1749. Maximum Absolute Sum of Any Subarray - Detailed Explanation

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Problem Statement

You are given an integer array nums. A subarray is a contiguous part of the array, and its sum is the sum of all the elements within it. The absolute sum of a subarray is defined as the absolute value of its sum. Your task is to find and return the maximum absolute sum among all possible subarrays of nums.

For example, if a subarray has a sum of -5, its absolute sum is 5. You need to maximize this absolute sum over every possible subarray.

Examples

Example 1:
- Input: nums = [1, -3, 2, 3, -4]
- Output: 5
- Explanation:
  - Compute prefix sums:
    P[0] = 0
    P[1] = 1
    P[2] = 1 + (-3) = -2
    P[3] = -2 + 2 = 0
    P[4] = 0 + 3 = 3
    P[5] = 3 + (-4) = -1
  - The maximum prefix sum is 3 and the minimum prefix sum is -2.
  - The maximum absolute sum of any subarray is 3 - (-2) = 5.
Example 2:
- Input: nums = [2, -5, 1, -4, 3, 0]
- Output: (Example analysis not provided, but the approach is the same)

Key Insights and Hints

Using Prefix Sums:
A subarray sum from index i to j can be expressed as the difference between two prefix sums:
$[ \text{sum}[i \ldots j] = \text{prefix}[j+1] - \text{prefix}[i] ]$
Maximizing the Difference:
To maximize the absolute value of any subarray sum, you need to maximize the difference between any two prefix sums. In particular, the maximum absolute sum equals: $[ \text{maxAbsSum} = \max(\text{prefix}) - \min(\text{prefix}) ]$ This works because the largest difference between any two prefix sums yields the maximum possible subarray sum in absolute value.
Optimal Approach:
You can compute the prefix sum while tracking the minimum and maximum values seen so far. The final answer is the difference between these two values.

Approaches to Solve the Problem

Approach: One-Pass with Prefix Sum Tracking

Initialize Variables:
- Set a variable prefix to 0 (to represent the current prefix sum).
- Set minPrefix to 0 and maxPrefix to 0.
Iterate Through the Array:
- For each number in nums, update the current prefix sum.
- Update minPrefix if the current prefix sum is smaller.
- Update maxPrefix if the current prefix sum is larger.
Calculate the Answer:
- The answer is the difference maxPrefix - minPrefix, which is the maximum absolute sum of any subarray.

Code Implementations

Python Implementation

Python3

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Java Implementation

Java