





"""simple docstring"""




def 	a_    (							_lowercase       ):
						_UpperCamelCase      : Any         =   len(_lowercase       )
						_UpperCamelCase      : Tuple         =   len(matrix[0]       )
						_UpperCamelCase      : str         =   min(_lowercase	,						_lowercase       )

						for row in range(_lowercase       ):
												# Check if diagonal element is not zero
												if matrix[row][row] != 0:
																		# Eliminate all the elements below the diagonal
																		for col in range(row + 1	,						_lowercase       ):
																								_UpperCamelCase      : Optional[int]         =   matrix[col][row] / matrix[row][row]
																								for i in range(_lowercase	,						_lowercase       ):
																														matrix[col][i] -= multiplier * matrix[row][i]
												else:
																		# Find a non-zero diagonal element to swap rows
																		_UpperCamelCase      : Dict         =   True
																		for i in range(row + 1	,						_lowercase       ):
																								if matrix[i][row] != 0:
																														_UpperCamelCase       ,							_UpperCamelCase      : Dict         =   matrix[i], matrix[row]
																														_UpperCamelCase      : int         =   False
																														break
																		if reduce:
																								rank -= 1
																								for i in range(_lowercase       ):
																														_UpperCamelCase      : Union[str, Any]         =   matrix[i][rank]

            # Reduce the row pointer by one to stay on the same row
																		row -= 1

						return rank


if __name__ == "__main__":
						import doctest

						doctest.testmod()










"""simple docstring"""




from __future__ import annotations

import math





def 	a_    (							_lowercase	,						_lowercase	,						_lowercase	,						_lowercase	,						_lowercase       ):

						if depth < 0:
												raise ValueError('''Depth cannot be less than 0'''       )

						if len(_lowercase       ) == 0:
												raise ValueError('''Scores cannot be empty'''       )

						if depth == height:
												return scores[node_index]

						if is_max:
												return max(
												    minimax(depth + 1	,						node_index * 2	,						_lowercase	,						_lowercase	,						_lowercase       )	,						minimax(depth + 1	,						node_index * 2 + 1	,						_lowercase	,						_lowercase	,						_lowercase       )	,						)

						return min(
						    minimax(depth + 1	,						node_index * 2	,						_lowercase	,						_lowercase	,						_lowercase       )	,						minimax(depth + 1	,						node_index * 2 + 1	,						_lowercase	,						_lowercase	,						_lowercase       )	,						)





def 	a_    (							):
						_UpperCamelCase      : Dict         =   [90, 23, 6, 33, 21, 65, 123, 3_4423]
						_UpperCamelCase      : int         =   math.log(len(_lowercase       )	,						2       )
						print('''Optimal value : '''	,						end=''''''       )
						print(minimax(0	,						0	,						_lowercase	,						_lowercase	,						_lowercase       )       )


if __name__ == "__main__":
						import doctest

						doctest.testmod()
						main()



