Maximum Subarray Sum After One Operation

Maximum Subarray Sum After One Operation

1. What is this problem about?

The Maximum Subarray Sum After One Operation problem is a fascinating variation of the classic Kadane's algorithm. In this version, you are given an array of integers and allowed to perform exactly one operation: square one of the elements in the array (i.e., replace x with x * x). Your goal is to find the maximum possible sum of a contiguous subarray after this single replacement. Since squaring a number (especially a negative one or a large positive one) can significantly boost the sum, the challenge lies in deciding which element to transform to maximize the overall result.

2. Why is this asked in interviews?

This "Maximum Subarray Sum After One Operation interview question" is popular at companies like Sprinklr because it tests a candidate's ability to extend a well-known algorithm (Kadane’s) with new constraints. It evaluates whether you can maintain multiple states in a Dynamic Programming solution. Interviewers want to see if you can distinguish between a subarray that hasn't used the "squaring" operation yet and one that has already used it, and how you transition between these states as you traverse the array.

3. Algorithmic pattern used

The "Array, Dynamic Programming interview pattern" is the most effective way to solve this. You typically maintain two DP states while iterating through the array. dp[i][0] represents the maximum subarray sum ending at index i without using the squaring operation. dp[i][1] represents the maximum subarray sum ending at index i after having used the squaring operation on exactly one element in the current subarray. The transitions involve choosing whether to start a new subarray, continue an existing one, or apply the square operation to the current element.

4. Example explanation

Consider the array: [2, -3, 1].

• If we don't square anything, the maximum sum is 2.
• If we square the -3, the array becomes [2, 9, 1]. The maximum subarray sum is 2 + 9 + 1 = 12.
• If we square the 2, the array becomes [4, -3, 1]. Max sum is 4. The algorithm would track these possibilities. At index 1 (-3), it would decide that dp[1][1] could be dp[0][0] + (-3)^2 (which is 2 + 9 = 11) or just (-3)^2 (which is 9). By the end of the array, the global maximum would be identified.

5. Common mistakes candidates make

A common mistake in the "Maximum Subarray Sum After One Operation coding problem" is forgetting that the squared element must be part of the contiguous subarray. You can't square an element at the beginning of the array and then add it to a subarray at the very end if they aren't connected. Another error is not correctly handling the transition where the current element is the one being squared, or failing to realize that once an element is squared, no subsequent elements in that subarray can be squared.

6. Interview preparation tip

When preparing for this problem, practice visualizing DP transitions as a state machine. You are in state "No Square Used," and you can either stay there or transition to state "Square Used" by squaring the current element. Once in "Square Used," you can never go back to "No Square Used" for that specific subarray. This mental model makes it much easier to write the recurrence relations without getting confused by the logic.