#1425

Hard

auto_awesomeState transition dynamic programming

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Constrained Subsequence Sum

Solve the Constrained Subsequence Sum problem using dynamic programming, sliding window, and priority queues to maximize subsequence sum with constraints.

array dynamic programming queue sliding window heap priority queue

Problem Statement

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

A subsequence is formed by deleting some elements from the array while keeping the remaining elements in their original order. The goal is to maximize the sum of the subsequence by respecting the constraint on the indices of the chosen elements.

Examples

Example 1

Input: nums = [10,2,-10,5,20], k = 2

Output: 37

The subsequence is [10, 2, 5, 20].

Example 2

Input: nums = [-1,-2,-3], k = 1

Output: -1

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

The subsequence is [10, -2, -5, 20].

Constraints

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

Solution Approach

Dynamic Programming with Sliding Window

We can solve this problem using dynamic programming, where each element represents the maximum sum of the subsequence ending at that position. We maintain a sliding window of size k to efficiently track the maximum sum of subsequences that are valid according to the constraint.

Priority Queue (Heap) Optimization

A priority queue (heap) can be used to optimize the dynamic programming solution. It allows quick retrieval of the maximum sum of subsequences from the last k elements, helping us efficiently compute the maximum sum for each position in the array.

State Transition Approach

The state transition for this problem depends on the maximum subsequence sum up to previous positions within a range of k indices. We update the current position based on the best possible subsequence sum up to a prior index, ensuring the subsequence constraint is met.

Complexity Analysis

Metric	Value
Time	O(n)
Space	O(n)

The time complexity is O(n) since each element is processed once, and heap operations take O(log k), which is bounded by the maximum value of k. The space complexity is O(n) due to the dynamic programming array and priority queue storage.

What Interviewers Usually Probe

The candidate demonstrates knowledge of dynamic programming with optimizations for sliding windows and heaps.
The candidate discusses the importance of efficient state transitions and boundary conditions in dynamic programming.
The candidate understands the trade-offs of using a heap for maintaining the maximum sum of subsequences in the context of a sliding window.

Common Pitfalls or Variants

Common pitfalls

Ignoring the need to maintain a sliding window of maximum subsequence sums, leading to inefficient solutions.
Overcomplicating the solution by using brute-force methods without taking advantage of dynamic programming and heap optimizations.
Failing to correctly handle edge cases like negative numbers or very small arrays, which can impact the correctness of the dynamic programming solution.

Follow-up variants

Allowing for an additional constraint where the sum of subsequence elements must be greater than a given threshold.
Modifying the problem to include more complex constraints on subsequence selection, such as allowing at most two elements to be skipped.
Changing the problem to find the minimum sum instead of the maximum sum, which would require reversing the optimization direction.

FAQ

What is the main technique used in the Constrained Subsequence Sum problem?

The main technique is dynamic programming with optimizations using a sliding window and priority queue to efficiently track and compute maximum subsequence sums.

How does the sliding window help in solving this problem?

The sliding window helps by limiting the search space for the maximum sum subsequence to the last k positions, ensuring the condition j - i <= k is met efficiently.

Can this problem be solved using a brute-force approach?

While a brute-force approach is possible, it is inefficient due to the large input size. Dynamic programming and priority queues provide a much more efficient solution.

How do priority queues (heaps) optimize the solution?

Priority queues help by maintaining the maximum sum of subsequences up to the last k elements, allowing for fast retrieval and updates while calculating the subsequence sum.

What edge cases should be considered in the Constrained Subsequence Sum problem?

Edge cases include arrays with negative values, very small arrays, and arrays where no valid subsequence can be formed under the given constraints.

terminal

Solution

Solution 1: Dynamic Programming + Monotonic Queue

We define $f[i]$ to represent the maximum sum of the subsequence ending at $\textit{nums}[i]$ that meets the conditions. Initially, $f[i] = 0$, and the answer is $\max_{0 \leq i \lt n} f(i)$.

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class Solution:
    def constrainedSubsetSum(self, nums: List[int], k: int) -> int:
        q = deque([0])
        n = len(nums)
        f = [0] * n
        ans = -inf
        for i, x in enumerate(nums):
            while i - q[0] > k:
                q.popleft()
            f[i] = max(0, f[q[0]]) + x
            ans = max(ans, f[i])
            while q and f[q[-1]] <= f[i]:
                q.pop()
            q.append(i)
        return ans

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