Preparando MOJI
Chiyuu has a bracket sequence$$$^\dagger$$$ $$$s$$$ of length $$$n$$$. Let $$$k$$$ be the minimum number of characters that Chiyuu has to remove from $$$s$$$ to make $$$s$$$ balanced$$$^\ddagger$$$.
Now, Koxia wants you to count the number of ways to remove $$$k$$$ characters from $$$s$$$ so that $$$s$$$ becomes balanced, modulo $$$998\,244\,353$$$.
Note that two ways of removing characters are considered distinct if and only if the set of indices removed is different.
$$$^\dagger$$$ A bracket sequence is a string containing only the characters "(" and ")".
$$$^\ddagger$$$ A bracket sequence is called balanced if one can turn it into a valid math expression by adding characters + and 1. For example, sequences (())(), (), (()(())) and the empty string are balanced, while )(, ((), and (()))( are not.
The first line of input contains a string $$$s$$$ ($$$1 \leq |s| \leq 5 \cdot {10}^5$$$) — the bracket sequence.
It is guaranteed that $$$s$$$ only contains the characters "(" and ")".
Output a single integer — the number of ways to remove $$$k$$$ characters from $$$s$$$ so that $$$s$$$ becomes balanced, modulo $$$998\,244\,353$$$.
())(()
4
(
1
In the first test case, it can be proved that the minimum number of characters that Chiyuu has to remove is $$$2$$$. There are $$$4$$$ ways to remove $$$2$$$ characters to make $$$s$$$ balanced as follows. Deleted characters are noted as red.
In the second test case, the only way to make $$$s$$$ balanced is by deleting the only character to get an empty bracket sequence, which is considered balanced.
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Provedor Codeforces
Código CF1770G
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Datas 09/05/2023 10:34:42
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