Bracket Coloring
Description
A regular bracket sequence 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:
- the bracket sequences "()()" and "(())" are regular (the resulting expressions are: "(1)+(1)" and "((1+1)+1)");
- the bracket sequences ")(", "(" and ")" are not.
A bracket sequence is called beautiful if one of the following conditions is satisfied:
- it is a regular bracket sequence;
- if the order of the characters in this sequence is reversed, it becomes a regular bracket sequence.
For example, the bracket sequences "()()", "(())", ")))(((", "))()((" are beautiful.
You are given a bracket sequence $s$. You have to color it in such a way that:
- every bracket is colored into one color;
- for every color, there is at least one bracket colored into that color;
- for every color, if you write down the sequence of brackets having that color in the order they appear, you will get a beautiful bracket sequence.
Color the given bracket sequence $s$ into the minimum number of colors according to these constraints, or report that it is impossible.
谷歌翻译
正则括号序列是通过在序列的原始字符之间插入字符“1”和“+”，可以将括号序列转化为正确的算术表达式。例如：
- 括号序列“()()”和“(())”是正则表达式（结果表达式为：“(1)+(1)”和“((1+1)+1)”）；
- 括号序列“)(”、“(”和“)”不是。
如果满足以下条件之一，则括号序列被称为漂亮的：
- 它是一个规则的括号序列；
- 如果这个序列中的字符顺序颠倒过来，它就变成了常规的括号序列。
例如，括号序列“()()”、“(())”、“)))(((”、“))()((”很漂亮。
您将获得一个括号序列 。你必须以这样的方式给它上色：
- 每个括号都涂成一种颜色；
- 对于每种颜色，至少有一个括号着色为该颜色；
- 对于每种颜色，如果您按照出现的顺序写下具有该颜色的括号序列，您将得到一个漂亮的括号序列。
根据这些约束将给定的括号序列 着色为最小数量的颜色，或者报告这是不可能的。
Input
The first line contains one integer $t$ ($1 \le t \le 10^4$) — the number of test cases.
Each test case consists of two lines. The first line contains one integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the number of characters in $s$. The second line contains $s$ — a string of $n$ characters, where each character is either "(" or ")".
Additional constraint on the input: the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
Output
For each test case, print the answer as follows:
- if it is impossible to color the brackets according to the problem statement, print $-1$;
- otherwise, print two lines. In the first line, print one integer $k$ ($1 \le k \le n$) — the minimum number of colors. In the second line, print $n$ integers $c_1, c_2, \dots, c_n$ ($1 \le c_i \le k$), where $c_i$ is the color of the $i$-th bracket. If there are multiple answers, print any of them.
4
8
((())))(
4
(())
4
))((
3
(()
2
2 2 2 1 2 2 2 1
1
1 1 1 1
1
1 1 1 1
-1
相关
在下列比赛中: