CF1264D2.Beautiful Bracket Sequence (hard version)

普及/提高-

通过率：0%

AC君温馨提醒

该题目为【codeforces】题库的题目，您提交的代码将被提交至codeforces进行远程评测，并由ACGO抓取测评结果后进行展示。由于远程测评的测评机由其他平台提供，我们无法保证该服务的稳定性，若提交后无反应，请等待一段时间后再进行重试。

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.

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 $9 = 2 + 2 + 3 + 2$ .

首页