




"""simple docstring"""

from __future__ import annotations

from math import pi, pow

def 		lowercase_   (		_UpperCAmelCase						):




	"""simple docstring"""






	if side_length < 0:
		raise ValueError('''vol_cube() only accepts non-negative values'''						)
	return pow(_UpperCAmelCase	,   3						)

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if height < 0 or radius < 0:
		raise ValueError('''vol_spherical_cap() only accepts non-negative values'''						)
	# Volume is 1/3 pi * height squared * (3 * radius - height)
	return 1 / 3 * pi * pow(_UpperCAmelCase	,   2						) * (3 * radius - height)

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if radius_a < 0 or radius_a < 0 or centers_distance < 0:
		raise ValueError('''vol_spheres_intersect() only accepts non-negative values'''						)
	if centers_distance == 0:
		return vol_sphere(min(_UpperCAmelCase	,   _UpperCAmelCase						)						)

	A_	:					Any						 =		(
	    (radius_a - radius_a + centers_distance)
	    * (radius_a + radius_a - centers_distance)
	    / (2 * centers_distance)
	)
	A_	:					int						 =		(
	    (radius_a - radius_a + centers_distance)
	    * (radius_a + radius_a - centers_distance)
	    / (2 * centers_distance)
	)

	return vol_spherical_cap(_UpperCAmelCase	,   _UpperCAmelCase						) + vol_spherical_cap(_UpperCAmelCase	,   _UpperCAmelCase						)

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if radius_a <= 0 or radius_a <= 0 or centers_distance < 0:
		raise ValueError(
		    '''vol_spheres_union() only accepts non-negative values, non-zero radius'''						)

	if centers_distance == 0:
		return vol_sphere(max(_UpperCAmelCase	,   _UpperCAmelCase						)						)

	return (
	    vol_sphere(_UpperCAmelCase						)
	    + vol_sphere(_UpperCAmelCase						)
	    - vol_spheres_intersect(_UpperCAmelCase	,   _UpperCAmelCase	,   _UpperCAmelCase						)
	)

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if width < 0 or height < 0 or length < 0:
		raise ValueError('''vol_cuboid() only accepts non-negative values'''						)
	return float(width * height * length						)

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if height < 0 or area_of_base < 0:
		raise ValueError('''vol_cone() only accepts non-negative values'''						)
	return area_of_base * height / 3.0

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if height < 0 or radius < 0:
		raise ValueError('''vol_right_circ_cone() only accepts non-negative values'''						)
	return pi * pow(_UpperCAmelCase	,   2						) * height / 3.0

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if height < 0 or area_of_base < 0:
		raise ValueError('''vol_prism() only accepts non-negative values'''						)
	return float(area_of_base * height						)

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if height < 0 or area_of_base < 0:
		raise ValueError('''vol_pyramid() only accepts non-negative values'''						)
	return area_of_base * height / 3.0

def 		lowercase_   (		_UpperCAmelCase						):




	"""simple docstring"""






	if radius < 0:
		raise ValueError('''vol_sphere() only accepts non-negative values'''						)
	# Volume is 4/3 * pi * radius cubed
	return 4 / 3 * pi * pow(_UpperCAmelCase	,   3						)

def 		lowercase_   (		_UpperCAmelCase						):




	"""simple docstring"""






	if radius < 0:
		raise ValueError('''vol_hemisphere() only accepts non-negative values'''						)
	# Volume is radius cubed * pi * 2/3
	return pow(_UpperCAmelCase	,   3						) * pi * 2 / 3

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if height < 0 or radius < 0:
		raise ValueError('''vol_circular_cylinder() only accepts non-negative values'''						)
	# Volume is radius squared * height * pi
	return pow(_UpperCAmelCase	,   2						) * height * pi

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if inner_radius < 0 or outer_radius < 0 or height < 0:
		raise ValueError(
		    '''vol_hollow_circular_cylinder() only accepts non-negative values'''						)
	if outer_radius <= inner_radius:
		raise ValueError('''outer_radius must be greater than inner_radius'''						)
	return pi * (pow(_UpperCAmelCase	,   2						) - pow(_UpperCAmelCase	,   2						)) * height

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if radius_a < 0 or radius_a < 0 or height < 0:
		raise ValueError('''vol_conical_frustum() only accepts non-negative values'''						)
	return (
	    1
	    / 3
	    * pi
	    * height
	    * (pow(_UpperCAmelCase	,   2						) + pow(_UpperCAmelCase	,   2						) + radius_a * radius_a)
	)

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if torus_radius < 0 or tube_radius < 0:
		raise ValueError('''vol_torus() only accepts non-negative values'''						)
	return 2 * pow(_UpperCAmelCase	,   2						) * torus_radius * pow(_UpperCAmelCase	,   2						)

def 		lowercase_   (		):




	"""simple docstring"""






	print('''Volumes:'''						)
	print(f"""Cube: {vol_cube(2						) = }"""						)  # = 8
	print(f"""Cuboid: {vol_cuboid(2	,   2	,   2						) = }"""						)  # = 8
	print(f"""Cone: {vol_cone(2	,   2						) = }"""						)  # ~= 1.33
	print(f"""Right Circular Cone: {vol_right_circ_cone(2	,   2						) = }"""						)  # ~= 8.38
	print(f"""Prism: {vol_prism(2	,   2						) = }"""						)  # = 4
	print(f"""Pyramid: {vol_pyramid(2	,   2						) = }"""						)  # ~= 1.33
	print(f"""Sphere: {vol_sphere(2						) = }"""						)  # ~= 33.5
	print(f"""Hemisphere: {vol_hemisphere(2						) = }"""						)  # ~= 16.75
	print(f"""Circular Cylinder: {vol_circular_cylinder(2	,   2						) = }"""						)  # ~= 25.1
	print(f"""Torus: {vol_torus(2	,   2						) = }"""						)  # ~= 157.9
	print(f"""Conical Frustum: {vol_conical_frustum(2	,   2	,   4						) = }"""						)  # ~= 58.6
	print(f"""Spherical cap: {vol_spherical_cap(1	,   2						) = }"""						)  # ~= 5.24
	print(f"""Spheres intersetion: {vol_spheres_intersect(2	,   2	,   1						) = }"""						)  # ~= 21.21
	print(f"""Spheres union: {vol_spheres_union(2	,   2	,   1						) = }"""						)  # ~= 45.81
	print(
	    f"""Hollow Circular Cylinder: {vol_hollow_circular_cylinder(1	,   2	,   3						) = }"""						)  # ~= 28.3


if __name__ == "__main__":
		main()











"""simple docstring"""

# This Python program implements an optimal binary search tree (abbreviated BST)
# building dynamic programming algorithm that delivers O(n^2) performance.
#
# The goal of the optimal BST problem is to build a low-cost BST for a
# given set of nodes, each with its own key and frequency. The frequency
# of the node is defined as how many time the node is being searched.
# The search cost of binary search tree is given by this formula:
#
# cost(1, n) = sum{i = 1 to n}((depth(node_i) + 1) * node_i_freq)
#
# where n is number of nodes in the BST. The characteristic of low-cost
# BSTs is having a faster overall search time than other implementations.
# The reason for their fast search time is that the nodes with high
# frequencies will be placed near the root of the tree while the nodes
# with low frequencies will be placed near the leaves of the tree thus
# reducing search time in the most frequent instances.
import sys
from random import randint

class 							lowercase   :







				def __init__(				self :       Optional[int]   ,  _lowerCamelCase :       str   ,  _lowerCamelCase :       Optional[Any]							):



					"""simple docstring"""

					A_	:					Union[str, Any]						 =		key
					A_	:					str						 =		freq







				def __str__(				self :       Optional[Any]							):



					"""simple docstring"""

					return F"""Node(key={self.key}, freq={self.freq})"""

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase	,   _UpperCAmelCase	,   _UpperCAmelCase	,   _UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	if i > j or i < 0 or j > len(_UpperCAmelCase						) - 1:
		return

	A_	:					Union[str, Any]						 =		root[i][j]
	if parent == -1:  # root does not have a parent
		print(f"""{key[node]} is the root of the binary search tree."""						)
	elif is_left:
		print(f"""{key[node]} is the left child of key {parent}."""						)
	else:
		print(f"""{key[node]} is the right child of key {parent}."""						)

	print_binary_search_tree(_UpperCAmelCase	,   _UpperCAmelCase	,   _UpperCAmelCase	,   node - 1	,   key[node]	,   _UpperCAmelCase						)
	print_binary_search_tree(_UpperCAmelCase	,   _UpperCAmelCase	,   node + 1	,   _UpperCAmelCase	,   key[node]	,   _UpperCAmelCase						)

def 		lowercase_   (		_UpperCAmelCase						):




	"""simple docstring"""






	nodes.sort(key=lambda _UpperCAmelCase						: node.key						)

	A_	:					Any						 =		len(_UpperCAmelCase						)

	A_	:					int						 =		[nodes[i].key for i in range(_UpperCAmelCase						)]
	A_	:					List[str]						 =		[nodes[i].freq for i in range(_UpperCAmelCase						)]

	# This 2D array stores the overall tree cost (which's as minimized as possible);
	# for a single key, cost is equal to frequency of the key.
	A_	:					List[str]						 =		[[freqs[i] if i == j else 0 for j in range(_UpperCAmelCase						)] for i in range(_UpperCAmelCase						)]
	# sum[i][j] stores the sum of key frequencies between i and j inclusive in nodes
	# array
	A_	:					Optional[int]						 =		[[freqs[i] if i == j else 0 for j in range(_UpperCAmelCase						)] for i in range(_UpperCAmelCase						)]
	# stores tree roots that will be used later for constructing binary search tree
	A_	:					List[Any]						 =		[[i if i == j else 0 for j in range(_UpperCAmelCase						)] for i in range(_UpperCAmelCase						)]

	for interval_length in range(2	,   n + 1						):
		for i in range(n - interval_length + 1						):
			A_	:					List[Any]						 =		i + interval_length - 1

			A_	:					List[Any]						 =		sys.maxsize  # set the value to "infinity"
			A_	:					Union[str, Any]						 =		total[i][j - 1] + freqs[j]

			# Apply Knuth's optimization
			# Loop without optimization: for r in range(i, j + 1):
			for r in range(root[i][j - 1]	,   root[i + 1][j] + 1						):  # r is a temporal root
				A_	:					Tuple						 =		dp[i][r - 1] if r != i else 0  # optimal cost for left subtree
				A_	:					List[str]						 =		dp[r + 1][j] if r != j else 0  # optimal cost for right subtree
				A_	:					Optional[int]						 =		left + total[i][j] + right

				if dp[i][j] > cost:
					A_	:					Dict						 =		cost
					A_	:					List[str]						 =		r

	print('''Binary search tree nodes:'''						)
	for node in nodes:
		print(_UpperCAmelCase						)

	print(f"""\nThe cost of optimal BST for given tree nodes is {dp[0][n - 1]}."""						)
	print_binary_search_tree(_UpperCAmelCase	,   _UpperCAmelCase	,   0	,   n - 1	,   -1	,   _UpperCAmelCase						)

def 		lowercase_   (		):




	"""simple docstring"""






	A_	:					List[Any]						 =		[Node(_UpperCAmelCase	,   randint(1	,   50						)						) for i in range(10	,   0	,   -1						)]
	find_optimal_binary_search_tree(_UpperCAmelCase						)


if __name__ == "__main__":
		main()





