C. Even Positions
constructive algorithms; data structures; greedy
Monocarp had a regular bracket sequence \(s\) of length \(n\) (\(n\) is even). He even came up with his own way to calculate its cost.He knows that in a regular bracket sequence (RBS), each opening bracket is paired up with the corresponding closing bracket. So he decided to calculate the cost of RBS as the sum of distances between pairs of corresponding bracket pairs.For example, let's look at RBS (())(). It has three pairs of brackets: (__)__: the distance between brackets at position \(1\) and at \(4\) is \(4 - 1 = 3\); _()___: the distance is \(3 - 2 = 1\); ____(): the distance is \(6 - 5 = 1\). So the cost of (())() is \(3 + 1 + 1 = 5\).Unfortunately, due to data corruption, Monocarp lost all characters on odd positions \(s_1, s_3, \dots, s_{n-1}\). Only characters on even positions (\(s_2, s_4, \dots, s_{n}\)) remain. For example, (())() turned to _(_)_).Monocarp wants to restore his RBS by placing brackets on the odd positions. But since the restored RBS may not be unique, he wants to choose one with minimum cost. It's too hard to do for Monocarp alone, so can you help him?Reminder: A regular bracket sequence is a string consisting of only brackets, such that this sequence, when inserted 1-s and +-s, gives a valid mathematical expression. For example, (), (()) or (()())() are RBS, while ), ()( or ())(() are not.
The first line contains a single integer \(t\) (\(1 \le t \le 5000\)) — the number of test cases. Next \(t\) cases follow.The first line of each test case contains a single integer \(n\) (\(2 \le n \le 2 \cdot 10^5\); \(n\) is even) — the length of string \(s\).The second line of each test case contains a string \(s\) of length \(n\), where all characters on the odd positions are '_' and all characters on the even positions are either '(' or ')'.Additional constraints: \(s\) can be restored to at least one regular bracket sequence; the total sum of \(n\) over all test cases doesn't exceed \(2 \cdot 10^5\).
For each test case, print one integer — the minimum cost of the regular bracket sequence that can be obtained from \(s\) by replacing '_'-s with brackets.
In the first test case, it's optimal to make \(s\) equal to (())(). The cost of \(s\) will be equal to \(3 + 1 + 1 = 5\).In the second test case, the only option is to make \(s\) equal to () with cost \(1\).In the third test case, the only possible RBS is ()()()() with cost \(1 + 1 + 1 + 1 = 4\).In the fourth test case, it's optimal to make \(s\) equal to (())(()) with cost \(3 + 1 + 3 + 1 = 8\).