






def 					_lowerCamelCase(			lowerCAmelCase__				:		Optional[int]    ,    lowerCAmelCase__				:		Optional[Any]    ,    lowerCAmelCase__				:		int      ):





				'''simple docstring'''







				SCREAMING_SNAKE_CASE_   :	Union[str, Any]													=   len(_UpperCAmelCase      )
				SCREAMING_SNAKE_CASE_   :	Optional[int]													=   [[0] * n for i in range(_UpperCAmelCase      )]
				for i in range(_UpperCAmelCase      ):
								SCREAMING_SNAKE_CASE_   :	str													=   y_points[i]

				for i in range(2    ,    _UpperCAmelCase      ):
								for j in range(_UpperCAmelCase    ,    _UpperCAmelCase      ):
												SCREAMING_SNAKE_CASE_   :	str													=   (
												    (xa - x_points[j - i + 1]) * q[j][i - 1]
												    - (xa - x_points[j]) * q[j - 1][i - 1]
												) / (x_points[j] - x_points[j - i + 1])

				return [q[n - 1][n - 1], q]


if __name__ == "__main__":
				import doctest

				doctest.testmod()







from __future__ import annotations

from collections import Counter
from random import random


class 							__a							:





		'''simple docstring'''



		def __init__(						self       ):
						SCREAMING_SNAKE_CASE_   :	List[str]													=   {}


		def   __snake_case			(						self ,    UpperCamelCase__       ):
						SCREAMING_SNAKE_CASE_   :	int													=   {}


		def   __snake_case			(						self ,    UpperCamelCase__ ,    UpperCamelCase__ ,    UpperCamelCase__       ):
						if nodea not in self.connections:
										self.add_node(UpperCamelCase__       )
						if nodea not in self.connections:
										self.add_node(UpperCamelCase__       )
						SCREAMING_SNAKE_CASE_   :	Tuple													=   probability


		def   __snake_case			(						self       ):
						return list(self.connections       )






		def   __snake_case			(						self ,    UpperCamelCase__       ):
						SCREAMING_SNAKE_CASE_   :	Dict													=   0
						SCREAMING_SNAKE_CASE_   :	List[Any]													=   random()

						for dest in self.connections[node]:
										current_probability += self.connections[node][dest]
										if current_probability > random_value:
														return dest
						return ""






def 					_lowerCamelCase(			lowerCAmelCase__				:		str    ,    lowerCAmelCase__				:		list[tuple[str, str, float]]    ,    lowerCAmelCase__				:		int      ):





				'''simple docstring'''







				SCREAMING_SNAKE_CASE_   :	Union[str, Any]													=   MarkovChainGraphUndirectedUnweighted()

				for nodea, nodea, probability in transitions:
								graph.add_transition_probability(lowerCAmelCase__    ,    lowerCAmelCase__    ,    lowerCAmelCase__      )

				SCREAMING_SNAKE_CASE_   :	List[str]													=   Counter(graph.get_nodes()      )
				SCREAMING_SNAKE_CASE_   :	Optional[Any]													=   start

				for _ in range(lowerCAmelCase__      ):
								SCREAMING_SNAKE_CASE_   :	Dict													=   graph.transition(lowerCAmelCase__      )
								visited[node] += 1

				return visited


if __name__ == "__main__":
				import doctest

				doctest.testmod()