



'''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 			_SCREAMING_SNAKE_CASE				:



			def __init__(     self		:						Optional[int]    ,			__UpperCamelCase		:						List[Any]    ,			__UpperCamelCase		:						Dict			)      ->      Any:
							"""simple docstring"""





							snake_case__      :   str      =    key
							snake_case__      :   List[str]      =    freq







			def __str__(     self		:						Union[str, Any]			)      ->      Optional[Any]:
							"""simple docstring"""





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



def 							__UpperCAmelCase				(				UpperCamelCase__				:str   ,				UpperCamelCase__				:Tuple   ,				UpperCamelCase__				:List[Any]   ,				UpperCamelCase__				:Optional[Any]   ,				UpperCamelCase__				:Optional[Any]   ,				UpperCamelCase__				:Any			)		->						int:
				if i > j or i < 0 or j > len(UpperCamelCase__			) - 1:
								return

				snake_case__      :   Optional[int]      =    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 							__UpperCAmelCase				(				UpperCamelCase__				:Any			)		->						Tuple:
				nodes.sort(key=lambda UpperCamelCase__			: node.key			)

				snake_case__      :   List[Any]      =    len(UpperCamelCase__			)

				snake_case__      :   Optional[Any]      =    [nodes[i].key for i in range(UpperCamelCase__			)]
				snake_case__      :   Optional[int]      =    [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.
				snake_case__      :   Dict      =    [[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
				snake_case__      :   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
				snake_case__      :   Tuple      =    [[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			):
												snake_case__      :   Any      =    i + interval_length - 1

												snake_case__      :   Dict      =    sys.maxsize  # set the value to "infinity"
												snake_case__      :   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
																snake_case__      :   str      =    dp[i][r - 1] if r != i else 0  # optimal cost for left subtree
																snake_case__      :   List[Any]      =    dp[r + 1][j] if r != j else 0  # optimal cost for right subtree
																snake_case__      :   Union[str, Any]      =    left + total[i][j] + right

																if dp[i][j] > cost:
																				snake_case__      :   List[str]      =    cost
																				snake_case__      :   Tuple      =    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 							__UpperCAmelCase				(				)		->						int:
				# A sample binary search tree
				snake_case__      :   Tuple      =    [Node(UpperCamelCase__   ,				randint(1   ,				50			)			) for i in range(10   ,				0   ,				-1			)]
				find_optimal_binary_search_tree(UpperCamelCase__			)


if __name__ == "__main__":
				main()









'''simple docstring'''



from __future__ import annotations

# Define the Gravitational Constant G and the function
_lowercase       :						List[str]						=6.6743E-11  # unit of G : m^3 * kg^-1 * s^-2


def 							__UpperCAmelCase				(				UpperCamelCase__				:float   ,				UpperCamelCase__				:float   ,				UpperCamelCase__				:float   ,				UpperCamelCase__				:float			)		->						dict[str, float]:
				snake_case__      :   str      =    mass_a * mass_a

				if (force, mass_a, mass_a, distance).count(0			) != 1:
								raise ValueError('''One and only one argument must be 0'''			)
				if force < 0:
								raise ValueError('''Gravitational force can not be negative'''			)
				if distance < 0:
								raise ValueError('''Distance can not be negative'''			)
				if mass_a < 0 or mass_a < 0:
								raise ValueError('''Mass can not be negative'''			)
				if force == 0:
								snake_case__      :   Dict      =    GRAVITATIONAL_CONSTANT * product_of_mass / (distance**2)
								return {"force": force}
				elif mass_a == 0:
								snake_case__      :   List[str]      =    (force) * (distance**2) / (GRAVITATIONAL_CONSTANT * mass_a)
								return {"mass_1": mass_a}
				elif mass_a == 0:
								snake_case__      :   List[Any]      =    (force) * (distance**2) / (GRAVITATIONAL_CONSTANT * mass_a)
								return {"mass_2": mass_a}
				elif distance == 0:
								snake_case__      :   str      =    (GRAVITATIONAL_CONSTANT * product_of_mass / (force)) ** 0.5
								return {"distance": distance}
				raise ValueError('''One and only one argument must be 0'''			)


# Run doctest
if __name__ == "__main__":
				import doctest

				doctest.testmod()




