



'''simple docstring'''
from __future__ import annotations

from collections import Counter
from random import random






class 							lowercase__	:





	def __init__( self					:				Optional[Any] ):







							'''simple docstring'''
							_UpperCamelCase					:		str 				=   {}





	def 	UpperCamelCase_  ( self					:				Any ,lowerCamelCase__					:				str ):







							'''simple docstring'''
							_UpperCamelCase					:		List[Any] 				=   {}





	def 	UpperCamelCase_  ( self					:				List[Any] ,lowerCamelCase__					:				str ,lowerCamelCase__					:				str ,lowerCamelCase__					:				float ):







							'''simple docstring'''
							if nodea not in self.connections:
													self.add_node(lowerCamelCase__ )
							if nodea not in self.connections:
													self.add_node(lowerCamelCase__ )
							_UpperCamelCase					:		str 				=   probability





	def 	UpperCamelCase_  ( self					:				Union[str, Any] ):







							'''simple docstring'''
							return list(self.connections )





	def 	UpperCamelCase_  ( self					:				Dict ,lowerCamelCase__					:				str ):







							'''simple docstring'''
							_UpperCamelCase					:		Optional[Any] 				=   0
							_UpperCamelCase					:		List[Any] 				=   random()

							for dest in self.connections[node]:
													current_probability += self.connections[node][dest]
													if current_probability > random_value:
																			return dest
							return ""






def A__    (     UpperCAmelCase_  ,  UpperCAmelCase_  ,  UpperCAmelCase_							):
						_UpperCamelCase					:		Dict 				=   MarkovChainGraphUndirectedUnweighted()

						for nodea, nodea, probability in transitions:
												graph.add_transition_probability(UpperCAmelCase_  ,  UpperCAmelCase_  ,  UpperCAmelCase_							)

						_UpperCamelCase					:		Union[str, Any] 				=   Counter(graph.get_nodes()							)
						_UpperCamelCase					:		Dict 				=   start

						for _ in range(UpperCAmelCase_							):
												_UpperCamelCase					:		Union[str, Any] 				=   graph.transition(UpperCAmelCase_							)
												visited[node] += 1

						return visited


if __name__ == "__main__":
					import doctest

					doctest.testmod()










from __future__ import annotations




def      lowercase__      (     __snake_case    :							tuple[int, int]     ,       __snake_case    :							int  ):


					'''simple docstring'''


					UpperCAmelCase_       ,  UpperCAmelCase_    :      Tuple            =						position
					UpperCAmelCase_    :      str            =						[
					    (y + 1, x + 2),
					    (y - 1, x + 2),
					    (y + 1, x - 2),
					    (y - 1, x - 2),
					    (y + 2, x + 1),
					    (y + 2, x - 1),
					    (y - 2, x + 1),
					    (y - 2, x - 1),
					]
					UpperCAmelCase_    :      Optional[Any]            =						[]

					for position in positions:
										UpperCAmelCase_       ,  UpperCAmelCase_    :      Tuple            =						position
										if 0 <= y_test < n and 0 <= x_test < n:
															permissible_positions.append(__snake_case  )

					return permissible_positions




def      lowercase__      (     __snake_case    :							list[list[int]]  ):


					'''simple docstring'''


					return not any(elem == 0 for row in board for elem in row  )




def      lowercase__      (     __snake_case    :							list[list[int]]     ,       __snake_case    :							tuple[int, int]     ,       __snake_case    :							int  ):


					'''simple docstring'''


					if is_complete(__snake_case  ):
										return True

					for position in get_valid_pos(__snake_case     ,       len(__snake_case  )  ):
										UpperCAmelCase_       ,  UpperCAmelCase_    :      Any            =						position

										if board[y][x] == 0:
															UpperCAmelCase_    :      Optional[Any]            =						curr + 1
															if open_knight_tour_helper(__snake_case     ,       __snake_case     ,       curr + 1  ):
																				return True
															UpperCAmelCase_    :      List[Any]            =						0

					return False




def      lowercase__      (     __snake_case    :							int  ):


					'''simple docstring'''


					UpperCAmelCase_    :      str            =						[[0 for i in range(__snake_case  )] for j in range(__snake_case  )]

					for i in range(__snake_case  ):
										for j in range(__snake_case  ):
															UpperCAmelCase_    :      Optional[Any]            =						1
															if open_knight_tour_helper(__snake_case     ,       (i, j)     ,       1  ):
																				return board
															UpperCAmelCase_    :      List[Any]            =						0

					UpperCAmelCase_    :      List[str]            =						F"Open Kight Tour cannot be performed on a board of size {n}"
					raise ValueError(__snake_case  )


if __name__ == "__main__":
		import doctest

		doctest.testmod()






