311316: CF1970A2. Balanced Unshuffle (Medium)
Description
The differences with the easy version of this problem are highlighted in bold.
A parentheses sequence is a string consisting of characters "(" and ")", for example "(()((".
A balanced parentheses sequence is a parentheses sequence which can become a valid mathematical expression after inserting numbers and operations into it, for example "(()(()))".
The balance of a parentheses sequence is defined as the number of opening parentheses "(" minus the number of closing parentheses ")". For example, the balance of the sequence "(()((" is 3.
A balanced parentheses sequence can also be defined as a parentheses sequence with balance 0 such that each of its prefixes has a non-negative balance.
We define the balanced shuffle operation that takes a parentheses sequence and returns a parentheses sequence as follows: first, for every character of the input sequence, we compute the balance of the prefix of the sequence before that character and write those down in a table together with the positions of the characters in the input sequence, for example:
| Prefix balance | 0 | 1 | 2 | 1 | 2 | 3 | 2 | 1 |
| Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Character | ( | ( | ) | ( | ( | ) | ) | ) |
Then, we sort the columns of this table in increasing order of prefix balance, breaking ties in decreasing order of position. In the above example, we get:
| Prefix balance | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 3 |
| Position | 1 | 8 | 4 | 2 | 7 | 5 | 3 | 6 |
| Character | ( | ) | ( | ( | ) | ( | ) | ) |
The last row of this table forms another parentheses sequence, in this case "()(()())". This sequence is called the result of applying the balanced shuffle operation to the input sequence, or in short just the balanced shuffle of the input sequence.
Surprisingly, it turns out that the balanced shuffle of any balanced parentheses sequence is always another balanced parentheses sequence (we will omit the proof for brevity). Even more surprisingly, the balanced shuffles of two different balanced parentheses sequences are always different, therefore the balanced shuffle operation is a bijection on the set of balanced parentheses sequences of any given length (we will omit this proof, too).
You are given a balanced parentheses sequence. Find its preimage: the balanced parentheses sequence the balanced shuffle of which is equal to the given sequence.
InputThe only line of input contains a string $s$ consisting only of characters "(" and ")". This string is guaranteed to be a non-empty balanced parentheses sequence with its length not exceeding $1\,000$.
OutputPrint the balanced parentheses sequence $t$ such that the balanced shuffle of $t$ is equal to $s$. It is guaranteed that the answer always exists and is unique.
ExampleInput()(()())Output
(()(()))
Output
这个问题是关于平衡括号序列的一个变种。给定一个平衡括号序列，你需要找到一个序列，其平衡洗牌（balanced shuffle）后的结果是给定的序列。
平衡洗牌操作定义为：对于输入序列的每个字符，我们计算序列中在该字符之前的前缀的平衡，并将其与字符在输入序列中的位置一起记录下来，然后按照前缀平衡的增加顺序对表格的列进行排序，如果前缀平衡相同，则按照位置的减少顺序排序。排序后的最后一行形成的新序列就是输入序列的平衡洗牌结果。
输入输出数据格式：
输入：
- 输入包含一个字符串 $ s $，该字符串只由字符 "(" 和 ")" 组成。
- 字符串 $ s $ 保证是一个非空的平衡括号序列，其长度不超过 $ 1,000 $。
输出：
- 输出平衡括号序列 $ t $，使得 $ t $ 的平衡洗牌等于 $ s $。
- 答案保证存在且唯一。
示例：
输入：
```
()(()())
```
输出：
```
(()(()))
```
请注意，这里只提供了题目大意和输入输出数据格式的翻译，没有提供完整的解题思路和算法实现。题目大意： 这个问题是关于平衡括号序列的一个变种。给定一个平衡括号序列，你需要找到一个序列，其平衡洗牌（balanced shuffle）后的结果是给定的序列。 平衡洗牌操作定义为：对于输入序列的每个字符，我们计算序列中在该字符之前的前缀的平衡，并将其与字符在输入序列中的位置一起记录下来，然后按照前缀平衡的增加顺序对表格的列进行排序，如果前缀平衡相同，则按照位置的减少顺序排序。排序后的最后一行形成的新序列就是输入序列的平衡洗牌结果。 输入输出数据格式： 输入： - 输入包含一个字符串 $ s $，该字符串只由字符 "(" 和 ")" 组成。 - 字符串 $ s $ 保证是一个非空的平衡括号序列，其长度不超过 $ 1,000 $。 输出： - 输出平衡括号序列 $ t $，使得 $ t $ 的平衡洗牌等于 $ s $。 - 答案保证存在且唯一。 示例： 输入： ``` ()(()()) ``` 输出： ``` (()(())) ``` 请注意，这里只提供了题目大意和输入输出数据格式的翻译，没有提供完整的解题思路和算法实现。