConquerDynamic ProgrammingBinary Indexed TreeSegment TreeQueueMonotonic Queue

Hints

Hint 1

We can use dynamic programming. Let dp[i][val] be the answer using only the first i + 1 elements, and the last element in the subsequence is equal to val.

Hint 2

The only value that might change between dp[i - 1] and dp[i] are dp[i - 1][val] and dp[i][val].

Hint 3

Try using dp[i - 1] and the fact that the second last element in the subsequence has to fall within a range to calculate dp[i][val].

Hint 4

We can use a segment tree to find the maximum value in dp[i - 1] within a certain range.

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Input