





'''simple docstring'''







from __future__ import annotations

from queue import Queue





def     lowerCamelCase_ (					SCREAMING_SNAKE_CASE__ :     int,     SCREAMING_SNAKE_CASE__ :     int				)	->		tuple[int, int]:
						a ^= b
						b ^= a
						a ^= b
						return a, b





def     lowerCamelCase_ (					SCREAMING_SNAKE_CASE__ :     int,     SCREAMING_SNAKE_CASE__ :     list[list[int]]				)	->		list[list[int]]:
						UpperCAmelCase_     : Union[str, Any]         =					1
						while (1 << j) < max_node:
												for i in range(1,     max_node + 1				):
																		UpperCAmelCase_     : List[Any]         =					parent[j - 1][parent[j - 1][i]]
												j += 1
						return parent





def     lowerCamelCase_ (					SCREAMING_SNAKE_CASE__ :     int,     SCREAMING_SNAKE_CASE__ :     int,     SCREAMING_SNAKE_CASE__ :     list[int],     SCREAMING_SNAKE_CASE__ :     list[list[int]]				)	->		int:
						# u must be deeper in the tree than v
						if level[u] < level[v]:
												UpperCAmelCase_     : int         =					swap(SCREAMING_SNAKE_CASE__,     SCREAMING_SNAKE_CASE__				)
						# making depth of u same as depth of v
						for i in range(18,     -1,     -1				):
												if level[u] - (1 << i) >= level[v]:
																		UpperCAmelCase_     : Tuple         =					parent[i][u]
    # at the same depth if u==v that mean lca is found
						if u == v:
												return u
						# moving both nodes upwards till lca in found
						for i in range(18,     -1,     -1				):
												if parent[i][u] not in [0, parent[i][v]]:
																		UpperCAmelCase_     : List[Any]         =					parent[i][u], parent[i][v]
    # returning longest common ancestor of u,v
						return parent[0][u]





def     lowerCamelCase_ (					SCREAMING_SNAKE_CASE__ :     list[int],     SCREAMING_SNAKE_CASE__ :     list[list[int]],     SCREAMING_SNAKE_CASE__ :     int,     SCREAMING_SNAKE_CASE__ :     dict[int, list[int]],     SCREAMING_SNAKE_CASE__ :     int = 1,     )	->		tuple[list[int], list[list[int]]]:
						UpperCAmelCase_     : str         =					0
						UpperCAmelCase_     : Queue[int]         =					Queue(maxsize=SCREAMING_SNAKE_CASE__				)
						q.put(SCREAMING_SNAKE_CASE__				)
						while q.qsize() != 0:
												UpperCAmelCase_     : Tuple         =					q.get()
												for v in graph[u]:
																		if level[v] == -1:
																								UpperCAmelCase_     : Dict         =					level[u] + 1
																								q.put(SCREAMING_SNAKE_CASE__				)
																								UpperCAmelCase_     : Optional[Any]         =					u
						return level, parent





def     lowerCamelCase_ (					)	->		None:
						UpperCAmelCase_     : Optional[int]         =					13
						# initializing with 0
						UpperCAmelCase_     : Any         =					[[0 for _ in range(max_node + 10				)] for _ in range(20				)]
						# initializing with -1 which means every node is unvisited
						UpperCAmelCase_     : str         =					[-1 for _ in range(max_node + 10				)]
						UpperCAmelCase_     : dict[int, list[int]]         =					{
						    1: [2, 3, 4],
						    2: [5],
						    3: [6, 7],
						    4: [8],
						    5: [9, 10],
						    6: [11],
						    7: [],
						    8: [12, 13],
						    9: [],
						    10: [],
						    11: [],
						    12: [],
						    13: [],
						}
						UpperCAmelCase_     : List[str]         =					breadth_first_search(SCREAMING_SNAKE_CASE__,     SCREAMING_SNAKE_CASE__,     SCREAMING_SNAKE_CASE__,     SCREAMING_SNAKE_CASE__,     1				)
						UpperCAmelCase_     : Optional[int]         =					create_sparse(SCREAMING_SNAKE_CASE__,     SCREAMING_SNAKE_CASE__				)
						print('''LCA of node 1 and 3 is: ''',     lowest_common_ancestor(1,     3,     SCREAMING_SNAKE_CASE__,     SCREAMING_SNAKE_CASE__				)				)
						print('''LCA of node 5 and 6 is: ''',     lowest_common_ancestor(5,     6,     SCREAMING_SNAKE_CASE__,     SCREAMING_SNAKE_CASE__				)				)
						print('''LCA of node 7 and 11 is: ''',     lowest_common_ancestor(7,     11,     SCREAMING_SNAKE_CASE__,     SCREAMING_SNAKE_CASE__				)				)
						print('''LCA of node 6 and 7 is: ''',     lowest_common_ancestor(6,     7,     SCREAMING_SNAKE_CASE__,     SCREAMING_SNAKE_CASE__				)				)
						print('''LCA of node 4 and 12 is: ''',     lowest_common_ancestor(4,     12,     SCREAMING_SNAKE_CASE__,     SCREAMING_SNAKE_CASE__				)				)
						print('''LCA of node 8 and 8 is: ''',     lowest_common_ancestor(8,     8,     SCREAMING_SNAKE_CASE__,     SCREAMING_SNAKE_CASE__				)				)


if __name__ == "__main__":
					main()







'''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()





