



'''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 _A	:





		def __init__(  self			,							__UpperCAmelCase			,							__UpperCAmelCase					)							->  Dict:

					'''simple docstring'''
					__UpperCAmelCase :   Optional[int]    					=  key
					__UpperCAmelCase :   Dict    					=  freq





		def __str__(  self					)							->  Dict:

					'''simple docstring'''
					return f'Node(key={self.key}, freq={self.freq})'


def 	lowercase_						( lowerCAmelCase__						:	List[Any]      ,				lowerCAmelCase__						:	str      ,				lowerCAmelCase__						:	Tuple      ,				lowerCAmelCase__						:	List[str]      ,				lowerCAmelCase__						:	Tuple      ,				lowerCAmelCase__						:	Union[str, Any]			):

			"""simple docstring"""





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

			__UpperCAmelCase :   Optional[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(a_      ,				a_      ,				a_      ,				node - 1      ,				key[node]      ,				a_			)
			print_binary_search_tree(a_      ,				a_      ,				node + 1      ,				a_      ,				key[node]      ,				a_			)


def 	lowercase_						( lowerCAmelCase__						:	Union[str, Any]			):

			"""simple docstring"""





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

			__UpperCAmelCase :   Optional[Any]    					=  len(a_			)

			__UpperCAmelCase :   Dict    					=  [nodes[i].key for i in range(a_			)]
			__UpperCAmelCase :   str    					=  [nodes[i].freq for i in range(a_			)]

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

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

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

												if dp[i][j] > cost:
															__UpperCAmelCase :   Union[str, Any]    					=  cost
															__UpperCAmelCase :   int    					=  r

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

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


def 	lowercase_						( ):

			"""simple docstring"""





			__UpperCAmelCase :   Optional[int]    					=  [Node(a_      ,				randint(1      ,				50			)			) for i in range(10      ,				0      ,				-1			)]
			find_optimal_binary_search_tree(a_			)


if __name__ == "__main__":
					main()












import os

from packaging import version

from .. import __version__
from .accelerate_utils import apply_forward_hook
from .constants import (
    CONFIG_NAME,
    DEPRECATED_REVISION_ARGS,
    DIFFUSERS_CACHE,
    DIFFUSERS_DYNAMIC_MODULE_NAME,
    FLAX_WEIGHTS_NAME,
    HF_MODULES_CACHE,
    HUGGINGFACE_CO_RESOLVE_ENDPOINT,
    ONNX_EXTERNAL_WEIGHTS_NAME,
    ONNX_WEIGHTS_NAME,
    SAFETENSORS_WEIGHTS_NAME,
    TEXT_ENCODER_ATTN_MODULE,
    WEIGHTS_NAME,
)
from .deprecation_utils import deprecate
from .doc_utils import replace_example_docstring
from .dynamic_modules_utils import get_class_from_dynamic_module
from .hub_utils import (
    HF_HUB_OFFLINE,
    _add_variant,
    _get_model_file,
    extract_commit_hash,
    http_user_agent,
)
from .import_utils import (
    BACKENDS_MAPPING,
    ENV_VARS_TRUE_AND_AUTO_VALUES,
    ENV_VARS_TRUE_VALUES,
    USE_JAX,
    USE_TF,
    USE_TORCH,
    DummyObject,
    OptionalDependencyNotAvailable,
    is_accelerate_available,
    is_accelerate_version,
    is_bsa_available,
    is_flax_available,
    is_ftfy_available,
    is_inflect_available,
    is_invisible_watermark_available,
    is_k_diffusion_available,
    is_k_diffusion_version,
    is_librosa_available,
    is_note_seq_available,
    is_omegaconf_available,
    is_onnx_available,
    is_safetensors_available,
    is_scipy_available,
    is_tensorboard_available,
    is_tf_available,
    is_torch_available,
    is_torch_version,
    is_torchsde_available,
    is_transformers_available,
    is_transformers_version,
    is_unidecode_available,
    is_wandb_available,
    is_xformers_available,
    requires_backends,
)
from .logging import get_logger
from .outputs import BaseOutput
from .pil_utils import PIL_INTERPOLATION, numpy_to_pil, pt_to_pil
from .torch_utils import is_compiled_module, randn_tensor


if is_torch_available():
							from .testing_utils import (
							    floats_tensor,
							    load_hf_numpy,
							    load_image,
							    load_numpy,
							    load_pt,
							    nightly,
							    parse_flag_from_env,
							    print_tensor_test,
							    require_torch_a,
							    require_torch_gpu,
							    skip_mps,
							    slow,
							    torch_all_close,
							    torch_device,
							)
							from .torch_utils import maybe_allow_in_graph

from .testing_utils import export_to_gif, export_to_video


A_			:List[str]	    =		get_logger(__name__)






def 				A	(			a_   )	->		Tuple:
					if version.parse(a_   ) < version.parse(a_   ):
										if "dev" in min_version:
															__UpperCamelCase :   List[str]									=(
															    'This example requires a source install from HuggingFace diffusers (see '
															    '`https://huggingface.co/docs/diffusers/installation#install-from-source`),'
															)
										else:
															__UpperCamelCase :   Tuple									=F'This example requires a minimum version of {min_version},'
										error_message += F' but the version found is {__version__}.\n'
										raise ImportError(a_   )

