Problem

You are part of a scientific team developing a new technique to image crystal structures at the molecular level. The technique involves blowing a very fine wind over the surface of the crystal at various angles to detect boundaries (indicated by molecules that are exposed to the wind). This is repeated with different wind directions and the boundaries observed for each direction are recorded. Your team has already collected the data, but – as is often the case with applied science – now the real work, analysis, must begin.

For a given crystal, you will receive the directions in which wind blew over the surface, and the locations of all boundaries encountered by each of these winds. For a wind blowing in direction (wx, wy), a boundary is defined as a location (x, y) such that a molecule exists at (x, y) and no molecule exists at (x − wx, y − wy). Note that for technical reasons wx and wy are not necessarily relatively prime.

The data might not uniquely determine the structure of the crystal. You must find the two unique structures with the minimal and maximal number of molecules consistent with the observations.

For example, in the first sample input, nine different molecules are directly encountered by the given winds. There must be a molecule at location (3, 3) because otherwise (4, 2) would be a boundary for the third wind. For similar reasons, there must be molecules at (4, 4) and (5, 5). There cannot be any further molecules as they would result in additional observations for some of the winds.

input

The first line of input contains three integers dx, dy, and k, where dx and dy (1 ≤ dx, dy ≤ 103) are the maximum dimensions of the crystal structure, and k (1 ≤ k ≤ 10) is the number of times wind was blown over the crystal.

Each of the remaining k lines specifies the data for one wind. These lines each start with two integers wx and wy (-dx ≤ wx ≤ dx and -dy ≤ wy ≤ dy, but not both zero) denoting the direction of the wind. Then comes an integer b (0 ≤ b ≤ 105) giving the number of boundaries encountered by this wind. The line finishes with b distinct pairs of integers x, y (1 ≤ x ≤ dx and 1 ≤ y ≤ dy) listing each observed boundary.

You may assume the input is consistent with at least one crystal and that no molecules exist outside the specified dimensions.

output

Output two textual representations of the crystal structure separated by an empty line. Each structure has dy rows of dx characters, with the top-left corner corresponding to location (1, 1). The first is the structure with the minimal number of molecules consistent with the observations, the second is the maximal one. Use ‘#’ for a location where a molecule exists and ‘.’ for a location where no molecule exists.

sample input

6 6 3
1 1 3 3 1 1 3 2 2
0 2 6 3 1 1 3 2 2 6 4 5 3 4 2
1 -1 4 1 3 2 4 3 5 4 6

sample output

..#...
.#.#..
#.#.#.
.#.#.#
..#.#.
...#..

..#...
.#.#..
#.#.#.
.#.#.#
..#.#.
...#..

solve that problem using with pythonShare Prompt
user
Sure, let's try to translate the problem requirements into Python code. Here's a high-level overview:

1. Parse the input, extracting the data from each wind direction.
2. Do some preliminary checks on the input data, such as ensuring there are no invalid wind directions and that the overlapping boundaries don't contradict each other.
3. Define a function to build and store the crystal structure. It should return a tuple of two lists, where each list represents one crystal structure (min and max respectively). The lists should be flattened into a 2D array that uses a string to mark occupied and empty positions, represented as '#' and '. '.
4. Define another function to use the data to build the crystal structures. It should generate all possible structures within a certain number of molecules, which is not specified in the problem, but I'll assume it's a small number, say 5. It calculates all possible ways to place a molecule by generating all points within the valid range with an offset of the wind direction.
5. To avoid duplicates while checking for existing boundaries, construct a set of point pairs, where points are composed of coordinates from 0 to dx-1.
6. The two functions should be called successively, reducing the number of occupied positions incrementally until a solution is found.

Here's a Python script that implements the functions:

```python
import random

def is_valid_wind(wind):
    """
    Check if the wind direction is valid,
    i.e. It's positive in one dimension
    and negative in the other.
    """
    x, y = wind
    return (x > 0 and y <= 0) or (x <= 0 and y > 0)

def build_crystal(
    dims, winds, boundaries, occupied_at={}, max_mols=5
):  # Returns tuple of structures with (min, max) indices
    def pos_to_str(pos, occupied):
        x, y = pos
        s = "." * dims[0] * dims[1]
        s = s[:x] + "#" + s[x + 1:x + 1 + y] + "." + s[x + 1 + y + 1:]
        return s

    def build_one(pos, occupied):
        # Util function to build the existing part of structure
        s = pos_to_str(pos, occupied)
        return [row for row in occupied.values()] + [s]

    occupied_wind_set = set(
        tuple(sorted((x, y) for x, y in occupied_at.keys()))
        for occupied_at in occupied_at.values())

    def gen_crystal(wind, occupied, max_mols):
        # Generate all the structures within max_mols
        sigma = "x" * max_mols
        occupied_str_list = [
            build_one((x, y), occupied) for x, y in occupied.keys()
        ]

        def put_molecule(
            wind, occupied, molecule, occupied_str_list, pos_str
        ):
            # Try to add the given molecule with the given wind direction
            # If succeeded, append string, and continue recursively
            x, y = pos
            while True:
                x += wind[0]
                y += wind[1]
                if not is_valid_wind((x, y)):
                    break
                if (x, y) in occupied:
                    occupied_str_list.append(occupied_str_list[occupied[x, y]])
                    break
                occupied_str_list.append(pos_str)
                break

        boundaries = [(x, y) for x, y in boundaries if x - wind[0] >= 0 and y - wind[1] >= 0]

        for _ in range(max_mols - len(occupied_str_list)):
            put_molecule(wind, occupied, sigma, occupied_str_list, pos_str)
            occupied_str_list.reverse()
            occupied = dict((x, y, occupied_str_list.pop()) for (x, y) in occupied_str_list)

        crystals = []
        for occupied_str in occupied_str_list:
            crystal = [row.replace("#", occupied_str) for row in occupied.values()]
            crystals.append(crystal)

        if not occupied or occupied == occupied_at:
            for crystal in crystals:
                yield crystal
        else:
            yield crystals

    for (x, y, point) in sorted(boundaries, key=lambda x: x[2]):
        # If I'm wrong, I still can get invalid boundaries by this implementation
        if point not in occupied_at:  # Neworeted boundary
            occupied_at.setdefault((x, y), set())).add(point)
            yield gen_crystal((x, y), occupied.copy(), max_mols)
        elif occupied_at[point] == occupied:
            yield gen_crystal((x, y), occupied.copy(), max_mols)

    yield crystals

def find_crystal(wen_dict):
    """
    Given wind directions and crystal structure.
    Finds the unique structure that satisfies points with the
    minimal and maximal number of molecules.
    """
    min_mols = 100
    max_mols = 0
    min_point_lst = []
    max_point_lst = []

    def check(points, curr_mols, point_lst, cur_max_point_lst):
        # Recursive function counts the number of occupied point.
        # If there are more than the maximum, it won't construct further.
        for point in sorted(points, key=lambda x: x[2]):
            if point not in point_lst:
                x, y, _, = point
                curr_mols += 1
                if curr_mols <= max_mols:
                    # Then build boundary, construct all crystal structures
                    cur_x, cur_y = point
                    cur_point_lst = [point for point in point_lst]
                    occupied_set = {point for point in cur_point_lst if cur_x <= point[0] < cur_x + wx and cur_y <= point[1] < cur_y + wy}
                    cur_points = list(
                        filterfalse(
                            lambda p: p not in occupied_set or cur_point_lst.count(p) == occupied_set.get(p, 0),
                            points,
                        )
                    )
                    yield build_crystal(wen_dict["dims"], wen_dict["winds_list"], cur_points, occupied_set)
                    cur_point_lst.append(point)
                    yield from check(cur_points, curr_mols, cur_point_lst, cur_max_point_lst)
                    cur_point_lst.pop()
                cur_max_point_lst.append(point)
                if curr_mols > max_mols:
                    max_mols = curr_mols
            else:
                # Decrement and exclude one point
                yield from check(points, curr_mols, point_lst.pop(), cur_max_point_lst)

        # Yield the crystal structure with the minimal one.
        if curr_mols < min_mols:
            min_mols = curr_mols
            min_point_lst = point_lst

        yield from check(points, curr_mols, point_lst, cur_max_point_lst)

    def is_valid_boundary(point, wind, occupied):
        # Check whether the given point satisfies the boundary
        x, y = wind
        if point[0] - x < 0 or point[1] - y < 0:
            return False
        if (
            point[0] - x >= 0
            and point[1] - y >= 0
            and occupied.get((point[0] - x, point[1] - y), set())
        ):
            return False
        return True

    for wind, b_points in zip(wen_dict["winds_list"], wen_dict["boundaries"]):
        for b_point in b_points:
            if not is_valid_boundary(b_point, wind, {(x[0], x[1]) for x in wen_dict["boundaries"]}):
                return None

    yield from check({(x, y, point, wind)
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Problem

You are part of a scientific team developing a new technique to image crystal structures at the molecular level. The technique involves blowing a very fine wind over the surface of the crystal at various angles to detect boundaries (indicated by molecules that are exposed to the wind). This is repeated with different wind directions and the boundaries observed for each direction are recorded. Your team has already collected the data, but – as is often the case with applied science – now the real work, analysis, must begin.

For a given crystal, you will receive the directions in which wind blew over the surface, and the locations of all boundaries encountered by each of these winds. For a wind blowing in direction (wx, wy), a boundary is defined as a location (x, y) such that a molecule exists at (x, y) and no molecule exists at (x − wx, y − wy). Note that for technical reasons wx and wy are not necessarily relatively prime.

The data might not uniquely determine the structure of the crystal. You must find the two unique structures with the minimal and maximal number of molecules consistent with the observations.

For example, in the first sample input, nine different molecules are directly encountered by the given winds. There must be a molecule at location (3, 3) because otherwise (4, 2) would be a boundary for the third wind. For similar reasons, there must be molecules at (4, 4) and (5, 5). There cannot be any further molecules as they would result in additional observations for some of the winds.

input

The first line of input contains three integers dx, dy, and k, where dx and dy (1 ≤ dx, dy ≤ 103) are the maximum dimensions of the crystal structure, and k (1 ≤ k ≤ 10) is the number of times wind was blown over the crystal.

Each of the remaining k lines specifies the data for one wind. These lines each start with two integers wx and wy (-dx ≤ wx ≤ dx and -dy ≤ wy ≤ dy, but not both zero) denoting the direction of the wind. Then comes an integer b (0 ≤ b ≤ 105) giving the number of boundaries encountered by this wind. The line finishes with b distinct pairs of integers x, y (1 ≤ x ≤ dx and 1 ≤ y ≤ dy) listing each observed boundary.

You may assume the input is consistent with at least one crystal and that no molecules exist outside the specified dimensions.

output

Output two textual representations of the crystal structure separated by an empty line. Each structure has dy rows of dx characters, with the top-left corner corresponding to location (1, 1). The first is the structure with the minimal number of molecules consistent with the observations, the second is the maximal one. Use ‘#’ for a location where a molecule exists and ‘.’ for a location where no molecule exists.

sample input

6 6 3
1 1 3 3 1 1 3 2 2
0 2 6 3 1 1 3 2 2 6 4 5 3 4 2
1 -1 4 1 3 2 4 3 5 4 6

sample output

..#...
.#.#..
#.#.#.
.#.#.#
..#.#.
...#..

..#...
.#.#..
#.#.#.
.#.#.#
..#.#.
...#..

solve that problem using with pythonShare Prompt
user
This problem involves finding the minimum and maximum crystal structures based on the given wind directions and observed boundaries. Here's a Python solution using backtracking to find all possible structures, then selecting the minimum and maximum ones.

```python
from collections import defaultdict
from itertools import product

def validate(structure, observations):
    for wind, boundaries in observations.items():
        wx, wy = wind
        for x, y in boundaries:
            if not (structure[x][y] and not structure[x - wx][y - wy]):
                return False
    return True

def backtrack(structure, observations, min_struct, max_struct):
    if validate(structure, observations):
        min_molecules = sum(sum(row) for row in min_struct)
        max_molecules = sum(sum(row in max_struct)))
        molecules = sum(sum(row) for row in structure)

        if molecules < min_molecules:
            min_struct[:] = [row[:] for row in structure]
        if molecules > max_molecules:
            max_struct[:] = [row[:] for row in structure]

    for x, y in product(range(dx), range(dy)):
        if structure[x][y] == -1:
            structure[x][y] = 0
            backtrack(structure, observations, min_struct, max_struct)
            structure[x][y] = 1
            backtrack(structure, observations, min_struct, max_struct)
            structure[x][y] = -1

dx, dy, k = map(int, input().split())
observations = defaultdict(list)

for _ in range(k):
    wx, wy, b = map(int, input().split())
    observations[(wx, wy)].extend([list(map(int, input().split())) for _ in range(b)])

structure = [[-1] * dy for _ in range(dx)]
min_struct, max_struct = [[0] * dy for _ in range(dx)], [[0] * dy for _ in range(dx)]
backtrack(structure, observations, min_struct, max_struct)

for row in min_struct:
    print(''.join('#' if cell else '.' for cell in row))
print()
for row in max_struct:
    print(''.join('#' if cell else '.' for cell in row))
```

This solution first reads the input, then uses backtracking to generate all possible structures. It checks each generated structure against the observations and updates the minimum and maximum structures if necessary. Finally, it prints the minimum and maximum structures as required.

This solution might not be the most efficient one due to the use of backtracking, but it should work for small input sizes. For larger input sizes, a more efficient algorithm might be needed.
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