


from __future__ import annotations


def     lowerCamelCase_	(      lowerCAmelCase:	list[int]    ,							lowerCAmelCase:	list[int]    ,							lowerCAmelCase:	list[int]    ,							lowerCAmelCase:	list[list[str]]    ,							lowerCAmelCase:	int    ,							)->		None:
  _snake_case   :				Optional[int]														=	len(UpperCAmelCase__				)

  # If row is equal to the size of the board it means there are a queen in each row in
  # the current board (possible_board)
  if row == n:
    # We convert the variable possible_board that looks like this: [1, 3, 0, 2] to
    # this: ['. Q . . ', '. . . Q ', 'Q . . . ', '. . Q . ']
    boards.append(['. ' * i + 'Q ' + '. ' * (n - 1 - i) for i in possible_board]				)
    return

  # We iterate each column in the row to find all possible results in each row
  for col in range(UpperCAmelCase__				):
    # We apply that we learned previously. First we check that in the current board
    # (possible_board) there are not other same value because if there is it means
    # that there are a collision in vertical. Then we apply the two formulas we
    # learned before:
    #
    # 45º: y - x = b or 45: row - col = b
    # 135º: y + x = b or row + col = b.
    #
    # And we verify if the results of this two formulas not exist in their variables
    # respectively.  (diagonal_right_collisions, diagonal_left_collisions)
    #
    # If any or these are True it means there is a collision so we continue to the
    # next value in the for loop.
    if (
        col in possible_board
        or row - col in diagonal_right_collisions
        or row + col in diagonal_left_collisions
    ):
      continue

    # If it is False we call dfs function again and we update the inputs
    depth_first_search(
        [*possible_board, col]    ,							[*diagonal_right_collisions, row - col]    ,							[*diagonal_left_collisions, row + col]    ,							UpperCAmelCase__    ,							UpperCAmelCase__    ,							)


def     lowerCamelCase_	(      lowerCAmelCase:	int				)->		None:
  _snake_case   :				Optional[int]														=	[]
  depth_first_search([]    ,							[]    ,							[]    ,							UpperCAmelCase__    ,							UpperCAmelCase__				)

  # Print all the boards
  for board in boards:
    for column in board:
      print(UpperCAmelCase__				)
    print(''				)

  print(len(UpperCAmelCase__				)    ,							'solutions were found.'				)


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      n_queens_solution(4)











import math
import random
from typing import Any

from .hill_climbing import SearchProblem


def     lowerCamelCase_	(      lowerCAmelCase:	Tuple    ,							lowerCAmelCase:	bool = True    ,							lowerCAmelCase:	float = math.inf    ,							lowerCAmelCase:	float = -math.inf    ,							lowerCAmelCase:	float = math.inf    ,							lowerCAmelCase:	float = -math.inf    ,							lowerCAmelCase:	bool = False    ,							lowerCAmelCase:	float = 1_00    ,							lowerCAmelCase:	float = 0.0_1    ,							lowerCAmelCase:	float = 1    ,							)->		Any:
  _snake_case   :				int														=	False
  _snake_case   :				Any														=	search_prob
  _snake_case   :				Tuple														=	start_temperate
  _snake_case   :				Any														=	[]
  _snake_case   :				List[str]														=	0
  _snake_case   :				Optional[Any]														=	None

  while not search_end:
    _snake_case   :				List[Any]														=	current_state.score()
    if best_state is None or current_score > best_state.score():
      _snake_case   :				Dict														=	current_state
    scores.append(lowerCAmelCase				)
    iterations += 1
    _snake_case   :				Optional[int]														=	None
    _snake_case   :				Union[str, Any]														=	current_state.get_neighbors()
    while (
        next_state is None and neighbors
    ):  # till we do not find a neighbor that we can move to
      _snake_case   :				Dict														=	random.randint(0    ,							len(lowerCAmelCase				) - 1				)  # picking a random neighbor
      _snake_case   :				int														=	neighbors.pop(lowerCAmelCase				)
      _snake_case   :				Union[str, Any]														=	picked_neighbor.score() - current_score

      if (
          picked_neighbor.x > max_x
          or picked_neighbor.x < min_x
          or picked_neighbor.y > max_y
          or picked_neighbor.y < min_y
      ):
        continue  # neighbor outside our bounds

      if not find_max:
        _snake_case   :				Union[str, Any]														=	change * -1  # in case we are finding minimum
      if change > 0:  # improves the solution
        _snake_case   :				Union[str, Any]														=	picked_neighbor
      else:
        _snake_case   :				Optional[Any]														=	(math.e) ** (
            change / current_temp
        )  # probability generation function
        if random.random() < probability:  # random number within probability
          _snake_case   :				int														=	picked_neighbor
    _snake_case   :				List[Any]														=	current_temp - (current_temp * rate_of_decrease)

    if current_temp < threshold_temp or next_state is None:
      # temperature below threshold, or could not find a suitable neighbor
      _snake_case   :				List[str]														=	True
    else:
      _snake_case   :				Union[str, Any]														=	next_state

  if visualization:
    from matplotlib import pyplot as plt

    plt.plot(range(lowerCAmelCase				)    ,							lowerCAmelCase				)
    plt.xlabel('Iterations'				)
    plt.ylabel('Function values'				)
    plt.show()
  return best_state


if __name__ == "__main__":


      def     lowerCamelCase_	(      lowerCAmelCase:	Any    ,							lowerCAmelCase:	List[Any]				)->		List[Any]:
        return (x**2) + (y**2)

      # starting the problem with initial coordinates (12, 47)
      lowerCAmelCase_						   =    SearchProblem(x=12, y=47, step_size=1, function_to_optimize=test_fa)
      lowerCAmelCase_						   =    simulated_annealing(
          prob, find_max=False, max_x=100, min_x=5, max_y=50, min_y=-5, visualization=True
      )
      print(
          """The minimum score for f(x, y) = x^2 + y^2 with the domain 100 > x > 5 """
          F"""and 50 > y > - 5 found via hill climbing: {local_min.score()}"""
      )

      # starting the problem with initial coordinates (12, 47)
      lowerCAmelCase_						   =    SearchProblem(x=12, y=47, step_size=1, function_to_optimize=test_fa)
      lowerCAmelCase_						   =    simulated_annealing(
          prob, find_max=True, max_x=100, min_x=5, max_y=50, min_y=-5, visualization=True
      )
      print(
          """The maximum score for f(x, y) = x^2 + y^2 with the domain 100 > x > 5 """
          F"""and 50 > y > - 5 found via hill climbing: {local_min.score()}"""
      )


      def     lowerCamelCase_	(      lowerCAmelCase:	Any    ,							lowerCAmelCase:	Dict				)->		Dict:
        return (3 * x**2) - (6 * y)

      lowerCAmelCase_						   =    SearchProblem(x=3, y=4, step_size=1, function_to_optimize=test_fa)
      lowerCAmelCase_						   =    simulated_annealing(prob, find_max=False, visualization=True)
      print(
          """The minimum score for f(x, y) = 3*x^2 - 6*y found via hill climbing: """
          F"""{local_min.score()}"""
      )

      lowerCAmelCase_						   =    SearchProblem(x=3, y=4, step_size=1, function_to_optimize=test_fa)
      lowerCAmelCase_						   =    simulated_annealing(prob, find_max=True, visualization=True)







      print(
          """The maximum score for f(x, y) = 3*x^2 - 6*y found via hill climbing: """
          F"""{local_min.score()}"""
      )




