





"""simple docstring"""


from __future__ import annotations

from decimal import Decimal

from numpy import array




def 					lowercase__    (					lowercase_			)					->		list[list[float]]:




      """simple docstring"""

      _UpperCamelCase      :	Dict						=       Decimal

      # Check if the provided matrix has 2 rows and 2 columns
      # since this implementation only works for 2x2 matrices
      if len(lowercase_			) == 2 and len(matrix[0]			) == 2 and len(matrix[1]			) == 2:
            # Calculate the determinant of the matrix
            _UpperCamelCase      :	Optional[int]						=       float(
                d(matrix[0][0]			) * d(matrix[1][1]			) - d(matrix[1][0]			) * d(matrix[0][1]			)			)
            if determinant == 0:
                  raise ValueError("This matrix has no inverse."			)

            # Creates a copy of the matrix with swapped positions of the elements
            _UpperCamelCase      :	List[Any]						=       [[0.0, 0.0], [0.0, 0.0]]
            _UpperCamelCase,  _UpperCamelCase      :	int						=       matrix[1][1], matrix[0][0]
            _UpperCamelCase,  _UpperCamelCase      :	List[str]						=       -matrix[1][0], -matrix[0][1]

            # Calculate the inverse of the matrix
            return [
                [(float(d(lowercase_			)			) / determinant) or 0.0 for n in row] for row in swapped_matrix
            ]
      elif (
          len(lowercase_			) == 3
          and len(matrix[0]			) == 3
          and len(matrix[1]			) == 3
          and len(matrix[2]			) == 3
      ):
            # Calculate the determinant of the matrix using Sarrus rule
            _UpperCamelCase      :	Optional[int]						=       float(
                (
                    (d(matrix[0][0]			) * d(matrix[1][1]			) * d(matrix[2][2]			))
                    + (d(matrix[0][1]			) * d(matrix[1][2]			) * d(matrix[2][0]			))
                    + (d(matrix[0][2]			) * d(matrix[1][0]			) * d(matrix[2][1]			))
                )
                - (
                    (d(matrix[0][2]			) * d(matrix[1][1]			) * d(matrix[2][0]			))
                    + (d(matrix[0][1]			) * d(matrix[1][0]			) * d(matrix[2][2]			))
                    + (d(matrix[0][0]			) * d(matrix[1][2]			) * d(matrix[2][1]			))
                )			)
            if determinant == 0:
                  raise ValueError("This matrix has no inverse."			)

            # Creating cofactor matrix
            _UpperCamelCase      :	Optional[Any]						=       [
                [d(0.0			), d(0.0			), d(0.0			)],
                [d(0.0			), d(0.0			), d(0.0			)],
                [d(0.0			), d(0.0			), d(0.0			)],
            ]
            _UpperCamelCase      :	Optional[Any]						=       (d(matrix[1][1]			) * d(matrix[2][2]			)) - (
                d(matrix[1][2]			) * d(matrix[2][1]			)
            )
            _UpperCamelCase      :	Optional[int]						=       -(
                (d(matrix[1][0]			) * d(matrix[2][2]			)) - (d(matrix[1][2]			) * d(matrix[2][0]			))
            )
            _UpperCamelCase      :	str						=       (d(matrix[1][0]			) * d(matrix[2][1]			)) - (
                d(matrix[1][1]			) * d(matrix[2][0]			)
            )
            _UpperCamelCase      :	int						=       -(
                (d(matrix[0][1]			) * d(matrix[2][2]			)) - (d(matrix[0][2]			) * d(matrix[2][1]			))
            )
            _UpperCamelCase      :	Optional[int]						=       (d(matrix[0][0]			) * d(matrix[2][2]			)) - (
                d(matrix[0][2]			) * d(matrix[2][0]			)
            )
            _UpperCamelCase      :	List[str]						=       -(
                (d(matrix[0][0]			) * d(matrix[2][1]			)) - (d(matrix[0][1]			) * d(matrix[2][0]			))
            )
            _UpperCamelCase      :	Any						=       (d(matrix[0][1]			) * d(matrix[1][2]			)) - (
                d(matrix[0][2]			) * d(matrix[1][1]			)
            )
            _UpperCamelCase      :	Union[str, Any]						=       -(
                (d(matrix[0][0]			) * d(matrix[1][2]			)) - (d(matrix[0][2]			) * d(matrix[1][0]			))
            )
            _UpperCamelCase      :	List[str]						=       (d(matrix[0][0]			) * d(matrix[1][1]			)) - (
                d(matrix[0][1]			) * d(matrix[1][0]			)
            )

            # Transpose the cofactor matrix (Adjoint matrix)
            _UpperCamelCase      :	str						=       array(lowercase_			)
            for i in range(3			):
                  for j in range(3			):
                        _UpperCamelCase      :	List[str]						=       cofactor_matrix[j][i]

        # Inverse of the matrix using the formula (1/determinant) * adjoint matrix
            _UpperCamelCase      :	List[str]						=       array(lowercase_			)
            for i in range(3			):
                  for j in range(3			):
                        inverse_matrix[i][j] /= d(lowercase_			)

        # Calculate the inverse of the matrix
            return [[float(d(lowercase_			)			) or 0.0 for n in row] for row in inverse_matrix]
      raise ValueError("Please provide a matrix of size 2x2 or 3x3."			)







"""simple docstring"""


def 					lowercase__    (					lowercase_			)					->		set:




      """simple docstring"""

      _UpperCamelCase      :	Union[str, Any]						=       set()
      # edges = list of graph's edges
      _UpperCamelCase      :	Union[str, Any]						=       get_edges(lowercase_			)

      # While there are still elements in edges list, take an arbitrary edge
      # (from_node, to_node) and add his extremity to chosen_vertices and then
      # remove all arcs adjacent to the from_node and to_node
      while edges:
            _UpperCamelCase,  _UpperCamelCase      :	str						=       edges.pop()
            chosen_vertices.add(lowercase_			)
            chosen_vertices.add(lowercase_			)
            for edge in edges.copy():
                  if from_node in edge or to_node in edge:
                        edges.discard(lowercase_			)
      return chosen_vertices




def 					lowercase__    (					lowercase_			)					->		set:




      """simple docstring"""

      _UpperCamelCase      :	List[str]						=       set()
      for from_node, to_nodes in graph.items():
            for to_node in to_nodes:
                  edges.add((from_node, to_node)			)
      return edges


if __name__ == "__main__":
  import doctest

  doctest.testmod()

  # graph = {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
  # print(f"Matching vertex cover:\n{matching_min_vertex_cover(graph)}")
