


import heapq



def     __lowerCAmelCase						(							snake_case	:    List[str] )       ->							List[str]:
   __lowerCamelCase:				List[Any]     =	[]

   # 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(lowercase__						,      [-1 * len(lowercase__ ), (key, value)] )

   # chosen_vertices = set of chosen vertices
   __lowerCamelCase:				str     =	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
      __lowerCamelCase:				Optional[int]     =	heapq.heappop(lowercase__ )[1][0]
      chosen_vertices.add(lowercase__ )

      # 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]:
            __lowerCamelCase:				Tuple     =	elem[1][1].index(lowercase__ )
            del elem[1][1][index]
            elem[0] += 1
        # re-order the queue
      heapq.heapify(lowercase__ )
   return chosen_vertices


if __name__ == "__main__":
   import doctest

   doctest.testmod()

   _A		:				Tuple									=       {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 __future__ import annotations

_A		:				List[str]									=       {
    '''A''': ['''B''', '''C''', '''E'''],
    '''B''': ['''A''', '''D''', '''E'''],
    '''C''': ['''A''', '''F''', '''G'''],
    '''D''': ['''B'''],
    '''E''': ['''A''', '''B''', '''D'''],
    '''F''': ['''C'''],
    '''G''': ['''C'''],
}



class        a					:





     def __init__(    self					:				Any    ,	SCREAMING_SNAKE_CASE_					:				dict[str, list[str]]    ,	SCREAMING_SNAKE_CASE_					:				str							):
        __lowerCamelCase:				Dict     =	graph
        # mapping node to its parent in resulting breadth first tree
        __lowerCamelCase:				dict[str, str | None]     =	{}
        __lowerCamelCase:				str     =	source_vertex





     def  SCREAMING_SNAKE_CASE__       (    self					:				int							):
        __lowerCamelCase:				Optional[int]     =	{self.source_vertex}
        __lowerCamelCase:				Optional[int]     =	None
        __lowerCamelCase:				Optional[int]     =	[self.source_vertex]  # first in first out queue

        while queue:
           __lowerCamelCase:				int     =	queue.pop(0							)
           for adjacent_vertex in self.graph[vertex]:
              if adjacent_vertex not in visited:
                 visited.add(SCREAMING_SNAKE_CASE_							)
                 __lowerCamelCase:				Dict     =	vertex
                 queue.append(SCREAMING_SNAKE_CASE_							)







     def  SCREAMING_SNAKE_CASE__       (    self					:				Optional[Any]    ,	SCREAMING_SNAKE_CASE_					:				str							):
        if target_vertex == self.source_vertex:
           return self.source_vertex

        __lowerCamelCase:				List[str]     =	self.parent.get(SCREAMING_SNAKE_CASE_							)
        if target_vertex_parent is None:
           __lowerCamelCase:				Optional[int]     =	(
               F'No path from vertex: {self.source_vertex} to vertex: {target_vertex}'
           )
           raise ValueError(SCREAMING_SNAKE_CASE_							)

        return self.shortest_path(SCREAMING_SNAKE_CASE_							) + F'->{target_vertex}'


if __name__ == "__main__":
   _A		:				int									=       Graph(graph, '''G''')
   g.breath_first_search()
   print(g.shortest_path('''D'''))
   print(g.shortest_path('''G'''))
   print(g.shortest_path('''Foo'''))


