All Divisions With the Highest Score of a Binary Array

All Divisions With the Highest Score of a Binary Array

What is this problem about?

The "All Divisions With the Highest Score of a Binary Array interview question" is a prefix-based array challenge. You are given a binary array (containing only 0s and 1s). You can "split" the array at any index $i$ (from $0$ to $n$) into two parts: a left part and a right part. The "score" of this division is the number of 0s in the left part plus the number of 1s in the right part. Your goal is to find all indices that result in the maximum possible score.

Why is this asked in interviews?

Google uses the "All Divisions With the Highest Score of a Binary Array coding problem" to check if a candidate can optimize a linear scan. The naive approach involves calculating the score at each index by counting 0s and 1s from scratch, which is $O(N^2)$. However, an efficient $O(N)$ solution exists. This tests the candidate's ability to use "Prefix Sum interview pattern" techniques to update scores in constant time as they move through the array.

Algorithmic pattern used

This problem utilizes the Prefix Sum or Running Total pattern.

1. Initial State: Calculate the total number of 1s in the entire array. This is the score for the division at index 0 (where the left part is empty).
2. Iteration: As you move the split point from index $i$ to $i+1$, you are moving one element from the right part to the left part.
   • If that element is a 0, the left score increases by 1.
   • If that element is a 1, the right score decreases by 1.
3. Tracking: Keep track of the maximum score encountered and store all indices that achieve it.

Example explanation

Array: [0, 0, 1, 0]

• Split at 0: Left [], Right [0, 0, 1, 0]. Score: (0 zeros) + (1 one) = 1.
• Split at 1: Left [0], Right [0, 1, 0]. Score: (1 zero) + (1 one) = 2.
• Split at 2: Left [0, 0], Right [1, 0]. Score: (2 zeros) + (1 one) = 3.
• Split at 3: Left [0, 0, 1], Right [0]. Score: (2 zeros) + (0 ones) = 2.
• Split at 4: Left [0, 0, 1, 0], Right []. Score: (3 zeros) + (0 ones) = 3. Max score is 3, achieved at indices [2, 4].

Common mistakes candidates make

• Index boundaries: Forgetting that there are $n+1$ possible split positions for an array of length $n$ (including before the first element and after the last).
• Recalculating every time: Using a nested loop to count 0s and 1s for every split point, resulting in $O(N^2)$ time complexity.
• Incorrect Score Logic: Confusing which side needs 0s and which side needs 1s.

Interview preparation tip

Practice the technique of "updating a result based on the previous state." Many array problems that look like they require $O(N^2)$ can be solved in $O(N)$ by observing how the result changes when moving from index $i$ to $i+1$. This is a core part of the "Array interview pattern."