
"""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)}""")








"""simple docstring"""





from collections import defaultdict






class     _UpperCAmelCase:



		def __init__(					self				,		__a				,		__a)       ->       Union[str, Any]:






							'''simple docstring'''
							_UpperCamelCase							 =  total  # total no of tasks (N)

							# DP table will have a dimension of (2^M)*N
							# initially all values are set to -1
							_UpperCamelCase							 =  [
							    [-1 for i in range(total + 1)] for j in range(2 ** len(__a))
							]

							_UpperCamelCase							 =  defaultdict(__a)  # stores the list of persons for each task

							# final_mask is used to check if all persons are included by setting all bits
							# to 1
							_UpperCamelCase							 =  (1 << len(__a)) - 1



		def 						UpperCAmelCase							(					self				,		__a				,		__a)       ->       Dict:






							'''simple docstring'''
							# if mask == self.finalmask all persons are distributed tasks, return 1
							if mask == self.final_mask:
												return 1

							# if not everyone gets the task and no more tasks are available, return 0
							if task_no > self.total_tasks:
												return 0

							# if case already considered
							if self.dp[mask][task_no] != -1:
												return self.dp[mask][task_no]

							# Number of ways when we don't this task in the arrangement
							_UpperCamelCase							 =  self.count_ways_until(__a				,		task_no + 1)

							# now assign the tasks one by one to all possible persons and recursively
							# assign for the remaining tasks.
							if task_no in self.task:
												for p in self.task[task_no]:
																	# if p is already given a task
																	if mask & (1 << p):
																						continue

																	# assign this task to p and change the mask value. And recursively
																	# assign tasks with the new mask value.
																	total_ways_util += self.count_ways_until(mask | (1 << p)				,		task_no + 1)

        # save the value.
							_UpperCamelCase							 =  total_ways_util

							return self.dp[mask][task_no]




		def 						UpperCAmelCase							(					self				,		__a)       ->       int:






							'''simple docstring'''
							# Store the list of persons for each task
							for i in range(len(__a)):
												for j in task_performed[i]:
																	self.task[j].append(__a)

        # call the function to fill the DP table, final answer is stored in dp[0][1]
							return self.count_ways_until(0				,		1)


if __name__ == "__main__":
				_a														=		5  # total no of tasks (the value of N)

				# the list of tasks that can be done by M persons.
				_a														=		[[1, 3, 4], [1, 2, 5], [3, 4]]
				print(
				    AssignmentUsingBitmask(task_performed, total_tasks).count_no_of_ways(
				        task_performed
				    )
				)






