Minimum Index of a Valid Split

Minimum Index of a Valid Split

1. What is this problem about?

The "Minimum Index of a Valid Split" interview question involves splitting an array into two non-empty parts such that both parts share a common "dominant" element. An element is considered dominant in an array if it appears more than half the time (frequency > length / 2).

In this problem, you are given an array that is guaranteed to have a dominant element. You need to find the smallest index i such that if you split the array into nums[0...i] and nums[i+1...n-1], the dominant element of the original array remains the dominant element in both the left and right sub-arrays. If no such split exists, the goal is to return -1.

2. Why is this asked in interviews?

This problem is a great test of a candidate's understanding of Boyer-Moore Voting Algorithm and Frequency Counting. Companies like Amazon and Google use it to see if candidates can:

• Identify the Core Constraint: Recognize that only the dominant element of the whole array can be the dominant element of both splits.
• Optimize Space and Time: Solve the problem in linear time $O(n)$ with minimal extra space.
• Handle Edge Cases: Manage the "more than half" condition accurately, especially with integer division.

3. Algorithmic pattern used

The solution typically follows these steps:

1. Find the Dominant Element: Use the Boyer-Moore Voting Algorithm or a hash map to identify the element that appears more than $n/2$ times in the total array.
2. Pre-calculate Total Frequency: Count exactly how many times this dominant element appears in the entire array.
3. Iterate and Split: Traverse the array from left to right, keeping a running count of the dominant element's appearances in the "left" part.
4. Validate the Split: At each index, calculate the frequency of the dominant element in the left and right parts. If both frequencies satisfy the count > length / 2 rule for their respective sub-array lengths, you've found the minimum index.

4. Example explanation

Consider the array [1, 2, 2, 2, 2, 1]. Total length is 6. The element 2 appears 4 times. Since $4 > 6/2$, 2 is the dominant element. Total frequency of 2 is 4.

• Split at index 0: Left [1], Right [2, 2, 2, 2, 1].
  • Left length 1: 2 appears 0 times. $0 \le 1/2$. Invalid.
• Split at index 1: Left [1, 2], Right [2, 2, 2, 1].
  • Left length 2: 2 appears 1 time. $1 \le 2/2$. Invalid.
• Split at index 2: Left [1, 2, 2], Right [2, 2, 1].
  • Left length 3: 2 appears 2 times. $2 > 3/2$ (1.5). Valid.
  • Right length 3: 2 appears 2 times. $2 > 3/2$. Valid. The minimum index is 2.

5. Common mistakes candidates make

• Not Identifying the Dominant Element: Trying to find a new dominant element for every possible split using a hash map inside a loop, resulting in $O(n^2)$ time.
• Off-by-one Errors: Miscalculating the length of the right sub-array or the split point.
• Strict Inequality: Forgetting that "dominant" means strictly greater than half. For an array of length 4, an element must appear 3 times, not 2.

6. Interview preparation tip

Always start by simplifying the problem. In this case, realizing that the "split" dominant element must be the original dominant element is the "Aha!" moment. Practice the Boyer-Moore Voting Algorithm as it is the most efficient way to find a majority element and is a common building block for many array-based interview questions.