Maximum Subarray Sum with One Deletion

Problem Description

Given an array of integers, return the maximum sum for a non-empty subarray (contiguous elements) with at most one element deletion. In other words, you want to choose a subarray and optionally delete one element from it so that there is still at least one element left and the sum of the remaining elements is maximum possible.

Note that the subarray needs to be non-empty after deleting one element.

Key Insights

• The problem can be tackled using dynamic programming.
• We can maintain two arrays or variables: one for the maximum sum without deletion and another for the maximum sum with one deletion.
• The maximum subarray sum can be derived from the maximum ending at each index, considering both cases of deleting and not deleting an element.
• Care must be taken to ensure that the subarray remains non-empty after a deletion.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve this problem, we will use a dynamic programming approach where we maintain two variables:

1. max_end_here: to store the maximum subarray sum ending at the current index without any deletions.
2. max_end_here_with_deletion: to store the maximum subarray sum ending at the current index with one deletion.

We iterate through the array, updating these variables based on the values of the current element and the maximum sums calculated so far. The final result will be the maximum of these two variables after processing all elements.

Code Solutions