7139: ABC268 —— G - Random Student ID
[Creator : ]
Description
Takahashi Elementary School has NN new students. For $i = 1, 2, \ldots, N$, the name of the $i$-th new student is $S_i$ (which is a string consisting of lowercase English letters). The names of the $N$ new students are distinct.
The $N$ students will be assigned a student ID $1, 2, 3, \ldots, N$ in ascending lexicographical order of their names. However, instead of the ordinary order of lowercase English letters where a is the minimum and z is the maximum, we use the following order:
What is the lexicographical order? A string $S = S_1S_2\ldots S_{|S|}$ is said to be lexicographically smaller than a string $T = T_1T_2\ldots T_{|T|}$ if one of the following 1. and 2. holds. Here, $|S|$ and $|T|$ denote the lengths of $S$ and $T$, respectively.
The $N$ students will be assigned a student ID $1, 2, 3, \ldots, N$ in ascending lexicographical order of their names. However, instead of the ordinary order of lowercase English letters where a is the minimum and z is the maximum, we use the following order:
- First, Principal Takahashi chooses a string PP from the $26!$ permutations of the string abcdefghijklmnopqrstuvwxyz of length $26$, uniformly at random.
- The lowercase English characters that occur earlier in $P$ are considered smaller.
What is the lexicographical order? A string $S = S_1S_2\ldots S_{|S|}$ is said to be lexicographically smaller than a string $T = T_1T_2\ldots T_{|T|}$ if one of the following 1. and 2. holds. Here, $|S|$ and $|T|$ denote the lengths of $S$ and $T$, respectively.
- $|S| \lt |T|$ and $S_1S_2\ldots S_{|S|} = T_1T_2\ldots T_{|S|}$.
-
There exists an integer $1 \leq i \leq \min\lbrace |S|, |T| \rbrace$ satisfying the following two conditions:
- $S_1S_2\ldots S_{i-1} = T_1T_2\ldots T_{i-1}$
- $S_i$ is a smaller character than $T_i$.
Notes
We can prove that the sought expected value is always a rational number. Moreover, under the Constraints of this problem, when the value is represented as $\frac{P}{Q}$ by two coprime integers $P$ and $Q$, we can prove that there is a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Find such $R$.Input
Input is given from Standard Input in the following format:
$N$
$S_1$
$S_2$
$\vdots$
$S_N$
$N$
$S_1$
$S_2$
$\vdots$
$S_N$
Output
Print $N$ lines. For each $i = 1, 2, \ldots, N$, the $i$-th line should contain the expected value, modulo 998244353, of the student ID assigned to Student $i$.
Constraints
$2≤N$
$N$ is an integer.
$S_i$ is a string of length at least 1 consisting of lowercase English letters.
The sum of lengths of the given strings is at most $5 \times 10^5$.
$i \neq j \Rightarrow S_i \neq S_j$
$N$ is an integer.
$S_i$ is a string of length at least 1 consisting of lowercase English letters.
The sum of lengths of the given strings is at most $5 \times 10^5$.
$i \neq j \Rightarrow S_i \neq S_j$
Sample 1 Input
3
a
aa
ab
Sample 1 Output
1
499122179
499122179
The expected value of the student ID assigned to Student 1 is 1; the expected values of the student ID assigned to Student 2 and 3 are $\frac{5}{2}$.
Note that the answer should be printed modulo 998244353.
For example, the sought expected value for Student 2 and 3 is $\frac{5}{2}$, and we have $2 \times 499122179 \equiv 5\pmod{998244353}$, so 499122179 should be printed.
Sample 2 Input
3
a
aa
aaa
Sample 2 Output
1
2
3