


# 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    lowerCAmelCase__				:






						"""simple docstring"""




						def __init__(   self				,							a_				,							a_     ):
							lowerCamelCase_			:							str      =       key
							lowerCamelCase_			:							str      =       freq




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







def     __magic_name__  (      lowerCAmelCase_      ,					lowerCAmelCase_      ,					lowerCAmelCase_      ,					lowerCAmelCase_      ,					lowerCAmelCase_      ,					lowerCAmelCase_):
	'''simple docstring'''


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

	lowerCamelCase_			:							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(lowerCAmelCase_      ,					lowerCAmelCase_      ,					lowerCAmelCase_      ,					node - 1      ,					key[node]      ,					lowerCAmelCase_)
	print_binary_search_tree(lowerCAmelCase_      ,					lowerCAmelCase_      ,					node + 1      ,					lowerCAmelCase_      ,					key[node]      ,					lowerCAmelCase_)







def     __magic_name__  (      lowerCAmelCase_):
	'''simple docstring'''


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

	lowerCamelCase_			:							Tuple      =       len(lowerCAmelCase_)

	lowerCamelCase_			:							int      =       [nodes[i].key for i in range(lowerCAmelCase_)]
	lowerCamelCase_			:							List[Any]      =       [nodes[i].freq for i in range(lowerCAmelCase_)]

	# 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.
	lowerCamelCase_			:							List[str]      =       [[freqs[i] if i == j else 0 for j in range(lowerCAmelCase_)] for i in range(lowerCAmelCase_)]
	# sum[i][j] stores the sum of key frequencies between i and j inclusive in nodes
	# array
	lowerCamelCase_			:							List[Any]      =       [[freqs[i] if i == j else 0 for j in range(lowerCAmelCase_)] for i in range(lowerCAmelCase_)]
	# stores tree roots that will be used later for constructing binary search tree
	lowerCamelCase_			:							Optional[Any]      =       [[i if i == j else 0 for j in range(lowerCAmelCase_)] for i in range(lowerCAmelCase_)]

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

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

				if dp[i][j] > cost:
					lowerCamelCase_			:							str      =       cost
					lowerCamelCase_			:							Optional[Any]      =       r

	print("Binary search tree nodes:")
	for node in nodes:
		print(lowerCAmelCase_)

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







def     __magic_name__  (      ):
	'''simple docstring'''


	lowerCamelCase_			:							Dict      =       [Node(lowerCAmelCase_      ,					randint(1      ,					50)) for i in range(10      ,					0      ,					-1)]
	find_optimal_binary_search_tree(lowerCAmelCase_)


if __name__ == "__main__":
							main()





from typing import TYPE_CHECKING

from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_torch_available, is_vision_available


__magic_name__       					=      {'''configuration_detr''': ['''DETR_PRETRAINED_CONFIG_ARCHIVE_MAP''', '''DetrConfig''', '''DetrOnnxConfig''']}

try:
							if not is_vision_available():
														raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
							pass
else:
							__magic_name__       					=      ['''DetrFeatureExtractor''']
							__magic_name__       					=      ['''DetrImageProcessor''']

try:
							if not is_torch_available():
														raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
							pass
else:
							__magic_name__       					=      [
							    '''DETR_PRETRAINED_MODEL_ARCHIVE_LIST''',
							    '''DetrForObjectDetection''',
							    '''DetrForSegmentation''',
							    '''DetrModel''',
							    '''DetrPreTrainedModel''',
							]


if TYPE_CHECKING:
							from .configuration_detr import DETR_PRETRAINED_CONFIG_ARCHIVE_MAP, DetrConfig, DetrOnnxConfig

							try:
														if not is_vision_available():
																					raise OptionalDependencyNotAvailable()
							except OptionalDependencyNotAvailable:
														pass
							else:
														from .feature_extraction_detr import DetrFeatureExtractor
														from .image_processing_detr import DetrImageProcessor

							try:
														if not is_torch_available():
																					raise OptionalDependencyNotAvailable()
							except OptionalDependencyNotAvailable:
														pass
							else:
														from .modeling_detr import (
														    DETR_PRETRAINED_MODEL_ARCHIVE_LIST,
														    DetrForObjectDetection,
														    DetrForSegmentation,
														    DetrModel,
														    DetrPreTrainedModel,
														)

else:
							import sys

							__magic_name__       					=      _LazyModule(__name__, globals()['''__file__'''], _import_structure, module_spec=__spec__)

