codeforces#P1837D. Bracket Coloring

    ID: 33852 远端评测题 2000ms 512MiB 尝试: 0 已通过: 0 难度: (无) 上传者: 标签>constructive algorithmsdata structuresgreedy

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.

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 $2 \cdot 10^5$.

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.

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