
"""simple docstring"""

import numpy as np
from numpy import floataa
from numpy.typing import NDArray
def 						UpperCAmelCase   (						a_,		a_     ):





		'''simple docstring'''


		lowerCamelCase      ,       lowerCamelCase	:     List[str]            =       np.shape(a_     )

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

				lowerCamelCase	:     Optional[Any]            =       (vector[row] - total) / coefficients[row, row]

		return x
def 						UpperCAmelCase   (						a_,		a_     ):





		'''simple docstring'''


		lowerCamelCase      ,       lowerCamelCase	:     List[Any]            =       np.shape(a_     )
		if rows != columns:
				return np.array((),		dtype=a_     )

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

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

		lowerCamelCase	:     Union[str, Any]            =       retroactive_resolution(
		    augmented_mat[:, 0:columns],		augmented_mat[:, columns : columns + 1]     )

		return x


if __name__ == "__main__":
					import doctest

					doctest.testmod()








"""simple docstring"""

from __future__ import annotations

from queue import Queue
def 						UpperCAmelCase   (						a_,		a_     ):





		'''simple docstring'''


		a ^= b
		b ^= a
		a ^= b
		return a, b
def 						UpperCAmelCase   (						a_,		a_     ):





		'''simple docstring'''


		lowerCamelCase	:     int            =       1
		while (1 << j) < max_node:
				for i in range(1,		max_node + 1     ):
						lowerCamelCase	:     Tuple            =       parent[j - 1][parent[j - 1][i]]
				j += 1
		return parent
def 						UpperCAmelCase   (						a_,		a_,		a_,		a_     ):





		'''simple docstring'''


		if level[u] < level[v]:
				lowerCamelCase      ,       lowerCamelCase	:     List[Any]            =       swap(a_,		a_     )
		# making depth of u same as depth of v
		for i in range(18,		-1,		-1     ):
				if level[u] - (1 << i) >= level[v]:
						lowerCamelCase	:     int            =       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]]:
						lowerCamelCase      ,       lowerCamelCase	:     str            =       parent[i][u], parent[i][v]
    # returning longest common ancestor of u,v
		return parent[0][u]
def 						UpperCAmelCase   (						a_,		a_,		a_,		a_,		a_ = 1,		):





		'''simple docstring'''


		lowerCamelCase	:     Any            =       0
		lowerCamelCase	:     Queue[int]            =       Queue(maxsize=a_     )
		q.put(a_     )
		while q.qsize() != 0:
				lowerCamelCase	:     Tuple            =       q.get()
				for v in graph[u]:
						if level[v] == -1:
								lowerCamelCase	:     Optional[Any]            =       level[u] + 1
								q.put(a_     )
								lowerCamelCase	:     str            =       u
		return level, parent
def 						UpperCAmelCase   (						):





		'''simple docstring'''


		lowerCamelCase	:     Any            =       13
		# initializing with 0
		lowerCamelCase	:     Dict            =       [[0 for _ in range(max_node + 10     )] for _ in range(20     )]
		# initializing with -1 which means every node is unvisited
		lowerCamelCase	:     List[Any]            =       [-1 for _ in range(max_node + 10     )]
		lowerCamelCase	:     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: [],
		}
		lowerCamelCase      ,       lowerCamelCase	:     Dict            =       breadth_first_search(a_,		a_,		a_,		a_,		1     )
		lowerCamelCase	:     List[Any]            =       create_sparse(a_,		a_     )
		print('LCA of node 1 and 3 is: ',		lowest_common_ancestor(1,		3,		a_,		a_     )     )
		print('LCA of node 5 and 6 is: ',		lowest_common_ancestor(5,		6,		a_,		a_     )     )
		print('LCA of node 7 and 11 is: ',		lowest_common_ancestor(7,		11,		a_,		a_     )     )
		print('LCA of node 6 and 7 is: ',		lowest_common_ancestor(6,		7,		a_,		a_     )     )
		print('LCA of node 4 and 12 is: ',		lowest_common_ancestor(4,		12,		a_,		a_     )     )
		print('LCA of node 8 and 8 is: ',		lowest_common_ancestor(8,		8,		a_,		a_     )     )


if __name__ == "__main__":
					main()






