
'''simple docstring'''



from __future__ import annotations




class   UpperCAmelCase_			:






						'''simple docstring'''






						def __init__( self   ,       _lowercase				):

												"""simple docstring"""



												_lowerCAmelCase  						=			order

												# a_{0} ... a_{k}
												_lowerCAmelCase  						=			[1.0] + [0.0] * order
												# b_{0} ... b_{k}
												_lowerCAmelCase  						=			[1.0] + [0.0] * order

												# x[n-1] ... x[n-k]
												_lowerCAmelCase  						=			[0.0] * self.order
												# y[n-1] ... y[n-k]
												_lowerCAmelCase  						=			[0.0] * self.order




						def  _lowercase     ( self   ,       _lowercase   ,       _lowercase				):

												"""simple docstring"""



												if len(_lowercase				) < self.order:
																		_lowerCAmelCase  						=			[1.0, *a_coeffs]

												if len(_lowercase				) != self.order + 1:
																		_lowerCAmelCase  						=			(
																		    F'Expected a_coeffs to have {self.order + 1} elements '
																		    F'for {self.order}-order filter, got {len(_lowercase				)}'
																		)
																		raise ValueError(_lowercase				)

												if len(_lowercase				) != self.order + 1:
																		_lowerCAmelCase  						=			(
																		    F'Expected b_coeffs to have {self.order + 1} elements '
																		    F'for {self.order}-order filter, got {len(_lowercase				)}'
																		)
																		raise ValueError(_lowercase				)

												_lowerCAmelCase  						=			a_coeffs
												_lowerCAmelCase  						=			b_coeffs




						def  _lowercase     ( self   ,       _lowercase				):

												"""simple docstring"""



												_lowerCAmelCase  						=			0.0

												# Start at index 1 and do index 0 at the end.
												for i in range(1   ,       self.order + 1				):
																		result += (
																		    self.b_coeffs[i] * self.input_history[i - 1]
																		    - self.a_coeffs[i] * self.output_history[i - 1]
																		)

												_lowerCAmelCase  						=			(result + self.b_coeffs[0] * sample) / self.a_coeffs[0]

												_lowerCAmelCase  						=			self.input_history[:-1]
												_lowerCAmelCase  						=			self.output_history[:-1]

												_lowerCAmelCase  						=			sample
												_lowerCAmelCase  						=			result

												return result


'''simple docstring'''



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




class   UpperCAmelCase_					:






						'''simple docstring'''






						def __init__( self   ,       _lowercase				):

												"""simple docstring"""



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




						def __lt__( self   ,       _lowercase				):

												"""simple docstring"""



												return self.key < other.key




						def __repr__( self				):

												"""simple docstring"""



												return self.id




						def  _lowercase     ( self   ,       _lowercase				):

												"""simple docstring"""



												self.neighbors.append(_lowercase				)




						def  _lowercase     ( self   ,       _lowercase   ,       _lowercase				):

												"""simple docstring"""



												_lowerCAmelCase  						=			weight






def A				(__lowerCamelCase				:List[Any]			,     __lowerCamelCase				:Union[str, Any]			,     __lowerCamelCase				:Dict			,     __lowerCamelCase				:Optional[int]		):
						# 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]			,     __lowerCamelCase		)
						graph[b - 1].add_edge(graph[a - 1]			,     __lowerCamelCase		)


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


def A				(__lowerCamelCase				:list			,     __lowerCamelCase				:Vertex		):
						for u in graph:
												_lowerCAmelCase  						=			math.inf
												_lowerCAmelCase  						=			None
						_lowerCAmelCase  						=			0

						_lowerCAmelCase  						=			list(__lowerCamelCase		)
						hq.heapify(__lowerCamelCase		)

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

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


def A				():
						pass


if __name__ == "__main__":
						import doctest

						doctest.testmod()
