





"""simple docstring"""





from __future__ import annotations

lowercase__				:Optional[int]				   =       list[list[int]]

# assigning initial values to the grid
lowercase__				:Matrix				   =       [
    [3, 0, 6, 5, 0, 8, 4, 0, 0],
    [5, 2, 0, 0, 0, 0, 0, 0, 0],
    [0, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]

# a grid with no solution
lowercase__				:Matrix				   =       [
    [5, 0, 6, 5, 0, 8, 4, 0, 3],
    [5, 2, 0, 0, 0, 0, 0, 0, 2],
    [1, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]


def       lowerCamelCase_       (   UpperCAmelCase_  ,			UpperCAmelCase_  ,			UpperCAmelCase_  ,			UpperCAmelCase_  )						->bool:





  """simple docstring"""


  for i in range(9  ):
    if grid[row][i] == n or grid[i][column] == n:
      return False

  for i in range(3  ):
    for j in range(3  ):
      if grid[(row - row % 3) + i][(column - column % 3) + j] == n:
        return False

  return True


def       lowerCamelCase_       (   UpperCAmelCase_  )						->tuple[int, int] | None:





  """simple docstring"""


  for i in range(9  ):
    for j in range(9  ):
      if grid[i][j] == 0:
        return i, j
  return None


def       lowerCamelCase_       (   UpperCAmelCase_  )						->Matrix | None:





  """simple docstring"""


  if location := find_empty_location(UpperCAmelCase_  ):
    __UpperCAmelCase    :						int       =						location
  else:
    # If the location is ``None``, then the grid is solved.
    return grid

  for digit in range(1  ,			10  ):
    if is_safe(UpperCAmelCase_  ,			UpperCAmelCase_  ,			UpperCAmelCase_  ,			UpperCAmelCase_  ):
      __UpperCAmelCase    :						int       =						digit

      if sudoku(UpperCAmelCase_  ) is not None:
        return grid

      __UpperCAmelCase    :						Union[str, Any]       =						0

  return None


def       lowerCamelCase_       (   UpperCAmelCase_  )						->None:





  """simple docstring"""


  for row in grid:
    for cell in row:
      print(UpperCAmelCase_  ,			end=''' '''  )
    print()


if __name__ == "__main__":
  # make a copy of grid so that you can compare with the unmodified grid
  for example_grid in (initial_grid, no_solution):
    print('\nExample grid:\n' + '=' * 2_0)
    print_solution(example_grid)
    print('\nExample grid solution:')
    lowercase__				:Tuple				   =       sudoku(example_grid)
    if solution is not None:
      print_solution(solution)
    else:
      print('Cannot find a solution.')






"""simple docstring"""





def       lowerCamelCase_       (   UpperCAmelCase_  ,			UpperCAmelCase_  )						->Tuple:





  """simple docstring"""


  return (pointa[0] - pointa[0]) ** 2 + (pointa[1] - pointa[1]) ** 2


def       lowerCamelCase_       (   UpperCAmelCase_  ,			UpperCAmelCase_=0  )						->Dict:





  """simple docstring"""


  return sorted(UpperCAmelCase_  ,			key=lambda UpperCAmelCase_  : x[column]  )


def       lowerCamelCase_       (   UpperCAmelCase_  ,			UpperCAmelCase_  ,			UpperCAmelCase_=float('''inf'''  )  )						->str:





  """simple docstring"""


  for i in range(points_counts - 1  ):
    for j in range(i + 1  ,			UpperCAmelCase_  ):
      __UpperCAmelCase    :						List[Any]       =						euclidean_distance_sqr(points[i]  ,			points[j]  )
      if current_dis < min_dis:
        __UpperCAmelCase    :						Tuple       =						current_dis
  return min_dis


def       lowerCamelCase_       (   UpperCAmelCase_  ,			UpperCAmelCase_  ,			UpperCAmelCase_=float('''inf'''  )  )						->str:





  """simple docstring"""


  for i in range(min(6  ,			points_counts - 1  )  ,			UpperCAmelCase_  ):
    for j in range(max(0  ,			i - 6  )  ,			UpperCAmelCase_  ):
      __UpperCAmelCase    :						Dict       =						euclidean_distance_sqr(points[i]  ,			points[j]  )
      if current_dis < min_dis:
        __UpperCAmelCase    :						Tuple       =						current_dis
  return min_dis


def       lowerCamelCase_       (   UpperCAmelCase_  ,			UpperCAmelCase_  ,			UpperCAmelCase_  )						->Any:





  """simple docstring"""


  if points_counts <= 3:
    return dis_between_closest_pair(UpperCAmelCase_  ,			UpperCAmelCase_  )

  # recursion
  __UpperCAmelCase    :						Any       =						points_counts // 2
  __UpperCAmelCase    :						Any       =						closest_pair_of_points_sqr(
      UpperCAmelCase_  ,			points_sorted_on_y[:mid]  ,			UpperCAmelCase_  )
  __UpperCAmelCase    :						Tuple       =						closest_pair_of_points_sqr(
      UpperCAmelCase_  ,			points_sorted_on_y[mid:]  ,			points_counts - mid  )
  __UpperCAmelCase    :						List[Any]       =						min(UpperCAmelCase_  ,			UpperCAmelCase_  )
  __UpperCAmelCase    :						int       =						[]
  for point in points_sorted_on_x:
    if abs(point[0] - points_sorted_on_x[mid][0]  ) < closest_pair_dis:
      cross_strip.append(UpperCAmelCase_  )

  __UpperCAmelCase    :						Union[str, Any]       =						dis_between_closest_in_strip(
      UpperCAmelCase_  ,			len(UpperCAmelCase_  )  ,			UpperCAmelCase_  )
  return min(UpperCAmelCase_  ,			UpperCAmelCase_  )


def       lowerCamelCase_       (   UpperCAmelCase_  ,			UpperCAmelCase_  )						->List[Any]:





  """simple docstring"""


  __UpperCAmelCase    :						str       =						column_based_sort(UpperCAmelCase_  ,			column=0  )
  __UpperCAmelCase    :						Any       =						column_based_sort(UpperCAmelCase_  ,			column=1  )
  return (
      closest_pair_of_points_sqr(
          UpperCAmelCase_  ,			UpperCAmelCase_  ,			UpperCAmelCase_  )
  ) ** 0.5


if __name__ == "__main__":
  lowercase__				:Optional[Any]				   =       [(2, 3), (1_2, 3_0), (4_0, 5_0), (5, 1), (1_2, 1_0), (3, 4)]
  print('Distance:', closest_pair_of_points(points, len(points)))