

"""simple docstring"""
import heapq







def       UpperCamelCase_					(     lowerCamelCase							:					Tuple     , lowerCamelCase							:					int     , lowerCamelCase							:					Optional[Any]   )						->     int:



       """simple docstring"""







       __magic_name__			:		Optional[Any]        =     [(0, start)]  # cost from start node,end node
       __magic_name__			:		Union[str, Any]        =     set()
       while heap:
              ((__magic_name__)       ,							(__magic_name__))			:		Tuple        =     heapq.heappop(lowerCamelCase   )
              if u in visited:
                     continue
              visited.add(lowerCamelCase   )
              if u == end:
                     return cost
              for v, c in graph[u]:
                     if v in visited:
                            continue
                     __magic_name__			:		Union[str, Any]        =     cost + c
                     heapq.heappush(lowerCamelCase     , (next_item, v)   )
       return -1


A									=						{
    """A""": [["""B""", 2], ["""C""", 5]],
    """B""": [["""A""", 2], ["""D""", 3], ["""E""", 1], ["""F""", 1]],
    """C""": [["""A""", 5], ["""F""", 3]],
    """D""": [["""B""", 3]],
    """E""": [["""B""", 4], ["""F""", 3]],
    """F""": [["""C""", 3], ["""E""", 3]],
}
A									=						{
    """B""": [["""C""", 1]],
    """C""": [["""D""", 1]],
    """D""": [["""F""", 1]],
    """E""": [["""B""", 1], ["""F""", 3]],
    """F""": [],
}
A									=						{
    """B""": [["""C""", 1]],
    """C""": [["""D""", 1]],
    """D""": [["""F""", 1]],
    """E""": [["""B""", 1], ["""G""", 2]],
    """F""": [],
    """G""": [["""F""", 1]],
}

A									=						dijkstra(G, """E""", """C""")
print(short_distance)  # E -- 3 --> F -- 3 --> C == 6

A									=						dijkstra(Ga, """E""", """F""")
print(short_distance)  # E -- 3 --> F == 3

A									=						dijkstra(Ga, """E""", """F""")
print(short_distance)  # E -- 2 --> G -- 1 --> F == 3

if __name__ == "__main__":
 import doctest

 doctest.testmod()



"""simple docstring"""
import math







def       UpperCamelCase_					(     lowerCamelCase							:					int   )						->     list:



       """simple docstring"""







       if n <= 0:
              raise ValueError('''Only positive integers have prime factors'''   )
       __magic_name__			:		Union[str, Any]        =     []
       while n % 2 == 0:
              pf.append(2   )
              __magic_name__			:		Union[str, Any]        =     int(n / 2   )
       for i in range(3     , int(math.sqrt(lowerCamelCase   )   ) + 1     , 2   ):
              while n % i == 0:
                     pf.append(lowerCamelCase   )
                     __magic_name__			:		Any        =     int(n / i   )
       if n > 2:
              pf.append(lowerCamelCase   )
       return pf







def       UpperCamelCase_					(     lowerCamelCase							:					int   )						->     int:



       """simple docstring"""







       if n <= 0:
              raise ValueError('''Only positive numbers are accepted'''   )
       __magic_name__			:		List[Any]        =     1
       __magic_name__			:		Dict        =     1
       while n % 2 == 0:
              temp += 1
              __magic_name__			:		Dict        =     int(n / 2   )
       div *= temp
       for i in range(3     , int(math.sqrt(lowerCamelCase   )   ) + 1     , 2   ):
              __magic_name__			:		Dict        =     1
              while n % i == 0:
                     temp += 1
                     __magic_name__			:		Union[str, Any]        =     int(n / i   )
              div *= temp
       if n > 1:
              div *= 2
       return div







def       UpperCamelCase_					(     lowerCamelCase							:					int   )						->     int:



       """simple docstring"""







       if n <= 0:
              raise ValueError('''Only positive numbers are accepted'''   )
       __magic_name__			:		int        =     1
       __magic_name__			:		Dict        =     1
       while n % 2 == 0:
              temp += 1
              __magic_name__			:		List[Any]        =     int(n / 2   )
       if temp > 1:
              s *= (2**temp - 1) / (2 - 1)
       for i in range(3     , int(math.sqrt(lowerCamelCase   )   ) + 1     , 2   ):
              __magic_name__			:		Any        =     1
              while n % i == 0:
                     temp += 1
                     __magic_name__			:		Any        =     int(n / i   )
              if temp > 1:
                     s *= (i**temp - 1) / (i - 1)
       return int(lowerCamelCase   )







def       UpperCamelCase_					(     lowerCamelCase							:					int   )						->     int:



       """simple docstring"""







       __magic_name__			:		str        =     n
       for x in set(prime_factors(lowerCamelCase   )   ):
              s *= (x - 1) / x
       return int(lowerCamelCase   )


if __name__ == "__main__":
 import doctest

 doctest.testmod()
