import random




def 							lowerCamelCase			(   UpperCAmelCase__ :      int   ,   UpperCAmelCase__ :      float   ,   UpperCAmelCase__ :      bool = False				)    ->   dict:
							lowercase_      :    dict       		= {i: [] for i in range(UpperCAmelCase__				)}

							# if probability is greater or equal than 1, then generate a complete graph
							if probability >= 1:
														return complete_graph(UpperCAmelCase__				)
							# if probability is lower or equal than 0, then return a graph without edges
							if probability <= 0:
														return graph

							# for each couple of nodes, add an edge from u to v
							# if the number randomly generated is greater than probability probability
							for i in range(UpperCAmelCase__				):
														for j in range(i + 1   ,   UpperCAmelCase__				):
																					if random.random() < probability:
																												graph[i].append(UpperCAmelCase__				)
																												if not directed:
																																			# if the graph is undirected, add an edge in from j to i, either
																																			graph[j].append(UpperCAmelCase__				)
							return graph




def 							lowerCamelCase			(   UpperCAmelCase__ :      int				)    ->   dict:
							return {
							    i: [j for j in range(UpperCAmelCase__				) if i != j] for i in range(UpperCAmelCase__				)
							}


if __name__ == "__main__":
						import doctest

						doctest.testmod()






'''simple docstring'''




import unittest

import numpy as np




def 							lowerCamelCase			(   UpperCAmelCase__ :      np.ndarray   ,   UpperCAmelCase__ :      np.ndarray   ,   UpperCAmelCase__ :      np.ndarray   ,   UpperCAmelCase__ :      np.ndarray | None = None   ,   )    ->   np.ndarray:
							lowercase_      :    List[Any]       		= np.shape(UpperCAmelCase__				)
							lowercase_      :    Dict       		= np.shape(UpperCAmelCase__				)
							lowercase_      :    int       		= np.shape(UpperCAmelCase__				)

							if shape_a[0] != shape_b[0]:
														lowercase_      :    Optional[int]       		= (
														    """Expected the same number of rows for A and B. """
														    F'''Instead found A of size {shape_a} and B of size {shape_b}'''
														)
														raise ValueError(UpperCAmelCase__				)

							if shape_b[1] != shape_c[1]:
														lowercase_      :    Optional[Any]       		= (
														    """Expected the same number of columns for B and C. """
														    F'''Instead found B of size {shape_b} and C of size {shape_c}'''
														)
														raise ValueError(UpperCAmelCase__				)

							lowercase_      :    Any       		= pseudo_inv
							if a_inv is None:
														try:
																					lowercase_      :    List[str]       		= np.linalg.inv(UpperCAmelCase__				)
														except np.linalg.LinAlgError:
																					raise ValueError(
																					    """Input matrix A is not invertible. Cannot compute Schur complement."""				)

							return mat_c - mat_b.T @ a_inv @ mat_b





class 					__magic_name__		(					unittest.TestCase):







			def 			SCREAMING_SNAKE_CASE_  (			self       :						Optional[int]					):
										lowercase_      :    Tuple       		= np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
										lowercase_      :    int       		= np.array([[0, 3], [3, 0], [2, 3]]					)
										lowercase_      :    Dict       		= np.array([[2, 1], [6, 3]]					)

										lowercase_      :    Union[str, Any]       		= schur_complement(lowercase_       ,    lowercase_       ,    lowercase_					)

										lowercase_      :    List[Any]       		= np.block([[a, b], [b.T, c]]					)

										lowercase_      :    Optional[int]       		= np.linalg.det(lowercase_					)
										lowercase_      :    int       		= np.linalg.det(lowercase_					)
										lowercase_      :    int       		= np.linalg.det(lowercase_					)

										self.assertAlmostEqual(lowercase_       ,    det_a * det_s					)







			def 			SCREAMING_SNAKE_CASE_  (			self       :						str					):
										lowercase_      :    int       		= np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
										lowercase_      :    Optional[Any]       		= np.array([[0, 3], [3, 0], [2, 3]]					)
										lowercase_      :    Union[str, Any]       		= np.array([[2, 1], [6, 3]]					)

										with self.assertRaises(lowercase_					):
																	schur_complement(lowercase_       ,    lowercase_       ,    lowercase_					)







			def 			SCREAMING_SNAKE_CASE_  (			self       :						Any					):
										lowercase_      :    List[Any]       		= np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
										lowercase_      :    List[Any]       		= np.array([[0, 3], [3, 0], [2, 3]]					)
										lowercase_      :    str       		= np.array([[2, 1, 3], [6, 3, 5]]					)

										with self.assertRaises(lowercase_					):
																	schur_complement(lowercase_       ,    lowercase_       ,    lowercase_					)


if __name__ == "__main__":
						import doctest

						doctest.testmod()
						unittest.main()





