Maximum Absolute Sum of Any Subarray - Video Solutions

Problem Overview: You are given an integer array nums. A subarray can have a positive or negative sum. The task is to return the maximum possible absolute value of any subarray sum. In other words, you must consider both the largest positive subarray and the most negative subarray.

Approach 1: Modified Kadane's Algorithm (O(n) time, O(1) space)

This approach adapts the classic Kadane’s algorithm used for maximum subarray problems. Since the goal is the absolute sum, you track two running values while iterating through the array: the maximum subarray sum ending at the current index and the minimum subarray sum ending at the current index. The maximum captures large positive segments, while the minimum captures strongly negative segments. At each step you update both values using max(current, current + prevMax) and min(current, current + prevMin). The answer becomes max(abs(maxSum), abs(minSum)). This works because the largest magnitude subarray must be either the largest positive sum or the smallest negative sum. The algorithm scans the array once, making it O(n) time with constant O(1) extra space. This technique is a common extension of problems involving dynamic programming on arrays.

Approach 2: Prefix Sum with Minimum/Maximum Tracking (O(n) time, O(1) space)

Another way to view the problem is through prefix sums. Maintain a running prefix sum while traversing the array. The absolute sum of a subarray between indices i and j equals |prefix[j] - prefix[i]|. To maximize this value, you track the smallest and largest prefix sums seen so far. During each iteration, compute the current prefix sum and compare it with the stored minimum and maximum prefixes. The difference between the current prefix and those extremes represents the largest possible subarray ending at the current position. Update the result with max(abs(prefix - minPrefix), abs(prefix - maxPrefix)). Because you only store a few variables while scanning once, the complexity remains O(n) time and O(1) space. This interpretation is especially helpful when practicing prefix sum techniques and understanding range sum relationships in an array.

Recommended for interviews: The modified Kadane’s algorithm is usually the expected solution. It shows that you recognize the connection to the classic maximum subarray problem and can adapt it to track both positive and negative extremes. The prefix sum method demonstrates deeper understanding of how subarray sums relate to differences between cumulative sums, which interviewers also appreciate when discussing reasoning.