RBS Deletion

3000ms

524288K

Description:

A bracket sequence is a string containing only characters "(" and ")". A regular bracket sequence (or, shortly, an RBS) is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters "1" and "+" between the original characters of the sequence. For example:

bracket sequences "()()" and "(())" are regular (the resulting expressions are: "(1)+(1)" and "((1+1)+1)");
bracket sequences ")(", "(" and ")" are not.

You are given a string $$$s$$$, which is an RBS. You can apply any number of operations to this string. Each operation can have one of the following types:

choose some non-empty prefix of $$$s$$$ and remove it from $$$s$$$, so $$$s$$$ is still an RBS. For example, we can apply this operation as follows: "(())()(())()()" $$$\to$$$ "()()" (the first $$$10$$$ characters are removed);
choose some contiguous non-empty substring of $$$s$$$ and remove it from $$$s$$$, so $$$s$$$ is still an RBS. For example, we can apply this operation as follows: "(())()(())()()" $$$\to$$$ "(())()()()" (the characters from the $$$7$$$-th to the $$$10$$$-th are removed).

The operation $$$2$$$ can be applied at most $$$k$$$ times. Calculate the maximum number of operations you can apply until $$$s$$$ becomes empty.

Input:

The first line contains one integer $$$t$$$ ($$$1 \le t \le 10^5$$$) — the number of test cases.

Each test case is described by two lines. The first line contains two integers $$$n$$$ and $$$k$$$ ($$$2 \le n \le 2 \cdot 10^5$$$; $$$1 \le k \le n$$$; $$$n$$$ is even) — the length of $$$s$$$ and the maximum number of operations of type $$$2$$$ you can apply.

The second line contains a string $$$s$$$ of $$$n$$$ characters '(' and ')'. This string is an RBS.

The sum of $$$n$$$ over all test cases doesn't exceed $$$2 \cdot 10^5$$$.

Output:

For each test case, print one integer — the maximum number of operations you can apply.

Sample Input:

3
12 2
(()())((()))
6 3
()()()
8 1
(((())))

Sample Output:

4
3
2

Informação

Provedor Codeforces

Código CF1488B