


'''simple docstring'''






from __future__ import annotations





def      lowerCamelCase      (		lowerCamelCase_  :int				,      lowerCamelCase_  :Optional[Any]				,      lowerCamelCase_  :Union[str, Any]				,      lowerCamelCase_  :str				,      lowerCamelCase_  :Optional[int]				,      ):







		'''simple docstring'''
		snake_case_		:							List[Any]			       =						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_  :List[Any] ):







		'''simple docstring'''
		snake_case_		:							Dict			       =						[]
		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)



'''simple docstring'''






from __future__ import annotations

import os
from collections.abc import Mapping

__A	:	str     =     tuple[int, int]
class 						__UpperCamelCase	:





							def __init__(  self		:Union[str, Any]  ,_UpperCamelCase		:set[int]  ,_UpperCamelCase		:Mapping[EdgeT, int]			):
									snake_case_		:							set[int]			       =						vertices
									snake_case_		:							dict[EdgeT, int]			       =						{
									    (min(_UpperCamelCase			), max(_UpperCamelCase			)): weight for edge, weight in edges.items()
									}





							def 		a__   (  self		:Tuple  ,_UpperCamelCase		:EdgeT  ,_UpperCamelCase		:int			):
									self.vertices.add(edge[0]			)
									self.vertices.add(edge[1]			)
									snake_case_		:							str			       =						weight





							def 		a__   (  self		:Tuple			):
									snake_case_		:							Graph			       =						Graph({min(self.vertices			)}  ,{}			)
									snake_case_		:							EdgeT
									snake_case_		:							int
									snake_case_		:							EdgeT
									snake_case_		:							int

									while len(subgraph.vertices			) < len(self.vertices			):
											snake_case_		:							int			       =						max(self.edges.values()			) + 1
											for edge, weight in self.edges.items():
													if (edge[0] in subgraph.vertices) ^ (edge[1] in subgraph.vertices):
															if weight < min_weight:
																	snake_case_		:							Dict			       =						edge
																	snake_case_		:							Dict			       =						weight

											subgraph.add_edge(_UpperCamelCase  ,_UpperCamelCase			)

									return subgraph





def      UpperCAmelCase      (		lowerCamelCase_  :str = "p107_network.txt" ):







		'''simple docstring'''
		snake_case_		:							str			       =						os.path.abspath(os.path.dirname(lowerCamelCase_ ) )
		snake_case_		:							str			       =						os.path.join(lowerCamelCase_				,      lowerCamelCase_ )
		snake_case_		:							dict[EdgeT, int]			       =						{}
		snake_case_		:							list[str]
		snake_case_		:							int
		snake_case_		:							int

		with open(lowerCamelCase_ ) as f:
				snake_case_		:							Optional[int]			       =						f.read().strip().split("""\n""" )

		snake_case_		:							Any			       =						[line.split(""",""" ) for line in data]

		for edgea in range(1				,      len(lowerCamelCase_ ) ):
				for edgea in range(lowerCamelCase_ ):
						if adjaceny_matrix[edgea][edgea] != "-":
								snake_case_		:							str			       =						int(adjaceny_matrix[edgea][edgea] )

		snake_case_		:							Graph			       =						Graph(set(range(len(lowerCamelCase_ ) ) )				,      lowerCamelCase_ )

		snake_case_		:							Graph			       =						graph.prims_algorithm()

		snake_case_		:							int			       =						sum(graph.edges.values() )
		snake_case_		:							int			       =						sum(subgraph.edges.values() )

		return initial_total - optimal_total


if __name__ == "__main__":
							print(F'{solution() = }')