




'''simple docstring'''






from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
_lowerCAmelCase				     = 20

# Time between time steps in seconds
_lowerCAmelCase				     = INTERVAL / 1000



class  A		:




				'''simple docstring'''





				def __init__(self			,  _UpperCAmelCase			,  _UpperCAmelCase			,  _UpperCAmelCase			,  _UpperCAmelCase			,  _UpperCAmelCase = 1.0			,  _UpperCAmelCase = 1.0			,  _UpperCAmelCase = "blue"			,  )     ->	None:
							__UpperCamelCase				:      Any									=       position_x
							__UpperCamelCase				:      Tuple									=       position_y
							__UpperCamelCase				:      List[str]									=       velocity_x
							__UpperCamelCase				:      str									=       velocity_y
							__UpperCamelCase				:      Dict									=       mass
							__UpperCamelCase				:      Optional[Any]									=       size
							__UpperCamelCase				:      Any									=       color
				@property
				def 	a_	(self      )     ->	tuple[float, float]:
							return self.position_x, self.position_y
				@property
				def 	a_	(self      )     ->	tuple[float, float]:
							return self.velocity_x, self.velocity_y
				def 	a_	(self			,  _UpperCAmelCase			,  _UpperCAmelCase			,  _UpperCAmelCase      )     ->	None:
							self.velocity_x += force_x * delta_time
							self.velocity_y += force_y * delta_time





				def 	a_	(self			,  _UpperCAmelCase      )     ->	None:
							self.position_x += self.velocity_x * delta_time
							self.position_y += self.velocity_y * delta_time







class  A		:




				'''simple docstring'''





				def __init__(self			,  _UpperCAmelCase			,  _UpperCAmelCase = 1.0			,  _UpperCAmelCase = 1.0			,  _UpperCAmelCase = 0.0			,  )     ->	None:
							__UpperCamelCase				:      Dict									=       bodies
							__UpperCamelCase				:      int									=       gravitation_constant
							__UpperCamelCase				:      int									=       time_factor
							__UpperCamelCase				:      str									=       softening_factor
				def __len__(self      )     ->	int:
							return len(self.bodies      )





				def 	a_	(self			,  _UpperCAmelCase      )     ->	None:
							for bodya in self.bodies:
										__UpperCamelCase				:      Any									=       0.0
										__UpperCamelCase				:      Optional[Any]									=       0.0
										for bodya in self.bodies:
													if bodya != bodya:
																__UpperCamelCase				:      Optional[int]									=       bodya.position_x - bodya.position_x
																__UpperCamelCase				:      Any									=       bodya.position_y - bodya.position_y

																# Calculation of the distance using Pythagoras's theorem
																# Extra factor due to the softening technique
																__UpperCamelCase				:      Any									=       (dif_x**2 + dif_y**2 + self.softening_factor) ** (
																    1 / 2
																)

																# Newton's law of universal gravitation.
																force_x += (
																    self.gravitation_constant * bodya.mass * dif_x / distance**3
																)
																force_y += (
																    self.gravitation_constant * bodya.mass * dif_y / distance**3
																)

            # Update the body's velocity once all the force components have been added
										bodya.update_velocity(_UpperCAmelCase			,  _UpperCAmelCase			,  delta_time * self.time_factor      )

							# Update the positions only after all the velocities have been updated
							for body in self.bodies:
										body.update_position(delta_time * self.time_factor      )







def       __lowerCAmelCase (   snake_case__ ,				snake_case__ ,				snake_case__						):
			body_system.update_system(snake_case__						)

			# Update the positions of the patches
			for patch, body in zip(snake_case__ ,				body_system.bodies						):
						__UpperCamelCase				:      List[str]									=       (body.position_x, body.position_y)

def       __lowerCAmelCase (   snake_case__ ,				snake_case__ ,				snake_case__ = -1 ,				snake_case__ = 1 ,				snake_case__ = -1 ,				snake_case__ = 1 ,				):
			__UpperCamelCase				:      Any									=       plt.figure()
			fig.canvas.set_window_title(snake_case__						)
			__UpperCamelCase				:      Optional[Any]									=       plt.axes(
			    xlim=(x_start, x_end) ,				ylim=(y_start, y_end)						)  # Set section to be plotted
			plt.gca().set_aspect("equal"						)  # Fix aspect ratio

			# Each body is drawn as a patch by the plt-function
			__UpperCamelCase				:      Optional[int]									=       [
			    plt.Circle((body.position_x, body.position_y) ,				body.size ,				fc=body.color						)
			    for body in body_system.bodies
			]

			for patch in patches:
						ax.add_patch(snake_case__						)

			# Function called at each step of the animation
			def update(snake_case__						) -> list[plt.Circle]:
						update_step(snake_case__ ,				snake_case__ ,				snake_case__						)
						return patches

			__UpperCamelCase				:      List[Any]									=       animation.FuncAnimation(  # noqa: F841
			    snake_case__ ,				snake_case__ ,				interval=snake_case__ ,				blit=snake_case__						)

			plt.show()

def       __lowerCAmelCase (   ):
			__UpperCamelCase				:      List[Any]									=       0.9700436
			__UpperCamelCase				:      Union[str, Any]									=       -0.24308753
			__UpperCamelCase				:      List[str]									=       0.466203685
			__UpperCamelCase				:      List[Any]									=       0.43236573

			__UpperCamelCase				:      Dict									=       [
			    Body(snake_case__ ,				snake_case__ ,				snake_case__ ,				snake_case__ ,				size=0.2 ,				color="red"						),
			    Body(-position_x ,				-position_y ,				snake_case__ ,				snake_case__ ,				size=0.2 ,				color="green"						),
			    Body(0 ,				0 ,				-2 * velocity_x ,				-2 * velocity_y ,				size=0.2 ,				color="blue"						),
			]
			return BodySystem(snake_case__ ,				time_factor=3						)

def       __lowerCAmelCase (   ):
			__UpperCamelCase				:      int									=       7.34_76E22
			__UpperCamelCase				:      List[Any]									=       5.9_72E24
			__UpperCamelCase				:      Optional[Any]									=       1_022
			__UpperCamelCase				:      Tuple									=       384_399_000
			__UpperCamelCase				:      str									=       6.6_74E-11

			# Calculation of the respective velocities so that total impulse is zero,
			# i.e. the two bodies together don't move
			__UpperCamelCase				:      Union[str, Any]									=       earth_mass * velocity_dif / (earth_mass + moon_mass)
			__UpperCamelCase				:      int									=       moon_velocity - velocity_dif

			__UpperCamelCase				:      Optional[int]									=       Body(-earth_moon_distance ,				0 ,				0 ,				snake_case__ ,				snake_case__ ,				10_000_000 ,				"grey"						)
			__UpperCamelCase				:      int									=       Body(0 ,				0 ,				0 ,				snake_case__ ,				snake_case__ ,				50_000_000 ,				"blue"						)
			return BodySystem([earth, moon] ,				snake_case__ ,				time_factor=1_000_000						)

def       __lowerCAmelCase (   ):
			__UpperCamelCase				:      Any									=       []
			for _ in range(10						):
						__UpperCamelCase				:      Optional[int]									=       random.uniform(-0.5 ,				0.5						)
						__UpperCamelCase				:      List[str]									=       random.uniform(-0.5 ,				0.5						)

						# Bodies are created pairwise with opposite velocities so that the
						# total impulse remains zero
						bodies.append(
						    Body(
						        random.uniform(-0.5 ,				0.5						) ,				random.uniform(-0.5 ,				0.5						) ,				snake_case__ ,				snake_case__ ,				size=0.05 ,				)						)
						bodies.append(
						    Body(
						        random.uniform(-0.5 ,				0.5						) ,				random.uniform(-0.5 ,				0.5						) ,				-velocity_x ,				-velocity_y ,				size=0.05 ,				)						)
			return BodySystem(snake_case__ ,				0.01 ,				10 ,				0.1						)


if __name__ == "__main__":
				plot('''Figure-8 solution to the 3-body-problem''', example_a(), -2, 2, -2, 2)
				plot(
				    '''Moon\'s orbit around the earth''',
				    example_a(),
				    -430000000,
				    430000000,
				    -430000000,
				    430000000,
				)
				plot('''Random system with many bodies''', example_a(), -1.5, 1.5, -1.5, 1.5)









'''simple docstring'''






from __future__ import annotations

import random
from collections.abc import Iterable
from typing import Any, Generic, TypeVar

_lowerCAmelCase				     = TypeVar('''T''', bound=bool)



class  A		(			Generic[T]		):




				'''simple docstring'''





				def __init__(self			,  _UpperCAmelCase      )     ->	None:
							__UpperCamelCase				:      T									=       value
							__UpperCamelCase				:      RandomizedHeapNode[T] | None									=       None
							__UpperCamelCase				:      RandomizedHeapNode[T] | None									=       None
				@property
				def 	a_	(self      )     ->	T:
							return self._value





				@staticmethod
				def 	a_	(_UpperCAmelCase			,  _UpperCAmelCase      )     ->	RandomizedHeapNode[T] | None:
							if not roota:
										return roota

							if not roota:
										return roota

							if roota.value > roota.value:
										__UpperCamelCase		, __UpperCamelCase				:      int									=       roota, roota

							if random.choice([True, False]      ):
										__UpperCamelCase		, __UpperCamelCase				:      Dict									=       roota.right, roota.left

							__UpperCamelCase				:      Any									=       RandomizedHeapNode.merge(roota.left			,  _UpperCAmelCase      )

							return roota







class  A		(			Generic[T]		):




				'''simple docstring'''





				def __init__(self			,  _UpperCAmelCase = ()      )     ->	None:
							__UpperCamelCase				:      RandomizedHeapNode[T] | None									=       None

							if data:
										for item in data:
													self.insert(_UpperCAmelCase      )
				def 	a_	(self			,  _UpperCAmelCase      )     ->	None:
							__UpperCamelCase				:      str									=       RandomizedHeapNode.merge(self._root			,  RandomizedHeapNode(_UpperCAmelCase      )      )
				def 	a_	(self      )     ->	T | None:
							__UpperCamelCase				:      Dict									=       self.top()

							if self._root is None:
										return None

							__UpperCamelCase				:      List[str]									=       RandomizedHeapNode.merge(self._root.left			,  self._root.right      )

							return result
				def 	a_	(self      )     ->	T:
							if not self._root:
										raise IndexError("Can't get top element for the empty heap."      )
							return self._root.value
				def 	a_	(self      )     ->	None:
							__UpperCamelCase				:      int									=       None
				def 	a_	(self      )     ->	list[Any]:
							__UpperCamelCase				:      int									=       []
							while self:
										result.append(self.pop()      )

							return result





				def __bool__(self      )     ->	bool:
							return self._root is not None


if __name__ == "__main__":
				import doctest

				doctest.testmod()



