import math





def 						__lowerCamelCase				(UpperCAmelCase__  :				Dict	):
					assert isinstance(__snake_case   ,			__snake_case	) and (
					    number >= 0
					), "'number' must been an int and positive"

					if 1 < number < 4:
										# 2 and 3 are primes
										return True
					elif number < 2 or not number % 2:
										# Negatives, 0, 1 and all even numbers are not primes
										return False

					SCREAMING_SNAKE_CASE           =							range(3   ,			int(math.sqrt(__snake_case	) + 1	)   ,			2	)
					return not any(not number % i for i in odd_numbers	)





def 						__lowerCamelCase				(UpperCAmelCase__  :				List[str]   ,			UpperCAmelCase__  :				Any=1   ,			**UpperCAmelCase__  :				List[Any]	):
					SCREAMING_SNAKE_CASE           =							factor * value
					SCREAMING_SNAKE_CASE           =							value

					while not is_prime(__snake_case	):
										value += 1 if not ("desc" in kwargs and kwargs["desc"] is True) else -1

					if value == first_value_val:
										return next_prime(value + 1   ,			**__snake_case	)
					return value






"""simple docstring"""





import heapq



def  lowerCamelCase__						(     __snake_case      )				->						set[int]:


					"""simple docstring"""





					_UpperCamelCase									=    []

					# 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(__snake_case,	[-1 * len(__snake_case      ), (key, value)]      )

					# chosen_vertices = set of chosen vertices
					_UpperCamelCase									=    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
										_UpperCamelCase									=    heapq.heappop(__snake_case      )[1][0]
										chosen_vertices.add(__snake_case      )

										# 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]:
																				_UpperCamelCase									=    elem[1][1].index(__snake_case      )
																				del elem[1][1][index]
																				elem[0] += 1
        # re-order the queue
										heapq.heapify(__snake_case      )
					return chosen_vertices


if __name__ == "__main__":
				import doctest

				doctest.testmod()

				_a								=						{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)}""")






