Jatayu's Balanced Bracket Sequence

题意翻译

### 题目大意 对于一个长度为 $2n$ 的**合法**的括号串 $s$，按照如下方法构造一张无向图： - 括号序列的所有位置都是无向图中的一个点。 - 对于该序列的任意位置 $l$，它能向另一个位置 $r$ 连边当且仅当满足子串 $s[l, \; \dots , \; r]$ 也是一个**合法**括号串。 求这张无向图的连通块个数。 ### 输入格式 第一行包含一个整数 $T \; (1 \leqslant T \leqslant 10^5)$ 表示测试样例组数。 对于每组测试样例，第一行包含一个整数 $n \; (1 \leqslant n \leqslant 10^5)$ 表示序列长度为 $2n$。 接下来的一行包含一个长度为 $2n$ 的合法括号串。 ### 输出格式 对于每组测试样例，包含一个整数表示该串构造的无向图的连通块数。 $Translated \; by \; Zigh$

题目描述

Last summer, Feluda gifted Lalmohan-Babu a balanced bracket sequence $ s $ of length $ 2 n $ . Topshe was bored during his summer vacations, and hence he decided to draw an undirected graph of $ 2 n $ vertices using the balanced bracket sequence $ s $ . For any two distinct vertices $ i $ and $ j $ ( $ 1 \le i < j \le 2 n $ ), Topshe draws an edge (undirected and unweighted) between these two nodes if and only if the subsegment $ s[i \ldots j] $ forms a balanced bracket sequence. Determine the number of connected components in Topshe's graph. See the Notes section for definitions of the underlined terms.

输入输出格式

输入格式

Each test contains multiple test cases. The first line contains a single integer $ t $ ( $ 1 \le t \le 10^5 $ ) — the number of test cases. Description of the test cases follows. The first line of each test case contains a single integer $ n $ ( $ 1 \le n \le 10^5 $ ) — the number of opening brackets in string $ s $ . The second line of each test case contains a string $ s $ of length $ 2 n $ — a balanced bracket sequence consisting of $ n $ opening brackets "(", and $ n $ closing brackets ")". It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 10^5 $ .

输出格式

For each test case, output a single integer — the number of connected components in Topshe's graph.

输入输出样例

输入样例 #1

4
1
()
3
()(())
3
((()))
4
(())(())

输出样例 #1

1
2
3
3

说明

Sample explanation: In the first test case, the graph constructed from the bracket sequence (), is just a graph containing nodes $ 1 $ and $ 2 $ connected by a single edge. In the second test case, the graph constructed from the bracket sequence ()(()) would be the following (containing two connected components): Definition of Underlined Terms: - A sequence of brackets is called balanced if one can turn it into a valid math expression by adding characters $ + $ and $ 1 $ . For example, sequences (())(), (), and (()(())) are balanced, while )(, ((), and (()))( are not. - The subsegment $ s[l \ldots r] $ denotes the sequence $ [s_l, s_{l + 1}, \ldots, s_r] $ . - A connected component is a set of vertices $ X $ such that for every two vertices from this set there exists at least one path in the graph connecting these vertices, but adding any other vertex to $ X $ violates this rule.