
"""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			,  __A			,  __A			)     ->	int:
						lowerCAmelCase_       :Tuple     	=    key
						lowerCAmelCase_       :Tuple     	=    freq






			def __str__(   self			)     ->	Tuple:
						return f"""Node(key={self.key}, freq={self.freq})"""






def 		_snake_case			(      lowercase__						:    str ,	lowercase__						:    List[str] ,	lowercase__						:    Any ,	lowercase__						:    Optional[Any] ,	lowercase__						:    Optional[int] ,	lowercase__						:    int						)    ->							int:


			'''simple docstring'''





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

			lowerCAmelCase_       :str     	=    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(lowercase__ ,	lowercase__ ,	lowercase__ ,	node - 1 ,	key[node] ,	lowercase__						)
			print_binary_search_tree(lowercase__ ,	lowercase__ ,	node + 1 ,	lowercase__ ,	key[node] ,	lowercase__						)






def 		_snake_case			(      lowercase__						:    Optional[int]						)    ->							str:


			'''simple docstring'''





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

			lowerCAmelCase_       :str     	=    len(lowercase__						)

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

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

			for interval_length in range(2 ,	n + 1						):
						for i in range(n - interval_length + 1						):
									lowerCAmelCase_       :Any     	=    i + interval_length - 1

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

												if dp[i][j] > cost:
															lowerCAmelCase_       :List[str]     	=    cost
															lowerCAmelCase_       :List[Any]     	=    r

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

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






def 		_snake_case			(      )    ->							List[Any]:


			'''simple docstring'''





			lowerCAmelCase_       :List[Any]     	=    [Node(lowercase__ ,	randint(1 ,	5_0						)						) for i in range(1_0 ,	0 ,	-1						)]
			find_optimal_binary_search_tree(lowercase__						)


if __name__ == "__main__":
				main()








"""simple docstring"""






from typing import List, Optional, Tuple, Union

from . import logging
from .import_utils import is_torch_available, is_torch_version


if is_torch_available():
				import torch

__UpperCAmelCase 					=    logging.get_logger(__name__)  # pylint: disable=invalid-name

try:
				from torch._dynamo import allow_in_graph as maybe_allow_in_graph
except (ImportError, ModuleNotFoundError):






				def 		_snake_case			(      cls						:    Tuple						)    ->							List[Any]:


							'''simple docstring'''





							return cls






def 		_snake_case			(      lowercase__						:    Union[Tuple, List] ,	lowercase__						:    Optional[Union[List["torch.Generator"], "torch.Generator"]] = None ,	lowercase__						:    Optional["torch.device"] = None ,	lowercase__						:    Optional["torch.dtype"] = None ,	lowercase__						:    Optional["torch.layout"] = None ,	)    ->							Optional[Any]:


			'''simple docstring'''





			lowerCAmelCase_       :Any     	=    device
			lowerCAmelCase_       :Any     	=    shape[0]

			lowerCAmelCase_       :Dict     	=    layout or torch.strided
			lowerCAmelCase_       :List[str]     	=    device or torch.device("""cpu"""						)

			if generator is not None:
						lowerCAmelCase_       :Optional[Any]     	=    generator.device.type if not isinstance(lowercase__ ,	lowercase__						) else generator[0].device.type
						if gen_device_type != device.type and gen_device_type == "cpu":
									lowerCAmelCase_       :Optional[int]     	=    """cpu"""
									if device != "mps":
												logger.info(
												    f"""The passed generator was created on 'cpu' even though a tensor on {device} was expected."""
												    f""" Tensors will be created on 'cpu' and then moved to {device}. Note that one can probably"""
												    f""" slighly speed up this function by passing a generator that was created on the {device} device."""						)
						elif gen_device_type != device.type and gen_device_type == "cuda":
									raise ValueError(f"""Cannot generate a {device} tensor from a generator of type {gen_device_type}."""						)

			if isinstance(lowercase__ ,	lowercase__						):
						lowerCAmelCase_       :List[str]     	=    (1,) + shape[1:]
						lowerCAmelCase_       :Optional[Any]     	=    [
						    torch.randn(lowercase__ ,	generator=generator[i] ,	device=lowercase__ ,	dtype=lowercase__ ,	layout=lowercase__						)
						    for i in range(lowercase__						)
						]
						lowerCAmelCase_       :str     	=    torch.cat(lowercase__ ,	dim=0						).to(lowercase__						)
			else:
						lowerCAmelCase_       :Dict     	=    torch.randn(lowercase__ ,	generator=lowercase__ ,	device=lowercase__ ,	dtype=lowercase__ ,	layout=lowercase__						).to(lowercase__						)

			return latents






def 		_snake_case			(      lowercase__						:    Union[str, Any]						)    ->							int:


			'''simple docstring'''





			if is_torch_version("""<""" ,	"""2.0.0"""						) or not hasattr(lowercase__ ,	"""_dynamo"""						):
						return False
			return isinstance(lowercase__ ,	torch._dynamo.eval_frame.OptimizedModule						)






