Maximum Nesting Depth of Two Valid Parentheses Strings

Introduction

A string is a valid parentheses string (denoted VPS) if and only if it consists of “(” and “)” characters only, and:

It is the empty string, or It can be written as AB (A concatenated with B), where A and B are VPS’s, or It can be written as (A), where A is a VPS. We can similarly define the nesting depth depth(S) of any VPS S as follows:

depth("") = 0
depth(A + B) = max(depth(A), depth(B)), where A and B are VPS's
depth("(" + A + ")") = 1 + depth(A), where A is a VPS.

For example, “”, “()()”, and “()(()())” are VPS’s (with nesting depths 0, 1, and 2), and “)(” and “(()” are not VPS’s.

Given a VPS seq, split it into two disjoint subsequences A and B, such that A and B are VPS’s (and A.length + B.length = seq.length).

Now choose any such A and B such that max(depth(A), depth(B)) is the minimum possible value.

Return an answer array (of length seq.length) that encodes such a choice of A and B: answer[i] = 0 if seq[i] is part of A, else answer[i] = 1. Note that even though multiple answers may exist, you may return any of them.

Example 1:

Input: seq = "(()())"
Output: [0,1,1,1,1,0]

Example 2:

Input: seq = "()(())()"
Output: [0,0,0,1,1,0,1,1]

Constraints:

1 <= seq.size <= 10000

Solution

func maxDepthAfterSplit(seq string) []int {
    n := len(seq)
    out := make([]int, n)
	stack, top := make([]int, n), -1
	for i, v := range seq {
		if v == '(' {
			top++
			stack[top], out[i] = i, top % 2
		} else {
			out[i] = out[stack[top]]
			top--
		}
	}
	return out
}

Explanation

Let’s create a virtual machine that will have stack and top reference. On each open paretness ‘(’ we push in to stack and increment top. We record out[i] to be true for even stack counts. When we exit function from stack, we record out as a value that was at the time of entering stack and decrement top.