
"""simple docstring"""





def  lowerCamelCase__						(     __snake_case      )				->						int:


					"""simple docstring"""





					_UpperCamelCase									=    [1]

					_UpperCamelCase					,    _UpperCamelCase					,    _UpperCamelCase									=    0, 0, 0
					_UpperCamelCase									=    ugly_nums[ia] * 2
					_UpperCamelCase									=    ugly_nums[ia] * 3
					_UpperCamelCase									=    ugly_nums[ia] * 5

					for _ in range(1,	__snake_case      ):
										_UpperCamelCase									=    min(__snake_case,	__snake_case,	__snake_case      )
										ugly_nums.append(__snake_case      )
										if next_num == next_a:
															ia += 1
															_UpperCamelCase									=    ugly_nums[ia] * 2
										if next_num == next_a:
															ia += 1
															_UpperCamelCase									=    ugly_nums[ia] * 3
										if next_num == next_a:
															ia += 1
															_UpperCamelCase									=    ugly_nums[ia] * 5
					return ugly_nums[-1]


if __name__ == "__main__":
				from doctest import testmod

				testmod(verbose=True)
				print(F"""{ugly_numbers(200) = }""")








"""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)}""")






