
"""simple docstring"""

from abc import ABC, abstractmethod
from argparse import ArgumentParser

class __lowerCAmelCase		(				__SCREAMING_SNAKE_CASE      ):





     @staticmethod
     @abstractmethod
     def 						UpperCAmelCase (     __UpperCAmelCase				):

      '''simple docstring'''




      raise NotImplementedError()







     @abstractmethod
     def 						UpperCAmelCase (     self				):

      '''simple docstring'''




      raise NotImplementedError()





"""simple docstring"""

import heapq as hq
import math
from collections.abc import Iterator

class __lowerCAmelCase		:





     def __init__(     self     ,					__UpperCAmelCase				):

      '''simple docstring'''




      __UpperCamelCase      =	str(id_				)
      __UpperCamelCase      =	None
      __UpperCamelCase      =	None
      __UpperCamelCase      =	[]
      __UpperCamelCase      =	{}  # {vertex:distance}





     def __lt__(     self     ,					__UpperCAmelCase				):

      '''simple docstring'''




      return self.key < other.key





     def __repr__(     self				):

      '''simple docstring'''




      return self.id





     def 						UpperCAmelCase (     self     ,					__UpperCAmelCase				):

      '''simple docstring'''




      self.neighbors.append(__UpperCAmelCase				)







     def 						UpperCAmelCase (     self     ,					__UpperCAmelCase     ,					__UpperCAmelCase				):

      '''simple docstring'''




      __UpperCamelCase      =	weight






def 							A						(    snake_case				:List[Any]    , snake_case				:Dict    , snake_case				:Any    , snake_case				:str		)				->     List[str]:
 # add the neighbors:
 graph[a - 1].add_neighbor(graph[b - 1]		)
 graph[b - 1].add_neighbor(graph[a - 1]		)
 # add the edges:
 graph[a - 1].add_edge(graph[b - 1]    , snake_case		)
 graph[b - 1].add_edge(graph[a - 1]    , snake_case		)






def 							A						(    snake_case				:list    , snake_case				:Vertex		)				->     list:
 __UpperCamelCase      =	[]
 for u in graph:
  __UpperCamelCase      =	math.inf
  __UpperCamelCase      =	None
 __UpperCamelCase      =	0
 __UpperCamelCase      =	graph[:]
 while q:
  __UpperCamelCase      =	min(snake_case		)
  q.remove(snake_case		)
  for v in u.neighbors:
   if (v in q) and (u.edges[v.id] < v.key):
    __UpperCamelCase      =	u
    __UpperCamelCase      =	u.edges[v.id]
 for i in range(1    , len(snake_case		)		):
  a.append((int(graph[i].id		) + 1, int(graph[i].pi.id		) + 1)		)
 return a






def 							A						(    snake_case				:list    , snake_case				:Vertex		)				->     Iterator[tuple]:
 for u in graph:
  __UpperCamelCase      =	math.inf
  __UpperCamelCase      =	None
 __UpperCamelCase      =	0

 __UpperCamelCase      =	list(snake_case		)
 hq.heapify(snake_case		)

 while h:
  __UpperCamelCase      =	hq.heappop(snake_case		)
  for v in u.neighbors:
   if (v in h) and (u.edges[v.id] < v.key):
    __UpperCamelCase      =	u
    __UpperCamelCase      =	u.edges[v.id]
    hq.heapify(snake_case		)

 for i in range(1    , len(snake_case		)		):
  yield (int(graph[i].id		) + 1, int(graph[i].pi.id		) + 1)






def 							A						(    )				->     None:
 pass


if __name__ == "__main__":
      import doctest

      doctest.testmod()



