





from collections import defaultdict

from graphs.minimum_spanning_tree_prims import prisms_algorithm as mst






def       _A  (      ):
   a__		,     a__	:						List[str]      		=  9, 14  # noqa: F841
   a__	:						Tuple      		=  [
       [0, 1, 4],
       [0, 7, 8],
       [1, 2, 8],
       [7, 8, 7],
       [7, 6, 1],
       [2, 8, 2],
       [8, 6, 6],
       [2, 3, 7],
       [2, 5, 4],
       [6, 5, 2],
       [3, 5, 14],
       [3, 4, 9],
       [5, 4, 10],
       [1, 7, 11],
   ]

   a__	:						Dict      		=  defaultdict(lowerCamelCase   )
   for nodea, nodea, cost in edges:
      adjancency[nodea].append([nodea, cost]   )
      adjancency[nodea].append([nodea, cost]   )

   a__	:						List[str]      		=  mst(lowerCamelCase   )

   a__	:						Dict      		=  [
       [7, 6, 1],
       [2, 8, 2],
       [6, 5, 2],
       [0, 1, 4],
       [2, 5, 4],
       [2, 3, 7],
       [0, 7, 8],
       [3, 4, 9],
   ]

   for answer in expected:
      a__	:						str      		=  tuple(answer[:2]   )
      a__	:						Tuple      		=  tuple(edge[::-1]   )
      assert edge in result or reverse in result








from collections.abc import Callable
from math import pi, sqrt
from random import uniform
from statistics import mean






def       _A  (      lowerCamelCase   ):

   # A local function to see if a dot lands in the circle.
   def is_in_circle(lowerCamelCase							, lowerCamelCase   ) -> bool:
      a__	:						Any      		=  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
   a__	:						Union[str, Any]      		=  mean(
       int(is_in_circle(uniform(-1.0							, 1.0   )							, uniform(-1.0							, 1.0   )   )   )
       for _ in range(lowerCamelCase   )   )
   # The ratio of the area for circle to square is pi/4.
   a__	:						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       _A  (      lowerCamelCase							, lowerCamelCase							, lowerCamelCase = 0.0							, lowerCamelCase = 1.0							, ):
   return mean(
       function_to_integrate(uniform(lowerCamelCase							, lowerCamelCase   )   ) for _ in range(lowerCamelCase   )   ) * (max_value - min_value)






def       _A  (      lowerCamelCase							, lowerCamelCase = 0.0							, lowerCamelCase = 1.0   ):

   def identity_function(lowerCamelCase   ) -> float:
      return x

   a__	:						int      		=  area_under_curve_estimator(
       lowerCamelCase							, lowerCamelCase							, lowerCamelCase							, lowerCamelCase   )
   a__	:						int      		=  (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       _A  (      lowerCamelCase   ):

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

   a__	:						int      		=  area_under_curve_estimator(
       lowerCamelCase							, lowerCamelCase							, 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()

