


"""simple docstring"""






import heapq


def __lowercase     (      lowerCamelCase_  :      dict  ):
				SCREAMING_SNAKE_CASE__     							=				[]

				# 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
				SCREAMING_SNAKE_CASE__     							=				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__     							=				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]:
																SCREAMING_SNAKE_CASE__     							=				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     =  {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 collections.abc import Sequence
from typing import Literal


def __lowercase     (      lowerCamelCase_  :      str						,					lowerCamelCase_  :      str  ):
				SCREAMING_SNAKE_CASE__     							=				list(lowerCamelCase_  )
				SCREAMING_SNAKE_CASE__     							=				list(lowerCamelCase_  )
				SCREAMING_SNAKE_CASE__     							=				0
				for i in range(len(lowerCamelCase_  )  ):
								if lista[i] != lista[i]:
												count += 1
												SCREAMING_SNAKE_CASE__     							=				"_"
				if count > 1:
								return False
				else:
								return "".join(lowerCamelCase_  )


def __lowercase     (      lowerCamelCase_  :      list[str]  ):
				SCREAMING_SNAKE_CASE__     							=				[]
				while True:
								SCREAMING_SNAKE_CASE__     							=				["$"] * len(lowerCamelCase_  )
								SCREAMING_SNAKE_CASE__     							=				[]
								for i in range(len(lowerCamelCase_  )  ):
												for j in range(i + 1						,					len(lowerCamelCase_  )  ):
																SCREAMING_SNAKE_CASE__     							=				compare_string(binary[i]						,					binary[j]  )
																if k is False:
																				SCREAMING_SNAKE_CASE__     							=				"*"
																				SCREAMING_SNAKE_CASE__     							=				"*"
																				temp.append("X"  )
								for i in range(len(lowerCamelCase_  )  ):
												if checka[i] == "$":
																pi.append(binary[i]  )
								if len(lowerCamelCase_  ) == 0:
												return pi
								SCREAMING_SNAKE_CASE__     							=				list(set(lowerCamelCase_  )  )


def __lowercase     (      lowerCamelCase_  :      int						,					lowerCamelCase_  :      Sequence[float]  ):
				SCREAMING_SNAKE_CASE__     							=				[]
				for minterm in minterms:
								SCREAMING_SNAKE_CASE__     							=				""
								for _ in range(lowerCamelCase_  ):
												SCREAMING_SNAKE_CASE__     							=				str(minterm % 2  ) + string
												minterm //= 2
								temp.append(lowerCamelCase_  )
				return temp


def __lowercase     (      lowerCamelCase_  :      str						,					lowerCamelCase_  :      str						,					lowerCamelCase_  :      int  ):
				SCREAMING_SNAKE_CASE__     							=				list(lowerCamelCase_  )
				SCREAMING_SNAKE_CASE__     							=				list(lowerCamelCase_  )
				SCREAMING_SNAKE_CASE__     							=				0
				for i in range(len(lowerCamelCase_  )  ):
								if lista[i] != lista[i]:
												count_n += 1
				return count_n == count


def __lowercase     (      lowerCamelCase_  :      list[list[int]]						,					lowerCamelCase_  :      list[str]  ):
				SCREAMING_SNAKE_CASE__     							=				[]
				SCREAMING_SNAKE_CASE__     							=				[0] * len(lowerCamelCase_  )
				for i in range(len(chart[0]  )  ):
								SCREAMING_SNAKE_CASE__     							=				0
								SCREAMING_SNAKE_CASE__     							=				-1
								for j in range(len(lowerCamelCase_  )  ):
												if chart[j][i] == 1:
																count += 1
																SCREAMING_SNAKE_CASE__     							=				j
								if count == 1:
												SCREAMING_SNAKE_CASE__     							=				1
				for i in range(len(lowerCamelCase_  )  ):
								if select[i] == 1:
												for j in range(len(chart[0]  )  ):
																if chart[i][j] == 1:
																				for k in range(len(lowerCamelCase_  )  ):
																								SCREAMING_SNAKE_CASE__     							=				0
												temp.append(prime_implicants[i]  )
				while True:
								SCREAMING_SNAKE_CASE__     							=				0
								SCREAMING_SNAKE_CASE__     							=				-1
								SCREAMING_SNAKE_CASE__     							=				0
								for i in range(len(lowerCamelCase_  )  ):
												SCREAMING_SNAKE_CASE__     							=				chart[i].count(1  )
												if count_n > max_n:
																SCREAMING_SNAKE_CASE__     							=				count_n
																SCREAMING_SNAKE_CASE__     							=				i

								if max_n == 0:
												return temp

								temp.append(prime_implicants[rem]  )

								for i in range(len(chart[0]  )  ):
												if chart[rem][i] == 1:
																for j in range(len(lowerCamelCase_  )  ):
																				SCREAMING_SNAKE_CASE__     							=				0


def __lowercase     (      lowerCamelCase_  :      list[str]						,					lowerCamelCase_  :      list[str]  ):
				SCREAMING_SNAKE_CASE__     							=				[[0 for x in range(len(lowerCamelCase_  )  )] for x in range(len(lowerCamelCase_  )  )]
				for i in range(len(lowerCamelCase_  )  ):
								SCREAMING_SNAKE_CASE__     							=				prime_implicants[i].count("_"  )
								for j in range(len(lowerCamelCase_  )  ):
												if is_for_table(prime_implicants[i]						,					binary[j]						,					lowerCamelCase_  ):
																SCREAMING_SNAKE_CASE__     							=				1

				return chart


def __lowercase     (      ):
				SCREAMING_SNAKE_CASE__     							=				int(input("Enter the no. of variables\n"  )  )
				SCREAMING_SNAKE_CASE__     							=				[
				    float(lowerCamelCase_  )
				    for x in input(
				        "Enter the decimal representation of Minterms 'Spaces Separated'\n"  ).split()
				]
				SCREAMING_SNAKE_CASE__     							=				decimal_to_binary(lowerCamelCase_						,					lowerCamelCase_  )

				SCREAMING_SNAKE_CASE__     							=				check(lowerCamelCase_  )
				print("Prime Implicants are:"  )
				print(lowerCamelCase_  )
				SCREAMING_SNAKE_CASE__     							=				prime_implicant_chart(lowerCamelCase_						,					lowerCamelCase_  )

				SCREAMING_SNAKE_CASE__     							=				selection(lowerCamelCase_						,					lowerCamelCase_  )
				print("Essential Prime Implicants are:"  )
				print(lowerCamelCase_  )


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	main()
