
# 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 						lowercase       :
					def __init__(		self ,	SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    ):
								"""simple docstring"""







								lowerCAmelCase__ :    str				      =			key
								lowerCAmelCase__ :    Union[str, Any]				      =			freq
					def __str__(		self    ):
								"""simple docstring"""







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





def      _a     (   __UpperCamelCase       :  int	,__UpperCamelCase       :  Union[str, Any]	,__UpperCamelCase       :  List[Any]	,__UpperCamelCase       :  Union[str, Any]	,__UpperCamelCase       :  str	,__UpperCamelCase       :  List[Any]    ):
			if i > j or i < 0 or j > len(__UpperCamelCase    ) - 1:
						return

			lowerCAmelCase__ :    Dict				      =			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      _a     (   __UpperCamelCase       :  Optional[int]    ):
			nodes.sort(key=lambda __UpperCamelCase    : node.key    )

			lowerCAmelCase__ :    Optional[int]				      =			len(__UpperCamelCase    )

			lowerCAmelCase__ :    str				      =			[nodes[i].key for i in range(__UpperCamelCase    )]
			lowerCAmelCase__ :    Dict				      =			[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.
			lowerCAmelCase__ :    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
			lowerCAmelCase__ :    str				      =			[[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
			lowerCAmelCase__ :    Optional[int]				      =			[[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    ):
									lowerCAmelCase__ :    Union[str, Any]				      =			i + interval_length - 1

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

												if dp[i][j] > cost:
															lowerCAmelCase__ :    Optional[Any]				      =			cost
															lowerCAmelCase__ :    Any				      =			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      _a     (   ):
			# A sample binary search tree
			lowerCAmelCase__ :    Union[str, Any]				      =			[Node(__UpperCamelCase	,randint(1	,50    )    ) for i in range(10	,0	,-1    )]
			find_optimal_binary_search_tree(__UpperCamelCase    )


if __name__ == "__main__":
							main()





import torch

from diffusers import HeunDiscreteScheduler
from diffusers.utils import torch_device

from .test_schedulers import SchedulerCommonTest





class 						lowercase       (  __UpperCamelCase					):
					__a					=    (HeunDiscreteScheduler,)
					__a					=    10
					def lowercase_			(		self ,	**SCREAMING_SNAKE_CASE__    ):
								"""simple docstring"""







								lowerCAmelCase__ :    Any				      =			{
								    '''num_train_timesteps''': 1100,
								    '''beta_start''': 0.0_001,
								    '''beta_end''': 0.02,
								    '''beta_schedule''': '''linear''',
								}

								config.update(**SCREAMING_SNAKE_CASE__    )
								return config
					def lowercase_			(		self    ):
								"""simple docstring"""







								for timesteps in [10, 50, 100, 1000]:
											self.check_over_configs(num_train_timesteps=SCREAMING_SNAKE_CASE__    )
					def lowercase_			(		self    ):
								"""simple docstring"""







								for beta_start, beta_end in zip([0.00_001, 0.0_001, 0.001] ,	[0.0_002, 0.002, 0.02]    ):
											self.check_over_configs(beta_start=SCREAMING_SNAKE_CASE__ ,	beta_end=SCREAMING_SNAKE_CASE__    )
					def lowercase_			(		self    ):
								"""simple docstring"""







								for schedule in ["linear", "scaled_linear", "exp"]:
											self.check_over_configs(beta_schedule=SCREAMING_SNAKE_CASE__    )
					def lowercase_			(		self    ):
								"""simple docstring"""







								for clip_sample_range in [1.0, 2.0, 3.0]:
											self.check_over_configs(clip_sample_range=SCREAMING_SNAKE_CASE__ ,	clip_sample=SCREAMING_SNAKE_CASE__    )
					def lowercase_			(		self    ):
								"""simple docstring"""







								for prediction_type in ["epsilon", "v_prediction", "sample"]:
											self.check_over_configs(prediction_type=SCREAMING_SNAKE_CASE__    )
					def lowercase_			(		self    ):
								"""simple docstring"""







								lowerCAmelCase__ :    List[str]				      =			self.scheduler_classes[0]
								lowerCAmelCase__ :    Optional[Any]				      =			self.get_scheduler_config()
								lowerCAmelCase__ :    Optional[Any]				      =			scheduler_class(**SCREAMING_SNAKE_CASE__    )

								scheduler.set_timesteps(self.num_inference_steps    )

								lowerCAmelCase__ :    str				      =			self.dummy_model()
								lowerCAmelCase__ :    Optional[int]				      =			self.dummy_sample_deter * scheduler.init_noise_sigma
								lowerCAmelCase__ :    Optional[Any]				      =			sample.to(SCREAMING_SNAKE_CASE__    )

								for i, t in enumerate(scheduler.timesteps    ):
											lowerCAmelCase__ :    List[Any]				      =			scheduler.scale_model_input(SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    )

											lowerCAmelCase__ :    str				      =			model(SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    )

											lowerCAmelCase__ :    Tuple				      =			scheduler.step(SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    )
											lowerCAmelCase__ :    Union[str, Any]				      =			output.prev_sample

								lowerCAmelCase__ :    Any				      =			torch.sum(torch.abs(SCREAMING_SNAKE_CASE__    )    )
								lowerCAmelCase__ :    Union[str, Any]				      =			torch.mean(torch.abs(SCREAMING_SNAKE_CASE__    )    )

								if torch_device in ["cpu", "mps"]:
											assert abs(result_sum.item() - 0.1_233    ) < 1E-2
											assert abs(result_mean.item() - 0.0_002    ) < 1E-3
								else:
											# CUDA
											assert abs(result_sum.item() - 0.1_233    ) < 1E-2
											assert abs(result_mean.item() - 0.0_002    ) < 1E-3
					def lowercase_			(		self    ):
								"""simple docstring"""







								lowerCAmelCase__ :    Optional[Any]				      =			self.scheduler_classes[0]
								lowerCAmelCase__ :    str				      =			self.get_scheduler_config(prediction_type='''v_prediction'''    )
								lowerCAmelCase__ :    Any				      =			scheduler_class(**SCREAMING_SNAKE_CASE__    )

								scheduler.set_timesteps(self.num_inference_steps    )

								lowerCAmelCase__ :    int				      =			self.dummy_model()
								lowerCAmelCase__ :    List[str]				      =			self.dummy_sample_deter * scheduler.init_noise_sigma
								lowerCAmelCase__ :    List[Any]				      =			sample.to(SCREAMING_SNAKE_CASE__    )

								for i, t in enumerate(scheduler.timesteps    ):
											lowerCAmelCase__ :    int				      =			scheduler.scale_model_input(SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    )

											lowerCAmelCase__ :    Optional[Any]				      =			model(SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    )

											lowerCAmelCase__ :    Optional[int]				      =			scheduler.step(SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    )
											lowerCAmelCase__ :    Tuple				      =			output.prev_sample

								lowerCAmelCase__ :    List[Any]				      =			torch.sum(torch.abs(SCREAMING_SNAKE_CASE__    )    )
								lowerCAmelCase__ :    List[str]				      =			torch.mean(torch.abs(SCREAMING_SNAKE_CASE__    )    )

								if torch_device in ["cpu", "mps"]:
											assert abs(result_sum.item() - 4.6934E-07    ) < 1E-2
											assert abs(result_mean.item() - 6.1112E-10    ) < 1E-3
								else:
											# CUDA
											assert abs(result_sum.item() - 4.693_4286_5017_0972E-07    ) < 1E-2
											assert abs(result_mean.item() - 0.0_002    ) < 1E-3
					def lowercase_			(		self    ):
								"""simple docstring"""







								lowerCAmelCase__ :    Optional[int]				      =			self.scheduler_classes[0]
								lowerCAmelCase__ :    Optional[Any]				      =			self.get_scheduler_config()
								lowerCAmelCase__ :    Union[str, Any]				      =			scheduler_class(**SCREAMING_SNAKE_CASE__    )

								scheduler.set_timesteps(self.num_inference_steps ,	device=SCREAMING_SNAKE_CASE__    )

								lowerCAmelCase__ :    Optional[Any]				      =			self.dummy_model()
								lowerCAmelCase__ :    Dict				      =			self.dummy_sample_deter.to(SCREAMING_SNAKE_CASE__    ) * scheduler.init_noise_sigma

								for t in scheduler.timesteps:
											lowerCAmelCase__ :    int				      =			scheduler.scale_model_input(SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    )

											lowerCAmelCase__ :    Dict				      =			model(SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    )

											lowerCAmelCase__ :    Any				      =			scheduler.step(SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    )
											lowerCAmelCase__ :    List[Any]				      =			output.prev_sample

								lowerCAmelCase__ :    Dict				      =			torch.sum(torch.abs(SCREAMING_SNAKE_CASE__    )    )
								lowerCAmelCase__ :    List[str]				      =			torch.mean(torch.abs(SCREAMING_SNAKE_CASE__    )    )

								if str(SCREAMING_SNAKE_CASE__    ).startswith('''cpu'''    ):
											# The following sum varies between 148 and 156 on mps. Why?
											assert abs(result_sum.item() - 0.1_233    ) < 1E-2
											assert abs(result_mean.item() - 0.0_002    ) < 1E-3
								elif str(SCREAMING_SNAKE_CASE__    ).startswith('''mps'''    ):
											# Larger tolerance on mps
											assert abs(result_mean.item() - 0.0_002    ) < 1E-2
								else:
											# CUDA
											assert abs(result_sum.item() - 0.1_233    ) < 1E-2
											assert abs(result_mean.item() - 0.0_002    ) < 1E-3
					def lowercase_			(		self    ):
								"""simple docstring"""







								lowerCAmelCase__ :    Any				      =			self.scheduler_classes[0]
								lowerCAmelCase__ :    Optional[Any]				      =			self.get_scheduler_config()
								lowerCAmelCase__ :    str				      =			scheduler_class(**SCREAMING_SNAKE_CASE__ ,	use_karras_sigmas=SCREAMING_SNAKE_CASE__    )

								scheduler.set_timesteps(self.num_inference_steps ,	device=SCREAMING_SNAKE_CASE__    )

								lowerCAmelCase__ :    str				      =			self.dummy_model()
								lowerCAmelCase__ :    Any				      =			self.dummy_sample_deter.to(SCREAMING_SNAKE_CASE__    ) * scheduler.init_noise_sigma
								lowerCAmelCase__ :    List[Any]				      =			sample.to(SCREAMING_SNAKE_CASE__    )

								for t in scheduler.timesteps:
											lowerCAmelCase__ :    Union[str, Any]				      =			scheduler.scale_model_input(SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    )

											lowerCAmelCase__ :    Any				      =			model(SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    )

											lowerCAmelCase__ :    str				      =			scheduler.step(SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__ ,	SCREAMING_SNAKE_CASE__    )
											lowerCAmelCase__ :    List[Any]				      =			output.prev_sample

								lowerCAmelCase__ :    int				      =			torch.sum(torch.abs(SCREAMING_SNAKE_CASE__    )    )
								lowerCAmelCase__ :    Dict				      =			torch.mean(torch.abs(SCREAMING_SNAKE_CASE__    )    )

								assert abs(result_sum.item() - 0.00_015    ) < 1E-2
								assert abs(result_mean.item() - 1.9869_5545_3503_4695E-07    ) < 1E-2



