


import sys
from random import randint



class  UpperCAmelCase_						:
       '''simple docstring'''






       def __init__(  self			,       __SCREAMING_SNAKE_CASE			,       __SCREAMING_SNAKE_CASE    ):




              """simple docstring"""




              UpperCamelCase     :						Tuple 						=		key
              UpperCamelCase     :						Tuple 						=		freq






       def __str__(  self    ):




              """simple docstring"""




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




def   a						(				SCREAMING_SNAKE_CASE_   :		str				,      SCREAMING_SNAKE_CASE_   :		List[str]				,      SCREAMING_SNAKE_CASE_   :		Any				,      SCREAMING_SNAKE_CASE_   :		Optional[Any]				,      SCREAMING_SNAKE_CASE_   :		Optional[int]				,      SCREAMING_SNAKE_CASE_   :		int							):






       """simple docstring"""





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

       UpperCamelCase     :						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   a						(				SCREAMING_SNAKE_CASE_   :		Optional[int]							):






       """simple docstring"""





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

       UpperCamelCase     :						str 						=		len(lowercase__							)

       UpperCamelCase     :						List[Any] 						=		[nodes[i].key for i in range(lowercase__							)]
       UpperCamelCase     :						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.
       UpperCamelCase     :						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
       UpperCamelCase     :						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
       UpperCamelCase     :						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							):
                     UpperCamelCase     :						Any 						=		i + interval_length - 1

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

                            if dp[i][j] > cost:
                                   UpperCamelCase     :						List[str] 						=		cost
                                   UpperCamelCase     :						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   a						(				):






       """simple docstring"""





       UpperCamelCase     :						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						)






