Maximum Sum Score of Array

Maximum Sum Score of Array

1. What is this problem about?

The Maximum Sum Score of Array problem asks you to evaluate an array and find the maximum "score" that can be achieved at any index i. For each index, the score is defined as the maximum of two values: the sum of the first i + 1 elements (prefix sum) and the sum of the last n - i elements (suffix sum). Your task is to iterate through all possible indices, calculate these scores, and return the largest one found.

2. Why is this asked in interviews?

This "Maximum Sum Score of Array interview question" is a popular Amazon problem used to test a candidate's understanding of prefix and suffix sums. It evaluates whether you can perform array summations efficiently without redundant calculations. Interviewers want to see if you can move from a naive O(N^2) solution (calculating sums from scratch for every index) to an optimal O(N) solution using pre-computation or a single running total.

3. Algorithmic pattern used

The "Array, Prefix Sum interview pattern" is the key to solving this in linear time. By first calculating the total sum of the array, you can determine any prefix or suffix sum in O(1) time as you iterate. For example, the suffix sum at index i is simply Total_Sum - Prefix_Sum_at_i_minus_1. This allows you to find the score for every index in a single pass through the array.

4. Example explanation

Array: [4, 3, -2, 5]

• Total Sum: 4 + 3 - 2 + 5 = 10.
• Index 0: Prefix [4] = 4, Suffix [4, 3, -2, 5] = 10. Score = max(4, 10) = 10.
• Index 1: Prefix [4, 3] = 7, Suffix [3, -2, 5] = 6. Score = max(7, 6) = 7.
• Index 2: Prefix [4, 3, -2] = 5, Suffix [-2, 5] = 3. Score = max(5, 3) = 5.
• Index 3: Prefix [4, 3, -2, 5] = 10, Suffix [5] = 5. Score = max(10, 5) = 10. Maximum score in this array is 10.

5. Common mistakes candidates make

In the "Maximum Sum Score of Array coding problem," a common mistake is recalculating the prefix and suffix sums for every index using a loop, which results in O(N^2) complexity. Another error is not handling negative numbers correctly—candidates might assume the prefix sum always increases or the suffix sum always decreases, which isn't true if the array contains negative values. Some also forget that the suffix sum at index i includes the element at index i.

6. Interview preparation tip

Master the "Total Sum Trick." For almost any problem involving prefix and suffix sums, knowing the total sum of the array allows you to derive one from the other instantly. This reduces the space complexity from O(N) (storing two extra arrays) to O(1) (storing just a few variables), which is a great way to impress an interviewer during the optimization phase of the discussion.