
from __future__ import annotations

import time

SCREAMING_SNAKE_CASE							:Dict							 =				list[tuple[int, int]]

SCREAMING_SNAKE_CASE							:Any							 =				[
    [0, 0, 0, 0, 0, 0, 0],
    [0, 1, 0, 0, 0, 0, 0],  # 0 are free path whereas 1's are obstacles
    [0, 0, 0, 0, 0, 0, 0],
    [0, 0, 1, 0, 0, 0, 0],
    [1, 0, 1, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 1, 0, 0],
]

SCREAMING_SNAKE_CASE							:List[Any]							 =				[[-1, 0], [0, -1], [1, 0], [0, 1]]  # up, left, down, right





class 							UpperCAmelCase      :




	'''simple docstring'''





	def __init__(       self					: List[Any]							,A					: int							,A					: int							,A					: int							,A					: int							,A					: Node | None ):
					__A   			= pos_x
					__A   			= pos_y
					__A   			= (pos_y, pos_x)
					__A   			= goal_x
					__A   			= goal_y
					__A   			= parent





class 							UpperCAmelCase      :




	'''simple docstring'''





	def __init__(       self					: List[str]							,A					: tuple[int, int]							,A					: tuple[int, int] ):
					__A   			= Node(start[1]							,start[0]							,goal[1]							,goal[0]							,A )
					__A   			= Node(goal[1]							,goal[0]							,goal[1]							,goal[0]							,A )

					__A   			= [self.start]
					__A   			= False






	def 				UpperCamelCase_  (       self					: Tuple ):
					while self.node_queue:
									__A   			= self.node_queue.pop(0 )

									if current_node.pos == self.target.pos:
													__A   			= True
													return self.retrace_path(A )

									__A   			= self.get_successors(A )

									for node in successors:
													self.node_queue.append(A )

					if not self.reached:
									return [self.start.pos]
					return None






	def 				UpperCamelCase_  (       self					: List[str]							,A					: Node ):
					__A   			= []
					for action in delta:
									__A   			= parent.pos_x + action[1]
									__A   			= parent.pos_y + action[0]
									if not (0 <= pos_x <= len(grid[0] ) - 1 and 0 <= pos_y <= len(A ) - 1):
													continue

									if grid[pos_y][pos_x] != 0:
													continue

									successors.append(
									    Node(A							,A							,self.target.pos_y							,self.target.pos_x							,A ) )
					return successors







	def 				UpperCamelCase_  (       self					: Optional[int]							,A					: Node | None ):
					__A   			= node
					__A   			= []
					while current_node is not None:
									path.append((current_node.pos_y, current_node.pos_x) )
									__A   			= current_node.parent
					path.reverse()
					return path





class 							UpperCAmelCase      :




	'''simple docstring'''





	def __init__(       self					: List[str]							,A					: Tuple							,A					: Union[str, Any] ):
					__A   			= BreadthFirstSearch(A							,A )
					__A   			= BreadthFirstSearch(A							,A )
					__A   			= False






	def 				UpperCamelCase_  (       self					: Tuple ):
					while self.fwd_bfs.node_queue or self.bwd_bfs.node_queue:
									__A   			= self.fwd_bfs.node_queue.pop(0 )
									__A   			= self.bwd_bfs.node_queue.pop(0 )

									if current_bwd_node.pos == current_fwd_node.pos:
													__A   			= True
													return self.retrace_bidirectional_path(
													    A							,A )

									__A   			= current_bwd_node
									__A   			= current_fwd_node

									__A   			= {
									    self.fwd_bfs: self.fwd_bfs.get_successors(A ),
									    self.bwd_bfs: self.bwd_bfs.get_successors(A ),
									}

									for bfs in [self.fwd_bfs, self.bwd_bfs]:
													for node in successors[bfs]:
																	bfs.node_queue.append(A )

					if not self.reached:
									return [self.fwd_bfs.start.pos]
					return None







	def 				UpperCamelCase_  (       self					: Dict							,A					: Node							,A					: Node ):
					__A   			= self.fwd_bfs.retrace_path(A )
					__A   			= self.bwd_bfs.retrace_path(A )
					bwd_path.pop()
					bwd_path.reverse()
					__A   			= fwd_path + bwd_path
					return path


if __name__ == "__main__":
					# all coordinates are given in format [y,x]
					import doctest

					doctest.testmod()
					SCREAMING_SNAKE_CASE							:Optional[int]							 =				(0, 0)
					SCREAMING_SNAKE_CASE							:List[Any]							 =				(len(grid) - 1, len(grid[0]) - 1)
					for elem in grid:
										print(elem)

					SCREAMING_SNAKE_CASE							:List[Any]							 =				time.time()
					SCREAMING_SNAKE_CASE							:List[str]							 =				BreadthFirstSearch(init, goal)
					SCREAMING_SNAKE_CASE							:str							 =				bfs.search()
					SCREAMING_SNAKE_CASE							:Optional[Any]							 =				time.time() - start_bfs_time

					print('Unidirectional BFS computation time : ', bfs_time)

					SCREAMING_SNAKE_CASE							:List[str]							 =				time.time()
					SCREAMING_SNAKE_CASE							:Dict							 =				BidirectionalBreadthFirstSearch(init, goal)
					SCREAMING_SNAKE_CASE							:Dict							 =				bd_bfs.search()
					SCREAMING_SNAKE_CASE							:Any							 =				time.time() - start_bd_bfs_time

					print('Bidirectional BFS computation time : ', bd_bfs_time)


import numpy as np


def 						UpperCAmelCase			(	a_		,   a_		,   a_ = 1E-12		,   a_ = 1_0_0		,   )		->	tuple[float, np.ndarray]:







				"""simple docstring"""



				assert np.shape(a_						)[0] == np.shape(a_						)[1]
				# Ensure proper dimensionality.
				assert np.shape(a_						)[0] == np.shape(a_						)[0]
				# Ensure inputs are either both complex or both real
				assert np.iscomplexobj(a_						) == np.iscomplexobj(a_						)
				__A   			= np.iscomplexobj(a_						)
				if is_complex:
								# Ensure complex input_matrix is Hermitian
								assert np.array_equal(a_		,   input_matrix.conj().T						)

				# Set convergence to False. Will define convergence when we exceed max_iterations
				# or when we have small changes from one iteration to next.

				__A   			= False
				__A   			= 0
				__A   			= 0
				__A   			= 1E12

				while not convergence:
								# Multiple matrix by the vector.
								__A   			= np.dot(a_		,   a_						)
								# Normalize the resulting output vector.
								__A   			= w / np.linalg.norm(a_						)
								# Find rayleigh quotient
								# (faster than usual b/c we know vector is normalized already)
								__A   			= vector.conj().T if is_complex else vector.T
								__A   			= np.dot(a_		,   np.dot(a_		,   a_						)						)

								# Check convergence.
								__A   			= np.abs(lambda_ - lambda_previous						) / lambda_
								iterations += 1

								if error <= error_tol or iterations >= max_iterations:
												__A   			= True

								__A   			= lambda_

				if is_complex:
								__A   			= np.real(lambda_						)

				return lambda_, vector


def 						UpperCAmelCase			(	)		->	None:







				"""simple docstring"""



				__A   			= np.array([[4_1, 4, 2_0], [4, 2_6, 3_0], [2_0, 3_0, 5_0]]						)
				__A   			= np.array([4_1, 4, 2_0]						)
				__A   			= real_input_matrix.astype(np.complexaaa						)
				__A   			= np.triu(1J * complex_input_matrix		,   1						)
				complex_input_matrix += imag_matrix
				complex_input_matrix += -1 * imag_matrix.T
				__A   			= np.array([4_1, 4, 2_0]						).astype(np.complexaaa						)

				for problem_type in ["real", "complex"]:
								if problem_type == "real":
												__A   			= real_input_matrix
												__A   			= real_vector
								elif problem_type == "complex":
												__A   			= complex_input_matrix
												__A   			= complex_vector

								# Our implementation.
								__A			,							__A   			= power_iteration(a_		,   a_						)

								# Numpy implementation.

								# Get eigenvalues and eigenvectors using built-in numpy
								# eigh (eigh used for symmetric or hermetian matrices).
								__A			,							__A   			= np.linalg.eigh(a_						)
								# Last eigenvalue is the maximum one.
								__A   			= eigen_values[-1]
								# Last column in this matrix is eigenvector corresponding to largest eigenvalue.
								__A   			= eigen_vectors[:, -1]

								# Check our implementation and numpy gives close answers.
								assert np.abs(eigen_value - eigen_value_max						) <= 1E-6
								# Take absolute values element wise of each eigenvector.
								# as they are only unique to a minus sign.
								assert np.linalg.norm(np.abs(a_						) - np.abs(a_						)						) <= 1E-6


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					test_power_iteration()
