





import heapq




def _a (  SCREAMING_SNAKE_CASE__      :  dict       )	->    set[int]:

							'''simple docstring'''






							SCREAMING_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(SCREAMING_SNAKE_CASE__			,				[-1 * len(SCREAMING_SNAKE_CASE__       ), (key, value)]       )

							# chosen_vertices = set of chosen vertices
							SCREAMING_SNAKE_CASE__       :      Dict     =						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
														SCREAMING_SNAKE_CASE__       :      Union[str, Any]     =						heapq.heappop(SCREAMING_SNAKE_CASE__       )[1][0]
														chosen_vertices.add(SCREAMING_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]:
																												SCREAMING_SNAKE_CASE__       :      Optional[Any]     =						elem[1][1].index(SCREAMING_SNAKE_CASE__       )
																												del elem[1][1][index]
																												elem[0] += 1
        # re-order the queue
														heapq.heapify(SCREAMING_SNAKE_CASE__       )
							return chosen_vertices


if __name__ == "__main__":
		import doctest

		doctest.testmod()

		_lowerCamelCase :    str  							= {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)}")













def _a (  SCREAMING_SNAKE_CASE__      :  int = 50_00_00_00       )	->    int:

							'''simple docstring'''






							SCREAMING_SNAKE_CASE__       :      str     =						set()
							SCREAMING_SNAKE_CASE__       :      Dict     =						int((limit - 24) ** (1 / 2)       )

							SCREAMING_SNAKE_CASE__       :      Any     =						set(range(3			,				prime_square_limit + 1			,				2       )       )
							primes.add(2       )
							for p in range(3			,				prime_square_limit + 1			,				2       ):
														if p not in primes:
																					continue
														primes.difference_update(set(range(p * p			,				prime_square_limit + 1			,				SCREAMING_SNAKE_CASE__       )       )       )

							for primea in primes:
														SCREAMING_SNAKE_CASE__       :      Optional[int]     =						primea * primea
														for primea in primes:
																					SCREAMING_SNAKE_CASE__       :      Union[str, Any]     =						primea * primea * primea
																					if square + cube >= limit - 16:
																												break
																					for primea in primes:
																												SCREAMING_SNAKE_CASE__       :      List[str]     =						primea * primea * primea * primea
																												SCREAMING_SNAKE_CASE__       :      Optional[int]     =						square + cube + tetr
																												if total >= limit:
																																			break
																												ret.add(SCREAMING_SNAKE_CASE__       )

							return len(SCREAMING_SNAKE_CASE__       )


if __name__ == "__main__":
		print(f"{solution() = }")






