

'''simple docstring'''




from __future__ import annotations



class        lowercase_	:

  '''simple docstring'''





  def __init__(     self   , a_     )							->				Tuple:


    """simple docstring"""
    UpperCAmelCase													=	vertices
    UpperCAmelCase													=	[[0] * vertices for _ in range(UpperCamelCase__     )]
  def 						snake_case_			(     self   , a_     )							->				Dict:


    """simple docstring"""
    print('Vertex \t Distance from Source'     )
    for vertex in range(self.vertices     ):
      print(UpperCamelCase__   , '\t\t'   , distances_from_source[vertex]     )
  def 						snake_case_			(     self   , a_   , a_     )							->				int:


    """simple docstring"""
    UpperCAmelCase													=	1E7
    UpperCAmelCase													=	0

    # Search not nearest vertex not in the shortest path tree
    for vertex in range(self.vertices     ):
      if distances_from_source[vertex] < minimum and visited[vertex] is False:
        UpperCAmelCase													=	distances_from_source[vertex]
        UpperCAmelCase													=	vertex
    return min_index





  def 						snake_case_			(     self   , a_     )							->				Union[str, Any]:


    """simple docstring"""
    UpperCAmelCase													=	[int(1E7     )] * self.vertices  # distances from the source
    UpperCAmelCase													=	0
    UpperCAmelCase													=	[False] * self.vertices

    for _ in range(self.vertices     ):
      UpperCAmelCase													=	self.minimum_distance(UpperCamelCase__   , UpperCamelCase__     )
      UpperCAmelCase													=	True

      # Update dist value of the adjacent vertices
      # of the picked vertex only if the current
      # distance is greater than new distance and
      # the vertex in not in the shortest path tree
      for v in range(self.vertices     ):
        if (
            self.graph[u][v] > 0
            and visited[v] is False
            and distances[v] > distances[u] + self.graph[u][v]
        ):
          UpperCAmelCase													=	distances[u] + self.graph[u][v]

    self.print_solution(UpperCamelCase__     )


if __name__ == "__main__":
  _a				: str 							=						Graph(9)
  _a				: Optional[Any] 							=						[
      [0, 4, 0, 0, 0, 0, 0, 8, 0],
      [4, 0, 8, 0, 0, 0, 0, 11, 0],
      [0, 8, 0, 7, 0, 4, 0, 0, 2],
      [0, 0, 7, 0, 9, 14, 0, 0, 0],
      [0, 0, 0, 9, 0, 10, 0, 0, 0],
      [0, 0, 4, 14, 10, 0, 2, 0, 0],
      [0, 0, 0, 0, 0, 2, 0, 1, 6],
      [8, 11, 0, 0, 0, 0, 1, 0, 7],
      [0, 0, 2, 0, 0, 0, 6, 7, 0],
  ]
  graph.dijkstra(0)





'''simple docstring'''




def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   list[list[int]]						):
  if len(SCREAMING_SNAKE_CASE						) == 0 or not all(len(SCREAMING_SNAKE_CASE						) == 2 for pair in coordinates						):
    raise ValueError('The program cannot work out a fitting polynomial.'						)

  if len({tuple(SCREAMING_SNAKE_CASE						) for pair in coordinates}						) != len(SCREAMING_SNAKE_CASE						):
    raise ValueError('The program cannot work out a fitting polynomial.'						)

  UpperCAmelCase													=	{x for x, _ in coordinates}
  if len(SCREAMING_SNAKE_CASE						) == 1:
    return f'''x={coordinates[0][0]}'''

  if len(SCREAMING_SNAKE_CASE						) != len(SCREAMING_SNAKE_CASE						):
    raise ValueError('The program cannot work out a fitting polynomial.'						)

  UpperCAmelCase													=	len(SCREAMING_SNAKE_CASE						)

  UpperCAmelCase													=	0
  UpperCAmelCase													=	[]
  # put the x and x to the power values in a matrix
  while count_of_line < x:
    UpperCAmelCase													=	0
    UpperCAmelCase													=	coordinates[count_of_line][0]
    UpperCAmelCase													=	[]
    while count_in_line < x:
      count_line.append(a ** (x - (count_in_line + 1))						)
      count_in_line += 1
    matrix.append(SCREAMING_SNAKE_CASE						)
    count_of_line += 1

  UpperCAmelCase													=	0
  # put the y values into a vector
  UpperCAmelCase													=	[]
  while count_of_line < x:
    vector.append(coordinates[count_of_line][1]						)
    count_of_line += 1

  UpperCAmelCase													=	0

  while count < x:
    UpperCAmelCase													=	0
    while zahlen < x:
      if count == zahlen:
        zahlen += 1
      if zahlen == x:
        break
      UpperCAmelCase													=	matrix[zahlen][count] / matrix[count][count]
      for counting_columns, item in enumerate(matrix[count]						):
        # manipulating all the values in the matrix
        matrix[zahlen][counting_columns] -= item * bruch
      # manipulating the values in the vector
      vector[zahlen] -= vector[count] * bruch
      zahlen += 1
    count += 1

  UpperCAmelCase													=	0
  # make solutions
  UpperCAmelCase													=	[]
  while count < x:
    solution.append(str(vector[count] / matrix[count][count]						)						)
    count += 1

  UpperCAmelCase													=	0
  UpperCAmelCase													=	'f(x)='

  while count < x:
    UpperCAmelCase													=	solution[count].split('E'						)
    if len(SCREAMING_SNAKE_CASE						) > 1:
      UpperCAmelCase													=	f'''{remove_e[0]}*10^{remove_e[1]}'''
    solved += f'''x^{x - (count + 1)}*{solution[count]}'''
    if count + 1 != x:
      solved += "+"
    count += 1

  return solved


if __name__ == "__main__":
  print(points_to_polynomial([]))
  print(points_to_polynomial([[]]))
  print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
  print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
  print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
  print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
  print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
  print(points_to_polynomial([[1, 3], [2, 6], [3, 11]]))
  print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
  print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))


