constrained-subsequence-sum

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1425. Constrained Subsequence Sum (Hard)

Given an integer array nums and an integer k, return the maximum sum of a non-empty subsequence of that array such that for every two consecutive integers in the subsequence, nums[i] and nums[j], where i < j, the condition j - i <= k is satisfied.

A subsequence of an array is obtained by deleting some number of elements (can be zero) from the array, leaving the remaining elements in their original order.

 

Example 1:

Input: nums = [10,2,-10,5,20], k = 2
Output: 37
Explanation: The subsequence is [10, 2, 5, 20].

Example 2:

Input: nums = [-1,-2,-3], k = 1
Output: -1
Explanation: The subsequence must be non-empty, so we choose the largest number.

Example 3:

Input: nums = [10,-2,-10,-5,20], k = 2
Output: 23
Explanation: The subsequence is [10, -2, -5, 20].

 

Constraints:

• 1 <= k <= nums.length <= 105
• -104 <= nums[i] <= 104

Related Topics

[Array] [Dynamic Programming] [Queue] [Sliding Window] [Heap (Priority Queue)] [Monotonic Queue]

Hints

Hint 1

Use dynamic programming.

Hint 2

Let dp[i] be the solution for the prefix of the array that ends at index i, if the element at index i is in the subsequence.

Hint 3

dp[i] = nums[i] + max(0, dp[i-k], dp[i-k+1], ..., dp[i-1])

Hint 4

Use a heap with the sliding window technique to optimize the dp.