Problem p02800 - Generation 2

Orig Description

Score : 1100 points
Problem StatementWe have an H \times W grid, where each square is painted white or black in the initial state.
Given are strings A_1, A_2, ..., A_H representing the colors of the squares in the initial state.
For each pair (i, j) (1 \leq i \leq H, 1 \leq j \leq W), if the j-th character of A_i is ., the square at the i-th row and j-th column is painted white; if that character is #, that square is painted black.
Among the 2^{HW} ways for each square in the grid to be painted white or black, how many can be obtained from the initial state by performing the operations below any number of times (possibly zero) in any order? Find this count modulo 998,244,353.
Choose one row, then paint all the squares in that row white.
Choose one row, then paint all the squares in that row black.
Choose one column, then paint all the squares in that column white.
Choose one column, then paint all the squares in that column black.
Constraints
1 \leq H, W \leq 10
|A_i| = W (1 \leq i \leq H)
All strings A_i consist of . and #.
H and W are integers.
InputInput is given from Standard Input in the following format:
H W
A_1
A_2
\vdots
A_H
OutputPrint the answer.
Sample Input 12 2
#.
.#
Sample Output 115
For example, if we paint the second row black, the grid becomes:
#.
##
Sample Input 23 3
...
...
...
Sample Output 2230
Sample Input 32 4
#...
...#
Sample Output 3150
Sample Input 46 7
.......
.......
.#.....
..#....
.#.#...
.......
Sample Output 4203949910

Extracted Specification

Canonicalized Input Description:

An integer H (1 ≤ H ≤ 10), representing some dimension.
An integer W (1 ≤ W ≤ 10), representing some dimension.
A list of H strings, each of length W, consisting of characters from the set {'.', '#'}. 

### Example Input:

```
2 2
#.
.#
```

### Function Signature:
Write a function f(H, W, A) that takes in the input. 
```python
def f(H: int, W: int, A: List[str]):
    '''
    H: an integer
    W: an integer
    A: a list of strings
    '''
```

Test Case Generator

Sure, I can help you write a test case generator for the given problem statement. Here's a Python script that generates test cases according to the constraints specified:

```python
import random

def generate_grid(H, W):
    grid = []
    for _ in range(H):
        row = ''.join(random.choice(['.', '#']) for _ in range(W))
        grid.append(row)
    return grid

def tcgen():
    H = random.randint(1, 10)
    W = random.randint(1, 10)
    grid = generate_grid(H, W)
    
    return H, W, grid

def print_test_case(H, W, grid):
    print(H, W)
    for row in grid:
        print(row)

# Example of generating a test case and printing it
H, W, grid = tcgen()
print_test_case(H, W, grid)
```

### Explanation:
1. **`generate_grid(H, W)`**: This function generates an H x W grid where each cell is either '.' (white) or '#' (black). Each character in a row is chosen randomly from the set {'.', '#'}.
2. **`tcgen()`**: This function generates random values for H and W within the constraints (1 ≤ H, W ≤ 10). It then uses `generate_grid(H, W)` to create the grid.
3. **`print_test_case(H, W, grid)`**: This function prints the test case in the format required by the problem statement.

### Example Output:
Running the script will produce output similar to the following:

```
7 5
.#...
##..#
.#.#.
..###
.#..#
#.###
.####
```

The script allows you to generate random valid test cases which can be useful for testing the solution to the problem statement. You can modify or extend the script to generate larger sets of test cases or add additional functionality as needed.

Extract Arguments

def extract_arguments(fh):
    H, W = map(int, fh.readline().strip().split())
    A = [fh.readline().strip() for _ in range(H)]
    return H, W, A