

'''simple docstring'''


import heapq




def 	_a       (				__lowerCAmelCase		:      dict     ):

		"""simple docstring"""

		snake_case__						:       list[list]       						=  []

		# for each node and his adjacency list add them and the rank of the node to queue
		# using heapq module the queue will be filled like a Priority Queue
		# heapq works with a min priority queue, so I used -1*len(v) to build it
		for key, value in graph.items():
				# O(log(n))
				heapq.heappush(__lowerCAmelCase   ,							[-1 * len(__lowerCAmelCase     ), (key, value)]     )

		# chosen_vertices = set of chosen vertices
		snake_case__						:       List[Any]       						=  set()

		# while queue isn't empty and there are still edges
		#   (queue[0][0] is the rank of the node with max rank)
		while queue and queue[0][0] != 0:
				# extract vertex with max rank from queue and add it to chosen_vertices
				snake_case__						:       Optional[Any]       						=  heapq.heappop(__lowerCAmelCase     )[1][0]
				chosen_vertices.add(__lowerCAmelCase     )

				# Remove all arcs adjacent to argmax
				for elem in queue:
						# if v haven't adjacent node, skip
						if elem[0] == 0:
								continue
						# if argmax is reachable from elem
						# remove argmax from elem's adjacent list and update his rank
						if argmax in elem[1][1]:
								snake_case__						:       int       						=  elem[1][1].index(__lowerCAmelCase     )
								del elem[1][1][index]
								elem[0] += 1
        # re-order the queue
				heapq.heapify(__lowerCAmelCase     )
		return chosen_vertices


if __name__ == "__main__":
					import doctest

					doctest.testmod()

					lowerCAmelCase__						:       List[Any]										=						{0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
					print(f"""Minimum vertex cover:\n{greedy_min_vertex_cover(graph)}""")






'''simple docstring'''


from __future__ import annotations

from typing import Any




class     a			:






						"""simple docstring"""




						def __init__(							self					:      Tuple			,			snake_case_					:      int			,			snake_case_					:      int			,			snake_case_					:      float = 0  ):



								'''simple docstring'''



								snake_case__						,       snake_case__						:       Optional[Any]       						=  row, column
								snake_case__						:       Dict       						=  [[default_value for c in range(snake_case_  )] for r in range(snake_case_  )]







						def __str__(							self					:      Optional[Any]  ):



								'''simple docstring'''



								snake_case__						:       Optional[int]       						=  F"""Matrix consist of {self.row} rows and {self.column} columns\n"""

								# Make string identifier
								snake_case__						:       List[str]       						=  0
								for row_vector in self.array:
										for obj in row_vector:
												snake_case__						:       str       						=  max(snake_case_			,			len(str(snake_case_  )  )  )
								snake_case__						:       Dict       						=  F"""%{max_element_length}s"""

								# Make string and return
								def single_line(snake_case_					:      list[float]  ) -> str:
										nonlocal string_format_identifier
										snake_case__						:       str       						=  '''['''
										line += ", ".join(string_format_identifier % (obj,) for obj in row_vector  )
										line += "]"
										return line

								s += "\n".join(single_line(snake_case_  ) for row_vector in self.array  )
								return s







						def __repr__(							self					:      Optional[int]  ):



								'''simple docstring'''



								return str(self  )







						def       __magic_name__							(							self					:      Optional[int]			,			snake_case_					:      tuple[int, int]  ):



								'''simple docstring'''



								if not (isinstance(snake_case_			,			(list, tuple)  ) and len(snake_case_  ) == 2):
										return False
								elif not (0 <= loc[0] < self.row and 0 <= loc[1] < self.column):
										return False
								else:
										return True







						def __getitem__(							self					:      List[str]			,			snake_case_					:      tuple[int, int]  ):



								'''simple docstring'''



								assert self.validate_indicies(snake_case_  )
								return self.array[loc[0]][loc[1]]







						def __setitem__(							self					:      int			,			snake_case_					:      tuple[int, int]			,			snake_case_					:      float  ):



								'''simple docstring'''



								assert self.validate_indicies(snake_case_  )
								snake_case__						:       Optional[Any]       						=  value







						def __add__(							self					:      Dict			,			snake_case_					:      Matrix  ):



								'''simple docstring'''



								assert isinstance(snake_case_			,			snake_case_  )
								assert self.row == another.row and self.column == another.column

								# Add
								snake_case__						:       str       						=  Matrix(self.row			,			self.column  )
								for r in range(self.row  ):
										for c in range(self.column  ):
												snake_case__						:       Optional[Any]       						=  self[r, c] + another[r, c]
								return result







						def __neg__(							self					:      Optional[Any]  ):



								'''simple docstring'''



								snake_case__						:       int       						=  Matrix(self.row			,			self.column  )
								for r in range(self.row  ):
										for c in range(self.column  ):
												snake_case__						:       str       						=  -self[r, c]
								return result







						def __sub__(							self					:      Optional[int]			,			snake_case_					:      Matrix  ):



								'''simple docstring'''



								return self + (-another)







						def __mul__(							self					:      int			,			snake_case_					:      int | float | Matrix  ):



								'''simple docstring'''



								if isinstance(snake_case_			,			(int, float)  ):  # Scalar multiplication
										snake_case__						:       Dict       						=  Matrix(self.row			,			self.column  )
										for r in range(self.row  ):
												for c in range(self.column  ):
														snake_case__						:       Tuple       						=  self[r, c] * another
										return result
								elif isinstance(snake_case_			,			snake_case_  ):  # Matrix multiplication
										assert self.column == another.row
										snake_case__						:       List[Any]       						=  Matrix(self.row			,			another.column  )
										for r in range(self.row  ):
												for c in range(another.column  ):
														for i in range(self.column  ):
																result[r, c] += self[r, i] * another[i, c]
										return result
								else:
										snake_case__						:       Union[str, Any]       						=  F"""Unsupported type given for another ({type(snake_case_  )})"""
										raise TypeError(snake_case_  )







						def       __magic_name__							(							self					:      List[str]  ):



								'''simple docstring'''



								snake_case__						:       Optional[int]       						=  Matrix(self.column			,			self.row  )
								for r in range(self.row  ):
										for c in range(self.column  ):
												snake_case__						:       List[Any]       						=  self[r, c]
								return result







						def       __magic_name__							(							self					:      Tuple			,			snake_case_					:      Matrix			,			snake_case_					:      Matrix  ):



								'''simple docstring'''



								assert isinstance(snake_case_			,			snake_case_  ) and isinstance(snake_case_			,			snake_case_  )
								assert self.row == self.column == u.row == v.row  # u, v should be column vector
								assert u.column == v.column == 1  # u, v should be column vector

								# Calculate
								snake_case__						:       Tuple       						=  v.transpose()
								snake_case__						:       Optional[Any]       						=  (v_t * self * u)[0, 0] + 1
								if numerator_factor == 0:
										return None  # It's not invertable
								return self - ((self * u) * (v_t * self) * (1.0 / numerator_factor))


# Testing
if __name__ == "__main__":




					def 	_a       (				):

							"""simple docstring"""

							snake_case__						:       Tuple       						=  Matrix(3   ,							3   ,							0     )
							for i in range(3     ):
									snake_case__						:       Any       						=  1
							print(F"""a^(-1) is {ainv}"""     )
							# u, v
							snake_case__						:       int       						=  Matrix(3   ,							1   ,							0     )
							snake_case__						,       snake_case__						,       snake_case__						:       List[Any]       						=  1, 2, -3
							snake_case__						:       Any       						=  Matrix(3   ,							1   ,							0     )
							snake_case__						,       snake_case__						,       snake_case__						:       Optional[int]       						=  4, -2, 5
							print(F"""u is {u}"""     )
							print(F"""v is {v}"""     )
							print(F"""uv^T is {u * v.transpose()}"""     )
							# Sherman Morrison
							print(F"""(a + uv^T)^(-1) is {ainv.sherman_morrison(__lowerCAmelCase   ,							__lowerCAmelCase     )}"""     )




					def 	_a       (				):

							"""simple docstring"""

							import doctest

							doctest.testmod()





					testa()



