






"""simple docstring"""



class 					_UpperCAmelCase       :




					def __init__(				self     :						Dict   )  ->  List[str]:
							lowercase_ : int							       =   0
							lowercase_ : Union[str, Any]							       =   0
							lowercase_ : Optional[int]							       =   {}




					def   A  (				self     :						Optional[Any]    ,						A     :						str   )  ->  Optional[Any]:
							if vertex not in self.adjacency:
									lowercase_ : int							       =   {}
									self.num_vertices += 1




					def   A  (				self     :						Any    ,						A     :						Tuple    ,						A     :						List[str]    ,						A     :						int   )  ->  int:
							self.add_vertex(a_   )
							self.add_vertex(a_   )

							if head == tail:
									return

							lowercase_ : List[Any]							       =   weight
							lowercase_ : Dict							       =   weight




					def   A  (				self     :						Tuple   )  ->  Dict:
							lowercase_ : Optional[int]							       =   self.get_edges()
							for edge in edges:
									lowercase_ : Dict							       =   edge
									edges.remove((tail, head, weight)   )
							for i in range(len(a_   )   ):
									lowercase_ : str							       =   list(edges[i]   )

							edges.sort(key=lambda A   : e[2]   )
							for i in range(len(a_   ) - 1   ):
									if edges[i][2] >= edges[i + 1][2]:
											lowercase_ : Optional[Any]							       =   edges[i][2] + 1
							for edge in edges:
									lowercase_ : Union[str, Any]							       =   edge
									lowercase_ : str							       =   weight
									lowercase_ : List[str]							       =   weight




					def __str__(				self     :						List[Any]   )  ->  Any:
							lowercase_ : List[Any]							       =   """"""
							for tail in self.adjacency:
									for head in self.adjacency[tail]:
											lowercase_ : List[str]							       =   self.adjacency[head][tail]
											string += F'''{head} -> {tail} == {weight}\n'''
							return string.rstrip('''\n'''   )




					def   A  (				self     :						List[Any]   )  ->  str:
							lowercase_ : int							       =   []
							for tail in self.adjacency:
									for head in self.adjacency[tail]:
											output.append((tail, head, self.adjacency[head][tail])   )
							return output




					def   A  (				self     :						Optional[Any]   )  ->  Optional[int]:
							return self.adjacency.keys()




					@staticmethod
					def   A  (				A     :						int=None    ,						A     :						List[Any]=None   )  ->  Tuple:
							lowercase_ : List[Any]							       =   Graph()
							if vertices is None:
									lowercase_ : List[str]							       =   []
							if edges is None:
									lowercase_ : Optional[Any]							       =   []
							for vertex in vertices:
									g.add_vertex(a_   )
							for edge in edges:
									g.add_edge(*a_   )
							return g





					class 					_UpperCAmelCase       :




										def __init__(				self     :						Tuple   )  ->  int:
												lowercase_ : List[str]							       =   {}
												lowercase_ : int							       =   {}




										def __len__(				self     :						Dict   )  ->  Tuple:
												return len(self.parent   )




										def   A  (				self     :						List[Any]    ,						A     :						Optional[Any]   )  ->  str:
												if item in self.parent:
														return self.find(a_   )

												lowercase_ : Optional[Any]							       =   item
												lowercase_ : List[Any]							       =   0
												return item




										def   A  (				self     :						Any    ,						A     :						Optional[Any]   )  ->  List[str]:
												if item not in self.parent:
														return self.make_set(a_   )
												if item != self.parent[item]:
														lowercase_ : List[Any]							       =   self.find(self.parent[item]   )
												return self.parent[item]






										def   A  (				self     :						List[Any]    ,						A     :						List[Any]    ,						A     :						int   )  ->  Union[str, Any]:
												lowercase_ : Any							       =   self.find(a_   )
												lowercase_ : List[str]							       =   self.find(a_   )

												if roota == roota:
														return roota

												if self.rank[roota] > self.rank[roota]:
														lowercase_ : Any							       =   roota
														return roota

												if self.rank[roota] < self.rank[roota]:
														lowercase_ : Any							       =   roota
														return roota

												if self.rank[roota] == self.rank[roota]:
														self.rank[roota] += 1
														lowercase_ : List[str]							       =   roota
														return roota
												return None




					@staticmethod
					def   A  (				A     :						List[Any]   )  ->  List[Any]:
							lowercase_ : int							       =   graph.num_vertices

							lowercase_ : int							       =   Graph.UnionFind()
							lowercase_ : Optional[int]							       =   []
							while num_components > 1:
									lowercase_ : int							       =   {}
									for vertex in graph.get_vertices():
											lowercase_ : Union[str, Any]							       =   -1

									lowercase_ : Tuple							       =   graph.get_edges()
									for edge in edges:
											lowercase_ : str							       =   edge
											edges.remove((tail, head, weight)   )
									for edge in edges:
											lowercase_ : Optional[Any]							       =   edge
											lowercase_ : Any							       =   union_find.find(a_   )
											lowercase_ : Any							       =   union_find.find(a_   )
											if seta != seta:
													if cheap_edge[seta] == -1 or cheap_edge[seta][2] > weight:
															lowercase_ : Tuple							       =   [head, tail, weight]

													if cheap_edge[seta] == -1 or cheap_edge[seta][2] > weight:
															lowercase_ : List[str]							       =   [head, tail, weight]
									for vertex in cheap_edge:
											if cheap_edge[vertex] != -1:
													lowercase_ : str							       =   cheap_edge[vertex]
													if union_find.find(a_   ) != union_find.find(a_   ):
															union_find.union(a_    ,						a_   )
															mst_edges.append(cheap_edge[vertex]   )
															lowercase_ : Tuple							       =   num_components - 1
							lowercase_ : Optional[int]							       =   Graph.build(edges=a_   )
							return mst








'''simple docstring'''






from numpy import exp, pi, sqrt




def     snake_case_					(     lowerCAmelCase_       ,   lowerCAmelCase_ = 0.0       ,   lowerCAmelCase_ = 1.0		)->		int:





       '''simple docstring'''


       return 1 / sqrt(2 * pi * sigma**2		) * exp(-((x - mu) ** 2) / (2 * sigma**2)		)


if __name__ == "__main__":
   import doctest

   doctest.testmod()



