

'''simple docstring'''






import math
from dataclasses import dataclass
from typing import List, Optional, Tuple, Union

import numpy as np
import torch

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import BaseOutput, logging, randn_tensor
from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin


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

@dataclass
# Copied from diffusers.schedulers.scheduling_ddpm.DDPMSchedulerOutput with DDPM->EulerDiscrete
class     SCREAMING_SNAKE_CASE(							lowercase__      ):







	"""simple docstring"""






	lowerCamelCase__      			=	42
	lowerCamelCase__      			=	None


def 			snake_case_					( _lowerCAmelCase  :						Optional[Any]       ,				_lowerCAmelCase  :						Union[str, Any]=0.9_9_9       ,				_lowerCAmelCase  :						Optional[int]="cosine"       ,				)				->       Dict:

							if alpha_transform_type == "cosine":

														def alpha_bar_fn(_lowerCAmelCase  :						List[str]  ):
																					return math.cos((t + 0.0_0_8) / 1.0_0_8 * math.pi / 2  ) ** 2

							elif alpha_transform_type == "exp":

														def alpha_bar_fn(_lowerCAmelCase  :						Tuple  ):
																					return math.exp(t * -12.0  )

							else:
														raise ValueError(f"""Unsupported alpha_tranform_type: {alpha_transform_type}"""  )

							UpperCAmelCase			:	Any										=		[]
							for i in range(lowercase_  ):
														UpperCAmelCase			:	str										=		i / num_diffusion_timesteps
														UpperCAmelCase			:	List[Any]										=		(i + 1) / num_diffusion_timesteps
														betas.append(min(1 - alpha_bar_fn(lowercase_  ) / alpha_bar_fn(lowercase_  )       ,				lowercase_  )  )
							return torch.tensor(lowercase_       ,				dtype=torch.floataa  )




class     SCREAMING_SNAKE_CASE(							lowercase__							,		lowercase__      ):







	"""simple docstring"""






	lowerCamelCase__      			=	[e.name for e in KarrasDiffusionSchedulers]
	lowerCamelCase__      			=	1







	@register_to_config
	def __init__(						self      :					List[Any]		,      __snake_case      :					int = 1000		,      __snake_case      :					float = 0.00_01		,      __snake_case      :					float = 0.02		,      __snake_case      :					str = "linear"		,      __snake_case      :					Optional[Union[np.ndarray, List[float]]] = None		,      __snake_case      :					str = "epsilon"		,      __snake_case      :					str = "linear"		,      __snake_case      :					Optional[bool] = False		,      __snake_case      :					str = "linspace"		,      __snake_case      :					int = 0		,      )      ->		Optional[Any]:
								if trained_betas is not None:
															UpperCAmelCase			:	Tuple										=		torch.tensor(lowerCAmelCase__		,      dtype=torch.floataa		)
								elif beta_schedule == "linear":
															UpperCAmelCase			:	Optional[Any]										=		torch.linspace(lowerCAmelCase__		,      lowerCAmelCase__		,      lowerCAmelCase__		,      dtype=torch.floataa		)
								elif beta_schedule == "scaled_linear":
															# this schedule is very specific to the latent diffusion model.
															UpperCAmelCase			:	Optional[Any]										=		(
															    torch.linspace(beta_start**0.5		,      beta_end**0.5		,      lowerCAmelCase__		,      dtype=torch.floataa		) ** 2
															)
								elif beta_schedule == "squaredcos_cap_v2":
															# Glide cosine schedule
															UpperCAmelCase			:	Dict										=		betas_for_alpha_bar(lowerCAmelCase__		)
								else:
															raise NotImplementedError(F"""{beta_schedule} does is not implemented for {self.__class__}"""		)

								UpperCAmelCase			:	List[str]										=		1.0 - self.betas
								UpperCAmelCase			:	Dict										=		torch.cumprod(self.alphas		,      dim=0		)

								UpperCAmelCase			:	List[Any]										=		np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5		)
								UpperCAmelCase			:	Optional[Any]										=		np.concatenate([sigmas[::-1], [0.0]]		).astype(np.floataa		)
								UpperCAmelCase			:	Tuple										=		torch.from_numpy(lowerCAmelCase__		)

								# setable values
								UpperCAmelCase			:	Optional[int]										=		None
								UpperCAmelCase			:	Optional[Any]										=		np.linspace(0		,      num_train_timesteps - 1		,      lowerCAmelCase__		,      dtype=lowerCAmelCase__		)[::-1].copy()
								UpperCAmelCase			:	int										=		torch.from_numpy(lowerCAmelCase__		)
								UpperCAmelCase			:	List[str]										=		False
								UpperCAmelCase			:	Tuple										=		use_karras_sigmas







	@property
	def 					A     (						self      :					List[Any]		)      ->		List[Any]:
								# standard deviation of the initial noise distribution
								if self.config.timestep_spacing in ["linspace", "trailing"]:
															return self.sigmas.max()

								return (self.sigmas.max() ** 2 + 1) ** 0.5







	def 					A     (						self      :					List[Any]		,      __snake_case      :					torch.FloatTensor		,      __snake_case      :					Union[float, torch.FloatTensor]		)      ->		torch.FloatTensor:
								if isinstance(lowerCAmelCase__		,      torch.Tensor		):
															UpperCAmelCase			:	int										=		timestep.to(self.timesteps.device		)
								UpperCAmelCase			:	Union[str, Any]										=		(self.timesteps == timestep).nonzero().item()
								UpperCAmelCase			:	Any										=		self.sigmas[step_index]

								UpperCAmelCase			:	str										=		sample / ((sigma**2 + 1) ** 0.5)

								UpperCAmelCase			:	Tuple										=		True
								return sample







	def 					A     (						self      :					str		,      __snake_case      :					int		,      __snake_case      :					Union[str, torch.device] = None		)      ->		List[str]:
								UpperCAmelCase			:	Optional[Any]										=		num_inference_steps

								# "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://arxiv.org/abs/2305.08891
								if self.config.timestep_spacing == "linspace":
															UpperCAmelCase			:	List[str]										=		np.linspace(0		,      self.config.num_train_timesteps - 1		,      lowerCAmelCase__		,      dtype=lowerCAmelCase__		)[
															    ::-1
															].copy()
								elif self.config.timestep_spacing == "leading":
															UpperCAmelCase			:	List[str]										=		self.config.num_train_timesteps // self.num_inference_steps
															# creates integer timesteps by multiplying by ratio
															# casting to int to avoid issues when num_inference_step is power of 3
															UpperCAmelCase			:	Dict										=		(np.arange(0		,      lowerCAmelCase__		) * step_ratio).round()[::-1].copy().astype(lowerCAmelCase__		)
															timesteps += self.config.steps_offset
								elif self.config.timestep_spacing == "trailing":
															UpperCAmelCase			:	Optional[Any]										=		self.config.num_train_timesteps / self.num_inference_steps
															# creates integer timesteps by multiplying by ratio
															# casting to int to avoid issues when num_inference_step is power of 3
															UpperCAmelCase			:	List[str]										=		(np.arange(self.config.num_train_timesteps		,      0		,      -step_ratio		)).round().copy().astype(lowerCAmelCase__		)
															timesteps -= 1
								else:
															raise ValueError(
															    F"""{self.config.timestep_spacing} is not supported. Please make sure to choose one of \'linspace\', \'leading\' or \'trailing\'."""		)

								UpperCAmelCase			:	Dict										=		np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5		)
								UpperCAmelCase			:	Optional[int]										=		np.log(lowerCAmelCase__		)

								if self.config.interpolation_type == "linear":
															UpperCAmelCase			:	Dict										=		np.interp(lowerCAmelCase__		,      np.arange(0		,      len(lowerCAmelCase__		)		)		,      lowerCAmelCase__		)
								elif self.config.interpolation_type == "log_linear":
															UpperCAmelCase			:	int										=		torch.linspace(np.log(sigmas[-1]		)		,      np.log(sigmas[0]		)		,      num_inference_steps + 1		).exp()
								else:
															raise ValueError(
															    F"""{self.config.interpolation_type} is not implemented. Please specify interpolation_type to either"""
															    ''' \'linear\' or \'log_linear\''''		)

								if self.use_karras_sigmas:
															UpperCAmelCase			:	List[Any]										=		self._convert_to_karras(in_sigmas=lowerCAmelCase__		,      num_inference_steps=self.num_inference_steps		)
															UpperCAmelCase			:	Optional[Any]										=		np.array([self._sigma_to_t(lowerCAmelCase__		,      lowerCAmelCase__		) for sigma in sigmas]		)

								UpperCAmelCase			:	str										=		np.concatenate([sigmas, [0.0]]		).astype(np.floataa		)
								UpperCAmelCase			:	Any										=		torch.from_numpy(lowerCAmelCase__		).to(device=lowerCAmelCase__		)
								if str(lowerCAmelCase__		).startswith('''mps'''		):
															# mps does not support float64
															UpperCAmelCase			:	int										=		torch.from_numpy(lowerCAmelCase__		).to(lowerCAmelCase__		,      dtype=torch.floataa		)
								else:
															UpperCAmelCase			:	List[Any]										=		torch.from_numpy(lowerCAmelCase__		).to(device=lowerCAmelCase__		)







	def 					A     (						self      :					List[str]		,      __snake_case      :					Union[str, Any]		,      __snake_case      :					List[Any]		)      ->		Optional[Any]:
								UpperCAmelCase			:	List[str]										=		np.log(lowerCAmelCase__		)

								# get distribution
								UpperCAmelCase			:	str										=		log_sigma - log_sigmas[:, np.newaxis]

								# get sigmas range
								UpperCAmelCase			:	List[Any]										=		np.cumsum((dists >= 0)		,      axis=0		).argmax(axis=0		).clip(max=log_sigmas.shape[0] - 2		)
								UpperCAmelCase			:	Dict										=		low_idx + 1

								UpperCAmelCase			:	Optional[int]										=		log_sigmas[low_idx]
								UpperCAmelCase			:	Tuple										=		log_sigmas[high_idx]

								# interpolate sigmas
								UpperCAmelCase			:	List[str]										=		(low - log_sigma) / (low - high)
								UpperCAmelCase			:	Dict										=		np.clip(lowerCAmelCase__		,      0		,      1		)

								# transform interpolation to time range
								UpperCAmelCase			:	Tuple										=		(1 - w) * low_idx + w * high_idx
								UpperCAmelCase			:	Any										=		t.reshape(sigma.shape		)
								return t







	def 					A     (						self      :					List[str]		,      __snake_case      :					torch.FloatTensor		,      __snake_case      :					List[Any]		)      ->		torch.FloatTensor:
								UpperCAmelCase			:	Optional[Any]										=		in_sigmas[-1].item()
								UpperCAmelCase			:	Dict										=		in_sigmas[0].item()

								UpperCAmelCase			:	List[Any]										=		7.0  # 7.0 is the value used in the paper
								UpperCAmelCase			:	Union[str, Any]										=		np.linspace(0		,      1		,      lowerCAmelCase__		)
								UpperCAmelCase			:	List[str]										=		sigma_min ** (1 / rho)
								UpperCAmelCase			:	List[str]										=		sigma_max ** (1 / rho)
								UpperCAmelCase			:	Dict										=		(max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
								return sigmas







	def 					A     (						self      :					Union[str, Any]		,      __snake_case      :					torch.FloatTensor		,      __snake_case      :					Union[float, torch.FloatTensor]		,      __snake_case      :					torch.FloatTensor		,      __snake_case      :					float = 0.0		,      __snake_case      :					float = 0.0		,      __snake_case      :					float = float('''inf'''		)		,      __snake_case      :					float = 1.0		,      __snake_case      :					Optional[torch.Generator] = None		,      __snake_case      :					bool = True		,      )      ->		Union[EulerDiscreteSchedulerOutput, Tuple]:

								if (
								    isinstance(lowerCAmelCase__		,      lowerCAmelCase__		)
								    or isinstance(lowerCAmelCase__		,      torch.IntTensor		)
								    or isinstance(lowerCAmelCase__		,      torch.LongTensor		)
								):
															raise ValueError(
															    (
															        '''Passing integer indices (e.g. from `enumerate(timesteps)`) as timesteps to'''
															        ''' `EulerDiscreteScheduler.step()` is not supported. Make sure to pass'''
															        ''' one of the `scheduler.timesteps` as a timestep.'''
															    )		,      )

								if not self.is_scale_input_called:
															logger.warning(
															    '''The `scale_model_input` function should be called before `step` to ensure correct denoising. '''
															    '''See `StableDiffusionPipeline` for a usage example.'''		)

								if isinstance(lowerCAmelCase__		,      torch.Tensor		):
															UpperCAmelCase			:	int										=		timestep.to(self.timesteps.device		)

								UpperCAmelCase			:	str										=		(self.timesteps == timestep).nonzero().item()
								UpperCAmelCase			:	List[Any]										=		self.sigmas[step_index]

								UpperCAmelCase			:	Optional[Any]										=		min(s_churn / (len(self.sigmas		) - 1)		,      2**0.5 - 1		) if s_tmin <= sigma <= s_tmax else 0.0

								UpperCAmelCase			:	List[Any]										=		randn_tensor(
								    model_output.shape		,      dtype=model_output.dtype		,      device=model_output.device		,      generator=lowerCAmelCase__		)

								UpperCAmelCase			:	str										=		noise * s_noise
								UpperCAmelCase			:	Tuple										=		sigma * (gamma + 1)

								if gamma > 0:
															UpperCAmelCase			:	Optional[int]										=		sample + eps * (sigma_hat**2 - sigma**2) ** 0.5

								# 1. compute predicted original sample (x_0) from sigma-scaled predicted noise
								# NOTE: "original_sample" should not be an expected prediction_type but is left in for
								# backwards compatibility
								if self.config.prediction_type == "original_sample" or self.config.prediction_type == "sample":
															UpperCAmelCase			:	Any										=		model_output
								elif self.config.prediction_type == "epsilon":
															UpperCAmelCase			:	List[Any]										=		sample - sigma_hat * model_output
								elif self.config.prediction_type == "v_prediction":
															# * c_out + input * c_skip
															UpperCAmelCase			:	List[Any]										=		model_output * (-sigma / (sigma**2 + 1) ** 0.5) + (sample / (sigma**2 + 1))
								else:
															raise ValueError(
															    F"""prediction_type given as {self.config.prediction_type} must be one of `epsilon`, or `v_prediction`"""		)

								# 2. Convert to an ODE derivative
								UpperCAmelCase			:	List[Any]										=		(sample - pred_original_sample) / sigma_hat

								UpperCAmelCase			:	List[Any]										=		self.sigmas[step_index + 1] - sigma_hat

								UpperCAmelCase			:	str										=		sample + derivative * dt

								if not return_dict:
															return (prev_sample,)

								return EulerDiscreteSchedulerOutput(prev_sample=lowerCAmelCase__		,      pred_original_sample=lowerCAmelCase__		)







	def 					A     (						self      :					Tuple		,      __snake_case      :					torch.FloatTensor		,      __snake_case      :					torch.FloatTensor		,      __snake_case      :					torch.FloatTensor		,      )      ->		torch.FloatTensor:
								UpperCAmelCase			:	Tuple										=		self.sigmas.to(device=original_samples.device		,      dtype=original_samples.dtype		)
								if original_samples.device.type == "mps" and torch.is_floating_point(lowerCAmelCase__		):
															# mps does not support float64
															UpperCAmelCase			:	int										=		self.timesteps.to(original_samples.device		,      dtype=torch.floataa		)
															UpperCAmelCase			:	int										=		timesteps.to(original_samples.device		,      dtype=torch.floataa		)
								else:
															UpperCAmelCase			:	Optional[int]										=		self.timesteps.to(original_samples.device		)
															UpperCAmelCase			:	List[Any]										=		timesteps.to(original_samples.device		)

								UpperCAmelCase			:	Dict										=		[(schedule_timesteps == t).nonzero().item() for t in timesteps]

								UpperCAmelCase			:	Dict										=		sigmas[step_indices].flatten()
								while len(sigma.shape		) < len(original_samples.shape		):
															UpperCAmelCase			:	Tuple										=		sigma.unsqueeze(-1		)

								UpperCAmelCase			:	int										=		original_samples + noise * sigma
								return noisy_samples







	def __len__(						self      :					str		)      ->		str:
								return self.config.num_train_timesteps










'''simple docstring'''






class     SCREAMING_SNAKE_CASE:







	"""simple docstring"""






	def __init__(						self      :					List[Any]		,      __snake_case      :					Dict		)      ->		str:
								UpperCAmelCase			:	str										=		val
								# Number of nodes in left subtree
								UpperCAmelCase			:	Optional[int]										=		0
								UpperCAmelCase			:	str										=		None
								UpperCAmelCase			:	Optional[Any]										=		None
								UpperCAmelCase			:	Dict										=		None







	def 					A     (						self      :					Tuple		,      __snake_case      :					Dict		)      ->		Dict:
								assert self.left_tree_size == other.left_tree_size, "Unequal Sizes of Blocks"

								if self.val < other.val:
															UpperCAmelCase			:	Optional[int]										=		self.right
															UpperCAmelCase			:	Optional[Any]										=		None
															if self.right:
																						UpperCAmelCase			:	Tuple										=		other
															UpperCAmelCase			:	int										=		other
															UpperCAmelCase			:	Dict										=		self.left_tree_size * 2 + 1
															return self
								else:
															UpperCAmelCase			:	int										=		other.right
															UpperCAmelCase			:	List[str]										=		None
															if other.right:
																						UpperCAmelCase			:	Dict										=		self
															UpperCAmelCase			:	Optional[Any]										=		self
															UpperCAmelCase			:	Optional[int]										=		other.left_tree_size * 2 + 1
															return other

class     SCREAMING_SNAKE_CASE:







	"""simple docstring"""






	def __init__(						self      :					Union[str, Any]		,      __snake_case      :					str=None		,      __snake_case      :					Any=None		,      __snake_case      :					Any=0		)      ->		Dict:
								UpperCAmelCase			:	str										=		heap_size
								UpperCAmelCase			:	Tuple										=		bottom_root
								UpperCAmelCase			:	str										=		min_node







	def 					A     (						self      :					Tuple		,      __snake_case      :					Union[str, Any]		)      ->		Optional[Any]:

								# Empty heaps corner cases
								if other.size == 0:
															return None
								if self.size == 0:
															UpperCAmelCase			:	List[str]										=		other.size
															UpperCAmelCase			:	Union[str, Any]										=		other.bottom_root
															UpperCAmelCase			:	Optional[int]										=		other.min_node
															return None
								# Update size
								UpperCAmelCase			:	Tuple										=		self.size + other.size

								# Update min.node
								if self.min_node.val > other.min_node.val:
															UpperCAmelCase			:	Union[str, Any]										=		other.min_node
								# Merge

								# Order roots by left_subtree_size
								UpperCAmelCase			:	List[str]										=		[]
								UpperCAmelCase							,		UpperCAmelCase			:	Optional[int]										=		self.bottom_root, other.bottom_root
								while i or j:
															if i and ((not j) or i.left_tree_size < j.left_tree_size):
																						combined_roots_list.append((i, True)		)
																						UpperCAmelCase			:	Union[str, Any]										=		i.parent
															else:
																						combined_roots_list.append((j, False)		)
																						UpperCAmelCase			:	List[str]										=		j.parent
        # Insert links between them
								for i in range(len(__snake_case		) - 1		):
															if combined_roots_list[i][1] != combined_roots_list[i + 1][1]:
																						UpperCAmelCase			:	Union[str, Any]										=		combined_roots_list[i + 1][0]
																						UpperCAmelCase			:	Optional[Any]										=		combined_roots_list[i][0]
        # Consecutively merge roots with same left_tree_size
								UpperCAmelCase			:	Optional[int]										=		combined_roots_list[0][0]
								while i.parent:
															if (
															    (i.left_tree_size == i.parent.left_tree_size) and (not i.parent.parent)
															) or (
															    i.left_tree_size == i.parent.left_tree_size
															    and i.left_tree_size != i.parent.parent.left_tree_size
															):
																						# Neighbouring Nodes
																						UpperCAmelCase			:	Optional[int]										=		i.left
																						UpperCAmelCase			:	List[Any]										=		i.parent.parent

																						# Merging trees
																						UpperCAmelCase			:	Optional[int]										=		i.merge_trees(i.parent		)

																						# Updating links
																						UpperCAmelCase			:	Any										=		previous_node
																						UpperCAmelCase			:	str										=		next_node
																						if previous_node:
																													UpperCAmelCase			:	Union[str, Any]										=		i
																						if next_node:
																													UpperCAmelCase			:	Optional[Any]										=		i
															else:
																						UpperCAmelCase			:	Union[str, Any]										=		i.parent
        # Updating self.bottom_root
								while i.left:
															UpperCAmelCase			:	str										=		i.left
								UpperCAmelCase			:	Optional[int]										=		i

								# Update other
								UpperCAmelCase			:	str										=		self.size
								UpperCAmelCase			:	Union[str, Any]										=		self.bottom_root
								UpperCAmelCase			:	Union[str, Any]										=		self.min_node

								# Return the merged heap
								return self







	def 					A     (						self      :					List[Any]		,      __snake_case      :					List[str]		)      ->		List[str]:
								if self.size == 0:
															UpperCAmelCase			:	List[Any]										=		Node(__snake_case		)
															UpperCAmelCase			:	List[str]										=		1
															UpperCAmelCase			:	str										=		self.bottom_root
								else:
															# Create new node
															UpperCAmelCase			:	int										=		Node(__snake_case		)

															# Update size
															self.size += 1

															# update min_node
															if val < self.min_node.val:
																						UpperCAmelCase			:	Optional[int]										=		new_node
															# Put new_node as a bottom_root in heap
															UpperCAmelCase			:	Optional[Any]										=		new_node
															UpperCAmelCase			:	Optional[Any]										=		self.bottom_root
															UpperCAmelCase			:	Union[str, Any]										=		new_node

															# Consecutively merge roots with same left_tree_size
															while (
															    self.bottom_root.parent
															    and self.bottom_root.left_tree_size
															    == self.bottom_root.parent.left_tree_size
															):
																						# Next node
																						UpperCAmelCase			:	List[str]										=		self.bottom_root.parent.parent

																						# Merge
																						UpperCAmelCase			:	int										=		self.bottom_root.merge_trees(self.bottom_root.parent		)

																						# Update Links
																						UpperCAmelCase			:	str										=		next_node
																						UpperCAmelCase			:	int										=		None
																						if next_node:
																													UpperCAmelCase			:	Dict										=		self.bottom_root







	def 					A     (						self      :					Tuple		)      ->		List[str]:
								return self.min_node.val







	def 					A     (						self      :					Optional[Any]		)      ->		int:
								return self.size == 0







	def 					A     (						self      :					List[str]		)      ->		int:
								UpperCAmelCase			:	Tuple										=		self.min_node.val

								# Last element in heap corner case
								if self.size == 1:
															# Update size
															UpperCAmelCase			:	Tuple										=		0

															# Update bottom root
															UpperCAmelCase			:	Optional[int]										=		None

															# Update min_node
															UpperCAmelCase			:	int										=		None

															return min_value
								# No right subtree corner case
								# The structure of the tree implies that this should be the bottom root
								# and there is at least one other root
								if self.min_node.right is None:
															# Update size
															self.size -= 1

															# Update bottom root
															UpperCAmelCase			:	Optional[Any]										=		self.bottom_root.parent
															UpperCAmelCase			:	List[Any]										=		None

															# Update min_node
															UpperCAmelCase			:	int										=		self.bottom_root
															UpperCAmelCase			:	Any										=		self.bottom_root.parent
															while i:
																						if i.val < self.min_node.val:
																													UpperCAmelCase			:	Dict										=		i
																						UpperCAmelCase			:	Union[str, Any]										=		i.parent
															return min_value
								# General case
								# Find the BinomialHeap of the right subtree of min_node
								UpperCAmelCase			:	List[str]										=		self.min_node.right
								UpperCAmelCase			:	Tuple										=		None
								UpperCAmelCase			:	Any										=		bottom_of_new
								UpperCAmelCase			:	str										=		1

								# Size, min_node and bottom_root
								while bottom_of_new.left:
															UpperCAmelCase			:	Tuple										=		size_of_new * 2 + 1
															UpperCAmelCase			:	Union[str, Any]										=		bottom_of_new.left
															if bottom_of_new.val < min_of_new.val:
																						UpperCAmelCase			:	Optional[int]										=		bottom_of_new
        # Corner case of single root on top left path
								if (not self.min_node.left) and (not self.min_node.parent):
															UpperCAmelCase			:	str										=		size_of_new
															UpperCAmelCase			:	str										=		bottom_of_new
															UpperCAmelCase			:	List[str]										=		min_of_new
															# print("Single root, multiple nodes case")
															return min_value
								# Remaining cases
								# Construct heap of right subtree
								UpperCAmelCase			:	int										=		BinomialHeap(
								    bottom_root=__snake_case		,      min_node=__snake_case		,      heap_size=__snake_case		)

								# Update size
								UpperCAmelCase			:	Union[str, Any]										=		self.size - 1 - size_of_new

								# Neighbour nodes
								UpperCAmelCase			:	Any										=		self.min_node.left
								UpperCAmelCase			:	Tuple										=		self.min_node.parent

								# Initialize new bottom_root and min_node
								UpperCAmelCase			:	Optional[int]										=		previous_node or next_node
								UpperCAmelCase			:	int										=		next_node

								# Update links of previous_node and search below for new min_node and
								# bottom_root
								if previous_node:
															UpperCAmelCase			:	List[Any]										=		next_node

															# Update bottom_root and search for min_node below
															UpperCAmelCase			:	Optional[Any]										=		previous_node
															UpperCAmelCase			:	Optional[Any]										=		previous_node
															while self.bottom_root.left:
																						UpperCAmelCase			:	Any										=		self.bottom_root.left
																						if self.bottom_root.val < self.min_node.val:
																													UpperCAmelCase			:	Tuple										=		self.bottom_root
								if next_node:
															UpperCAmelCase			:	Tuple										=		previous_node

															# Search for new min_node above min_node
															UpperCAmelCase			:	Tuple										=		next_node
															while i:
																						if i.val < self.min_node.val:
																													UpperCAmelCase			:	List[str]										=		i
																						UpperCAmelCase			:	Optional[int]										=		i.parent
        # Merge heaps
								self.merge_heaps(__snake_case		)

								return min_value







	def 					A     (						self      :					Union[str, Any]		)      ->		Union[str, Any]:
								UpperCAmelCase			:	Any										=		self.bottom_root
								while top_root.parent:
															UpperCAmelCase			:	Optional[Any]										=		top_root.parent
								# preorder
								UpperCAmelCase			:	List[Any]										=		[]
								self.__traversal(__snake_case		,      __snake_case		)
								return heap_pre_order







	def 					A     (						self      :					List[str]		,      __snake_case      :					Tuple		,      __snake_case      :					int		,      __snake_case      :					Union[str, Any]=0		)      ->		Optional[int]:
								if curr_node:
															preorder.append((curr_node.val, level)		)
															self.__traversal(curr_node.left		,      __snake_case		,      level + 1		)
															self.__traversal(curr_node.right		,      __snake_case		,      level + 1		)
								else:
															preorder.append(('''#''', level)		)







	def __str__(						self      :					Tuple		)      ->		Union[str, Any]:
								if self.is_empty():
															return ""
								UpperCAmelCase			:	Dict										=		self.pre_order()

								return "\n".join(('''-''' * level + str(__snake_case		)) for value, level in preorder_heap		)


# Unit Tests
if __name__ == "__main__":
						import doctest

						doctest.testmod()





