

import heapq
def 						_snake_case(    SCREAMING_SNAKE_CASE__			:	dict      )							->      set[int]:





    '''simple docstring'''






    A__          =   []

    # 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
    A__          =   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
        A__          =   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]:
                A__          =   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()

     lowercase_       		=  {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)}""")





from typing import Any
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						)     ->			list:







 """simple docstring"""


 _validation(
     __snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						)
 # Creates data structures and fill initial step
 _lowercase							      ={}
 _lowercase							      ={}
 for state in states_space:
  _lowercase							      =observations_space[0]
  _lowercase							      =(
      initial_probabilities[state] * emission_probabilities[state][observation]
  )
  _lowercase							      =None

 # Fills the data structure with the probabilities of
 # different transitions and pointers to previous states
 for o in range(1     ,						len(__snake_case						)						):
  _lowercase							      =observations_space[o]
  _lowercase							      =observations_space[o - 1]
  for state in states_space:
   # Calculates the argmax for probability function
   _lowercase							      =''''''
   _lowercase							      =-1
   for k_state in states_space:
    _lowercase							      =(
        probabilities[(k_state, prior_observation)]
        * transition_probabilities[k_state][state]
        * emission_probabilities[state][observation]
    )
    if probability > max_probability:
     _lowercase							      =probability
     _lowercase							      =k_state

            # Update probabilities and pointers dicts
   _lowercase							      =(
       probabilities[(arg_max, prior_observation)]
       * transition_probabilities[arg_max][state]
       * emission_probabilities[state][observation]
   )

   _lowercase							      =arg_max

    # The final observation
 _lowercase							      =observations_space[len(__snake_case						) - 1]

 # argmax for given final observation
 _lowercase							      =''''''
 _lowercase							      =-1
 for k_state in states_space:
  _lowercase							      =probabilities[(k_state, final_observation)]
  if probability > max_probability:
   _lowercase							      =probability
   _lowercase							      =k_state
 _lowercase							      =arg_max

 # Process pointers backwards
 _lowercase							      =last_state
 _lowercase							      =[]
 for o in range(len(__snake_case						) - 1     ,						-1     ,						-1						):
  result.append(__snake_case						)
  _lowercase							      =pointers[previous, observations_space[o]]
 result.reverse()

 return result
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						)     ->			None:







 """simple docstring"""


 _validate_not_empty(
     __snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						)
 _validate_lists(__snake_case     ,						__snake_case						)
 _validate_dicts(
     __snake_case     ,						__snake_case     ,						__snake_case						)
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case     ,						)     ->			None:







 """simple docstring"""


 if not all(
     [
         observations_space,
         states_space,
         initial_probabilities,
         transition_probabilities,
         emission_probabilities,
     ]						):
  raise ValueError('''There\'s an empty parameter'''						)
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case						)     ->			None:







 """simple docstring"""


 _validate_list(__snake_case     ,						'''observations_space'''						)
 _validate_list(__snake_case     ,						'''states_space'''						)
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case						)     ->			None:







 """simple docstring"""


 if not isinstance(_object     ,						__snake_case						):
  _lowercase							      =F"{var_name} must be a list"
  raise ValueError(__snake_case						)
 else:
  for x in _object:
   if not isinstance(__snake_case     ,						__snake_case						):
    _lowercase							      =F"{var_name} must be a list of strings"
    raise ValueError(__snake_case						)
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case     ,						__snake_case     ,						)     ->			None:







 """simple docstring"""


 _validate_dict(__snake_case     ,						'''initial_probabilities'''     ,						__snake_case						)
 _validate_nested_dict(__snake_case     ,						'''transition_probabilities'''						)
 _validate_nested_dict(__snake_case     ,						'''emission_probabilities'''						)
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case						)     ->			None:







 """simple docstring"""


 _validate_dict(_object     ,						__snake_case     ,						__snake_case						)
 for x in _object.values():
  _validate_dict(__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case						)
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case     ,						__snake_case     ,						__snake_case = False						)     ->			None:







 """simple docstring"""


 if not isinstance(_object     ,						__snake_case						):
  _lowercase							      =F"{var_name} must be a dict"
  raise ValueError(__snake_case						)
 if not all(isinstance(__snake_case     ,						__snake_case						) for x in _object						):
  _lowercase							      =F"{var_name} all keys must be strings"
  raise ValueError(__snake_case						)
 if not all(isinstance(__snake_case     ,						__snake_case						) for x in _object.values()						):
  _lowercase							      ='''nested dictionary ''' if nested else ''''''
  _lowercase							      =F"{var_name} {nested_text}all values must be {value_type.__name__}"
  raise ValueError(__snake_case						)


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

  testmod()

