Maximum Nesting Depth of Two Valid Parentheses Strings

The problem requires us to split a valid parentheses string (VPS) into two disjoint subsequences A and B in such a way that they are both VPS’s, and the maximum depth of A and B is minimized.

We can achieve this by maintaining a variable depth to keep track of the current depth of the parentheses, and then use this depth to decide which subsequence the current character belongs to.

The idea is to distribute the parentheses in an alternating fashion, so that the overall depth is evenly distributed between A and B.

Here’s the Python code:

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class Solution:
    def maxDepthAfterSplit(self, seq: str) -> List[int]:
        answer = [0] * len(seq)
        depth = 0
        for i, char in enumerate(seq):
            if char == '(':
                # Increment the depth and assign the current character based on the depth
                depth += 1
                answer[i] = depth % 2
            else:
                # Assign the current character based on the depth and then decrement the depth
                answer[i] = depth % 2
                depth -= 1
        return answer

Explanation:

• We iterate through the characters of seq, and for each opening parenthesis '(', we increment the depth and assign the character to either A or B based on whether the current depth is odd or even.
• For each closing parenthesis ')', we first assign the character based on the current depth and then decrement the depth.
• By alternating the characters based on the current depth, we ensure that both A and B have balanced and minimized depths.

The time complexity of this solution is O(N), where N is the length of the given string, and the space complexity is O(N) for storing the answer array.