#1749

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximum Absolute Sum of Any Subarray

Solve for the maximum absolute sum of any subarray using dynamic programming and understanding state transitions.

Problem Statement

Given an integer array nums, your task is to calculate the maximum absolute sum of any subarray within nums. The absolute sum of a subarray is defined as the absolute value of the sum of its elements.

For example, if nums = [1,-3,2,3,-4], the subarray [2,3] has the maximum absolute sum of 5. The problem requires an efficient solution that handles subarrays within the constraints of the array length and element range.

Examples

Example 1

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

Output: 5

The subarray [2,3] has absolute sum = abs(2+3) = abs(5) = 5.

Example 2

Input: nums = [2,-5,1,-4,3,-2]

Output: 8

The subarray [-5,1,-4] has absolute sum = abs(-5+1-4) = abs(-8) = 8.

Constraints

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

Solution Approach

Track Running Sums

Use dynamic programming to track both the maximum and minimum running sums. At each element, calculate the running sum and update the global maximum absolute sum by considering both the maximum and minimum running sums at each step.

State Transitions for Optimal Result

Apply state transitions to maintain the current running sum and adjust it by either adding or resetting based on the signs of the elements. This ensures that we always keep the potential maximum absolute sum while iterating through the array.

Constant Space Optimization

Although dynamic programming typically requires additional space, this problem can be solved in O(1) space by only keeping track of the current sums and updating them as we progress through the array.

Complexity Analysis

Metric	Value
Time	O(N)
Space	O(1)

The time complexity is O(N) because we only need to traverse the array once. The space complexity is O(1) since we only store a few variables to track running sums and the maximum absolute sum, without requiring extra space proportional to the input size.

What Interviewers Usually Probe

Can the candidate efficiently optimize space usage with dynamic programming?
How well does the candidate handle state transitions in dynamic programming problems?
Does the candidate understand how to compute absolute sums within subarrays efficiently?

Common Pitfalls or Variants

Common pitfalls

Forgetting to consider the possibility of empty subarrays, which can affect the result when calculating the maximum absolute sum.
Not handling negative numbers correctly in the subarray sums and absolute value calculations.
Overcomplicating the problem by using unnecessary data structures, when a solution with constant space can be achieved.

Follow-up variants

What if we asked for the maximum sum, rather than the absolute sum? This would eliminate the need to consider the negative sum possibilities.
How would the solution change if the array contained only positive numbers or only negative numbers?
What if we asked for the subarray with the smallest absolute sum instead?

FAQ

What is the maximum absolute sum of any subarray problem about?

The problem asks for the maximum absolute sum of any subarray in a given integer array, using dynamic programming to compute this in linear time.

How does dynamic programming apply to the maximum absolute sum problem?

Dynamic programming tracks the running sums of subarrays and uses state transitions to efficiently compute the maximum absolute sum, optimizing both time and space.

What are the key challenges when solving this problem?

The challenges lie in handling negative sums and ensuring that the solution works within O(N) time and O(1) space constraints.

What is the space complexity of the maximum absolute sum problem?

The space complexity is O(1), as the solution only requires a few variables to store the current sums and maximum absolute sum.

How does the problem change if we ask for the maximum sum instead of the absolute sum?

If we ask for the maximum sum, the absolute value operation would not be necessary, and the problem would focus on finding the subarray with the largest sum directly.

terminal

Solution

Solution 1: Dynamic Programming

We define $f[i]$ to represent the maximum value of the subarray ending with $nums[i]$, and define $g[i]$ to represent the minimum value of the subarray ending with $nums[i]$. Then the state transition equation of $f[i]$ and $g[i]$ is as follows:

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class Solution:
    def maxAbsoluteSum(self, nums: List[int]) -> int:
        f = g = 0
        ans = 0
        for x in nums:
            f = max(f, 0) + x
            g = min(g, 0) + x
            ans = max(ans, f, abs(g))
        return ans

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