Maximum Nesting Depth of Two Valid Parentheses Strings

Description

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). The subsequences may not necessarily be contiguous.

For example, for the sequence 123456789, one possible split is:


A = {1, 3, 5, 7, 9}`,




B = {2, 4, 6, 8}`.



This corresponds to the output [0, 1, 0, 1, 0, 1, 0, 1, 0] where 0 indicates membership in A and 1 indicates membership in B.

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.