A13163. Beautiful Bracket Sequence (hard version)
编程题
普及/提高-
知识点
题目描述
This is the hard version of this problem. The only difference is the limit of $n$ - the length of the input string. In this version, $1 \leq n \leq 10^6$ .
Let's define a correct bracket sequence and its depth as follow:
- An empty string is a correct bracket sequence with depth $0$ .
- If "s" is a correct bracket sequence with depth $d$ then "(s)" is a correct bracket sequence with depth $d + 1$ .
- If "s" and "t" are both correct bracket sequences then their concatenation "st" is a correct bracket sequence with depth equal to the maximum depth of $s$ and $t$ .
For a (not necessarily correct) bracket sequence $s$ , we define its depth as the maximum depth of any correct bracket sequence induced by removing some characters from $s$ (possibly zero). For example: the bracket sequence $s = $ "())(())" has depth $2$ , because by removing the third character we obtain a correct bracket sequence "()(())" with depth $2$ .
Given a string $a$ consists of only characters '(', ')' and '?'. Consider all (not necessarily correct) bracket sequences obtained by replacing all characters '?' in $a$ by either '(' or ')'. Calculate the sum of all the depths of all these bracket sequences. As this number can be large, find it modulo $998244353$ .
Hacks in this problem can be done only if easy and hard versions of this problem was solved.
Let's define a correct bracket sequence and its depth as follow:
- An empty string is a correct bracket sequence with depth $0$ .
- If "s" is a correct bracket sequence with depth $d$ then "(s)" is a correct bracket sequence with depth $d + 1$ .
- If "s" and "t" are both correct bracket sequences then their concatenation "st" is a correct bracket sequence with depth equal to the maximum depth of $s$ and $t$ .
For a (not necessarily correct) bracket sequence $s$ , we define its depth as the maximum depth of any correct bracket sequence induced by removing some characters from $s$ (possibly zero). For example: the bracket sequence $s = $ "())(())" has depth $2$ , because by removing the third character we obtain a correct bracket sequence "()(())" with depth $2$ .
Given a string $a$ consists of only characters '(', ')' and '?'. Consider all (not necessarily correct) bracket sequences obtained by replacing all characters '?' in $a$ by either '(' or ')'. Calculate the sum of all the depths of all these bracket sequences. As this number can be large, find it modulo $998244353$ .
Hacks in this problem can be done only if easy and hard versions of this problem was solved.
输入格式
The only line contains a non-empty string consist of only '(', ')' and '?'. The length of the string is at most $10^6$ .
输出格式
Print the answer modulo $998244353$ in a single line.
输入输出样例
输入 #1
??
输出 #1
1
输入 #2
(?(?))
输出 #2
9
说明/提示
In the first test case, we can obtain $4$ bracket sequences by replacing all characters '?' with either '(' or ')':
- "((". Its depth is $0$ ;
- "))". Its depth is $0$ ;
- ")(". Its depth is $0$ ;
- "()". Its depth is $1$ .
So, the answer is $1 = 0 + 0 + 0 + 1$ .
In the second test case, we can obtain $4$ bracket sequences by replacing all characters '?' with either '(' or ')':
- "(((())". Its depth is $2$ ;
- "()()))". Its depth is $2$ ;
- "((()))". Its depth is $3$ ;
- "()(())". Its depth is $2$ .
So, the answer is $9 = 2 + 2 + 3 + 2$ .
- "((". Its depth is $0$ ;
- "))". Its depth is $0$ ;
- ")(". Its depth is $0$ ;
- "()". Its depth is $1$ .
So, the answer is $1 = 0 + 0 + 0 + 1$ .
In the second test case, we can obtain $4$ bracket sequences by replacing all characters '?' with either '(' or ')':
- "(((())". Its depth is $2$ ;
- "()()))". Its depth is $2$ ;
- "((()))". Its depth is $3$ ;
- "()(())". Its depth is $2$ .
So, the answer is $9 = 2 + 2 + 3 + 2$ .