#3196

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximize Total Cost of Alternating Subarrays

Maximize the total cost of alternating subarrays using dynamic programming to efficiently split an array into optimal subarrays.

array dynamic programming

Problem Statement

You are given an integer array nums of length n. The goal is to maximize the total cost of alternating subarrays within nums. A valid alternating subarray consists of adjacent elements that alternate in sign, and the total cost is the sum of all values within these subarrays.

To calculate the cost, consider the subarrays of nums and sum their elements. For each valid split of nums into alternating subarrays, compute the total cost as the sum of the individual subarray sums. The task is to find the split that results in the maximum total cost.

Examples

Example 1

Input: nums = [1,-2,3,4]

Output: 10

One way to maximize the total cost is by splitting [1, -2, 3, 4] into subarrays [1, -2, 3] and [4] . The total cost will be (1 + 2 + 3) + 4 = 10 .

Example 2

Input: nums = [1,-1,1,-1]

Output: 4

One way to maximize the total cost is by splitting [1, -1, 1, -1] into subarrays [1, -1] and [1, -1] . The total cost will be (1 + 1) + (1 + 1) = 4 .

Example 3

Input: nums = [0]

Output: 0

We cannot split the array further, so the answer is 0.

Constraints

1 <= nums.length <= 105
-109 <= nums[i] <= 109

Solution Approach

State Transition Dynamic Programming

The core approach relies on dynamic programming with state transitions. We maintain a DP table where each entry tracks the maximum cost achievable up to a certain index. The transition depends on whether the current element continues an alternating sequence or starts a new one.

Greedy Subarray Splits

For each element, we evaluate whether it should extend the current subarray or start a new one, based on its sign. This greedy decision ensures that we maximize the sum of each alternating subarray.

Optimal Subarray Cost Calculation

Once the subarrays are identified, the total cost is computed by summing the costs of each valid subarray. The goal is to select splits that maximize the sum of these alternating subarrays.

Complexity Analysis

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

The time complexity depends on the dynamic programming approach, which generally runs in O(n) time due to the linear scan of the array. Space complexity may vary based on the implementation, but an optimal solution can achieve O(n) space usage.

What Interviewers Usually Probe

Look for an understanding of dynamic programming and state transitions.
Evaluate the ability to optimize a solution based on greedy subarray splits.
Test the candidate's ability to calculate and maximize subarray costs effectively.

Common Pitfalls or Variants

Common pitfalls

Not properly identifying valid alternating subarrays.
Using a brute force method that doesn't take advantage of dynamic programming.
Failing to consider all possible valid splits of the array.

Follow-up variants

Adjust the problem to handle larger arrays efficiently.
Implement a solution that handles various edge cases like single-element arrays.
Test with arrays containing both positive and negative values to explore different split patterns.

FAQ

What is the primary pattern used in the "Maximize Total Cost of Alternating Subarrays" problem?

The problem utilizes state transition dynamic programming to solve for the maximum total cost of alternating subarrays.

How can dynamic programming help in solving the problem efficiently?

Dynamic programming helps by storing intermediate results for subarray sums, ensuring that overlapping subproblems are solved only once, thus optimizing the overall solution.

What are some common pitfalls to avoid when solving this problem?

Avoid not correctly identifying alternating subarrays, using inefficient brute force methods, and missing valid subarray splits that maximize the cost.

How do you handle edge cases like a single-element array?

For a single-element array, the maximum cost is 0, as no valid alternating subarray split is possible.

What is the time complexity of the optimal solution for this problem?

The time complexity of the optimal solution is O(n), where n is the length of the array, due to the linear scan of the array and dynamic programming transitions.

terminal

Solution

Solution 1: Memoization

Based on the problem description, if the current number has not been flipped, then the next one can either be flipped or not flipped; if the current number has been flipped, then the next one can only remain unflipped.

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class Solution:
    def maximumTotalCost(self, nums: List[int]) -> int:
        @cache
        def dfs(i: int, j: int) -> int:
            if i >= len(nums):
                return 0
            ans = nums[i] + dfs(i + 1, 1)
            if j == 1:
                ans = max(ans, -nums[i] + dfs(i + 1, 0))
            return ans

        return dfs(0, 0)

Solution 2: Dynamic Programming

We can transform the memoization search from Solution 1 into dynamic programming.

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class Solution:
    def maximumTotalCost(self, nums: List[int]) -> int:
        @cache
        def dfs(i: int, j: int) -> int:
            if i >= len(nums):
                return 0
            ans = nums[i] + dfs(i + 1, 1)
            if j == 1:
                ans = max(ans, -nums[i] + dfs(i + 1, 0))
            return ans

        return dfs(0, 0)

Continue Practicing

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