
"""simple docstring"""





from __future__ import annotations

from collections import Counter
from random import random






class        _UpperCAmelCase:



		def __init__(   self)			->			Dict:






							'''simple docstring'''
							_UpperCamelCase									=    {}



		def 		UpperCAmelCase				(   self							,						__a)			->			None:






							'''simple docstring'''
							_UpperCamelCase									=    {}



		def 		UpperCAmelCase				(   self							,						__a							,						__a							,						__a)			->			None:






							'''simple docstring'''
							if nodea not in self.connections:
												self.add_node(__a)
							if nodea not in self.connections:
												self.add_node(__a)
							_UpperCamelCase									=    probability



		def 		UpperCAmelCase				(   self)			->			list[str]:






							'''simple docstring'''
							return list(self.connections)




		def 		UpperCAmelCase				(   self							,						__a)			->			str:






							'''simple docstring'''
							_UpperCamelCase									=    0
							_UpperCamelCase									=    random()

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





def  lowerCamelCase__						(     __snake_case,	__snake_case,	__snake_case      )				->						dict[str, int]:


					"""simple docstring"""





					_UpperCamelCase									=    MarkovChainGraphUndirectedUnweighted()

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

					_UpperCamelCase									=    Counter(graph.get_nodes()      )
					_UpperCamelCase									=    start

					for _ in range(__snake_case      ):
										_UpperCamelCase									=    graph.transition(__snake_case      )
										visited[node] += 1

					return visited


if __name__ == "__main__":
				import doctest

				doctest.testmod()








"""simple docstring"""





from collections.abc import Callable

import numpy as np



def  lowerCamelCase__						(     __snake_case,	__snake_case,	__snake_case,	__snake_case,	__snake_case      )				->						np.array:


					"""simple docstring"""





					_UpperCamelCase									=    int(np.ceil((x_end - xa) / step_size      )      )
					_UpperCamelCase									=    np.zeros((n + 1,)      )
					_UpperCamelCase									=    ya
					_UpperCamelCase									=    xa

					for k in range(__snake_case      ):
										_UpperCamelCase									=    y[k] + step_size * ode_func(__snake_case,	y[k]      )
										_UpperCamelCase									=    y[k] + (
										    (step_size / 2) * (ode_func(__snake_case,	y[k]      ) + ode_func(x + step_size,	__snake_case      ))
										)
										x += step_size

					return y


if __name__ == "__main__":
				import doctest

				doctest.testmod()






