





'''simple docstring'''







# A Bipartite Graph is a graph whose vertices can be divided into two independent sets,
# U and V such that every edge (u, v) either connects a vertex from U to V or a vertex
# from V to U. In other words, for every edge (u, v), either u belongs to U and v to V,
# or u belongs to V and v to U. We can also say that there is no edge that connects
# vertices of same set.
from queue import Queue





def     lowerCamelCase_ (					SCREAMING_SNAKE_CASE__ :     int				)	->		List[Any]:
						UpperCAmelCase_     : List[Any]         =					Queue()
						UpperCAmelCase_     : Union[str, Any]         =					[False] * len(SCREAMING_SNAKE_CASE__				)
						UpperCAmelCase_     : Optional[int]         =					[-1] * len(SCREAMING_SNAKE_CASE__				)

						def bfs():
												while not queue.empty():
																		UpperCAmelCase_     : Optional[int]         =					queue.get()
																		UpperCAmelCase_     : Any         =					True

																		for neighbour in graph[u]:
																								if neighbour == u:
																														return False

																								if color[neighbour] == -1:
																														UpperCAmelCase_     : List[str]         =					1 - color[u]
																														queue.put(SCREAMING_SNAKE_CASE__				)

																								elif color[neighbour] == color[u]:
																														return False

												return True

						for i in range(len(SCREAMING_SNAKE_CASE__				)				):
												if not visited[i]:
																		queue.put(SCREAMING_SNAKE_CASE__				)
																		UpperCAmelCase_     : Union[str, Any]         =					0
																		if bfs() is False:
																								return False

						return True


if __name__ == "__main__":
					# Adjacency List of graph
					print(check_bipartite({0: [1, 3], 1: [0, 2], 2: [1, 3], 3: [0, 2]}))












'''simple docstring'''







import numpy as np
from numpy import floataa
from numpy.typing import NDArray





def     lowerCamelCase_ (					SCREAMING_SNAKE_CASE__ :     NDArray[floataa],     SCREAMING_SNAKE_CASE__ :     NDArray[floataa]				)	->		NDArray[floataa]:
						UpperCAmelCase_  ,    UpperCAmelCase_     : Any         =					np.shape(SCREAMING_SNAKE_CASE__				)

						UpperCAmelCase_     : NDArray[floataa]         =					np.zeros((rows, 1),     dtype=SCREAMING_SNAKE_CASE__				)
						for row in reversed(range(SCREAMING_SNAKE_CASE__				)				):
												UpperCAmelCase_     : Union[str, Any]         =					0
												for col in range(row + 1,     SCREAMING_SNAKE_CASE__				):
																		total += coefficients[row, col] * x[col]

												UpperCAmelCase_     : str         =					(vector[row] - total) / coefficients[row, row]

						return x





def     lowerCamelCase_ (					SCREAMING_SNAKE_CASE__ :     NDArray[floataa],     SCREAMING_SNAKE_CASE__ :     NDArray[floataa]				)	->		NDArray[floataa]:
						UpperCAmelCase_  ,    UpperCAmelCase_     : List[Any]         =					np.shape(SCREAMING_SNAKE_CASE__				)
						if rows != columns:
												return np.array((),     dtype=SCREAMING_SNAKE_CASE__				)

						# augmented matrix
						UpperCAmelCase_     : NDArray[floataa]         =					np.concatenate((coefficients, vector),     axis=1				)
						UpperCAmelCase_     : Optional[Any]         =					augmented_mat.astype('''float64'''				)

						# scale the matrix leaving it triangular
						for row in range(rows - 1				):
												UpperCAmelCase_     : Optional[int]         =					augmented_mat[row, row]
												for col in range(row + 1,     SCREAMING_SNAKE_CASE__				):
																		UpperCAmelCase_     : Optional[Any]         =					augmented_mat[col, row] / pivot
																		augmented_mat[col, :] -= factor * augmented_mat[row, :]

						UpperCAmelCase_     : Optional[Any]         =					retroactive_resolution(
						    augmented_mat[:, 0:columns],     augmented_mat[:, columns : columns + 1]				)

						return x


if __name__ == "__main__":
					import doctest

					doctest.testmod()





