



"""simple docstring"""





def 	__lowerCAmelCase						(		__UpperCamelCase      :			Union[str, Any]    ):


						'''simple docstring'''






						snake_case_   : int						=			[0] * len(__UpperCamelCase    )
						snake_case_   : List[str]						=			[]
						snake_case_   : Any						=			[1] * len(__UpperCamelCase    )

						for values in graph.values():
												for i in values:
																		indegree[i] += 1

						for i in range(len(__UpperCamelCase    )    ):
												if indegree[i] == 0:
																		queue.append(__UpperCamelCase    )

						while queue:
												snake_case_   : Optional[int]						=			queue.pop(0    )
												for x in graph[vertex]:
																		indegree[x] -= 1

																		if long_dist[vertex] + 1 > long_dist[x]:
																								snake_case_   : Union[str, Any]						=			long_dist[vertex] + 1

																		if indegree[x] == 0:
																								queue.append(__UpperCamelCase    )

						print(max(__UpperCamelCase    )    )


# Adjacency list of Graph
__lowerCAmelCase       :					str      					=					{0: [2, 3, 4], 1: [2, 7], 2: [5], 3: [5, 7], 4: [7], 5: [6], 6: [7], 7: []}






longest_distance(graph)






"""simple docstring"""





from collections.abc import Callable
from math import pi, sqrt
from random import uniform
from statistics import mean
def 	__lowerCAmelCase						(		__UpperCamelCase      :			int    ):


						'''simple docstring'''






						def is_in_circle(__UpperCamelCase      :			float					, __UpperCamelCase      :			float    ) -> bool:
												snake_case_   : Dict						=			sqrt((x**2) + (y**2)    )
												# Our circle has a radius of 1, so a distance
												# greater than 1 would land outside the circle.
												return distance_from_centre <= 1

						# The proportion of guesses that landed in the circle
						snake_case_   : Tuple						=			mean(
						    int(is_in_circle(uniform(-1.0					, 1.0    )					, uniform(-1.0					, 1.0    )    )    )
						    for _ in range(__UpperCamelCase    )    )
						# The ratio of the area for circle to square is pi/4.
						snake_case_   : Union[str, Any]						=			proportion * 4
						print(F'The estimated value of pi is {pi_estimate}'    )
						print(F'The numpy value of pi is {pi}'    )
						print(F'The total error is {abs(pi - pi_estimate    )}'    )
def 	__lowerCAmelCase						(		__UpperCamelCase      :			int					, __UpperCamelCase      :			Callable[[float], float]					, __UpperCamelCase      :			float = 0.0					, __UpperCamelCase      :			float = 1.0					, ):


						'''simple docstring'''






						return mean(
						    function_to_integrate(uniform(__UpperCamelCase					, __UpperCamelCase    )    ) for _ in range(__UpperCamelCase    )    ) * (max_value - min_value)
def 	__lowerCAmelCase						(		__UpperCamelCase      :			int					, __UpperCamelCase      :			float = 0.0					, __UpperCamelCase      :			float = 1.0    ):


						'''simple docstring'''






						def identity_function(__UpperCamelCase      :			float    ) -> float:
												return x

						snake_case_   : int						=			area_under_curve_estimator(
						    __UpperCamelCase					, __UpperCamelCase					, __UpperCamelCase					, __UpperCamelCase    )
						snake_case_   : str						=			(max_value * max_value - min_value * min_value) / 2

						print("""******************"""    )
						print(F'Estimating area under y=x where x varies from {min_value} to {max_value}'    )
						print(F'Estimated value is {estimated_value}'    )
						print(F'Expected value is {expected_value}'    )
						print(F'Total error is {abs(estimated_value - expected_value    )}'    )
						print("""******************"""    )
def 	__lowerCAmelCase						(		__UpperCamelCase      :			int    ):


						'''simple docstring'''






						def function_to_integrate(__UpperCamelCase      :			float    ) -> float:
												return sqrt(4.0 - x * x    )

						snake_case_   : List[Any]						=			area_under_curve_estimator(
						    __UpperCamelCase					, __UpperCamelCase					, 0.0					, 2.0    )

						print("""******************"""    )
						print("""Estimating pi using area_under_curve_estimator"""    )
						print(F'Estimated value is {estimated_value}'    )
						print(F'Expected value is {pi}'    )
						print(F'Total error is {abs(estimated_value - pi    )}'    )
						print("""******************"""    )


if __name__ == "__main__":
					import doctest

					doctest.testmod()

