code: 
  
  
  
  
  
  
  # 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 					snake_case_							:
  
  
  
  
  
  
  
     """simple docstring"""
  
  
  
     def __init__(    self:	int   ,       snake_case__:	List[Any]   ,       snake_case__:	Optional[Any]	)  ->		Tuple:
  
  
         """simple docstring"""
  
  
  
         _lowercase     =	key
         _lowercase     =	freq
  
     def __str__(    self:	Any	)  ->		Optional[int]:
  
  
         """simple docstring"""
  
  
  
         return f'''Node(key={self.key}, freq={self.freq})'''
  
  
  
  
  
  
  def        UpperCAmelCase      (   _lowercase,      _lowercase,      _lowercase,      _lowercase,      _lowercase,      _lowercase      ):
  
  
      '''simple docstring'''
  
  
  
      if i > j or i < 0 or j > len(_lowercase      ) - 1:
          return
  
      _lowercase     =	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        UpperCAmelCase      (   _lowercase      ):
  
  
      '''simple docstring'''
  
  
  
      nodes.sort(key=lambda _lowercase      : node.key      )
  
      _lowercase     =	len(_lowercase      )
  
      _lowercase     =	[nodes[i].key for i in range(_lowercase      )]
      _lowercase     =	[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.
      _lowercase     =	[[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
      _lowercase     =	[[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
      _lowercase     =	[[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      ):
              _lowercase     =	i + interval_length - 1
  
              _lowercase     =	sys.maxsize  # set the value to "infinity"
              _lowercase     =	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
                  _lowercase     =	dp[i][r - 1] if r != i else 0  # optimal cost for left subtree
                  _lowercase     =	dp[r + 1][j] if r != j else 0  # optimal cost for right subtree
                  _lowercase     =	left + total[i][j] + right
  
                  if dp[i][j] > cost:
                      _lowercase     =	cost
                      _lowercase     =	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        UpperCAmelCase      (   ):
  
  
      '''simple docstring'''
  
  
  
      _lowercase     =	[Node(_lowercase,      randint(1,      50      )      ) for i in range(10,      0,      -1      )]
      find_optimal_binary_search_tree(_lowercase      )
  
  
  if __name__ == "__main__":
     main()
  
style_context: 
  
  
  
  
  
  
  from typing import Dict, List, Optional, Union
  
  import numpy as np
  
  from ...image_processing_utils import BaseImageProcessor, BatchFeature, get_size_dict
  from ...image_transforms import center_crop, normalize, rescale, resize, to_channel_dimension_format
  from ...image_utils import (
      IMAGENET_STANDARD_MEAN,
      IMAGENET_STANDARD_STD,
      ChannelDimension,
      ImageInput,
      PILImageResampling,
      make_list_of_images,
      to_numpy_array,
      valid_images,
  )
  from ...utils import TensorType, is_vision_available, logging
  
  
  if is_vision_available():
     import PIL
  
  
  lowerCamelCase										=  logging.get_logger(__name__)
  
  class 					snake_case_							(       lowercase_      ):
  
  
  
  
  
  
  
     """simple docstring"""
  
  
  
     __lowerCAmelCase      =						["""pixel_values"""]
  
     def __init__(    self:	int   ,       snake_case__:	bool = True   ,       snake_case__:	Dict[str, int] = None   ,       snake_case__:	PILImageResampling = PIL.Image.NEAREST   ,       snake_case__:	bool = False   ,       snake_case__:	Dict[str, int] = None   ,       snake_case__:	Union[int, float] = 1 / 255   ,       snake_case__:	bool = False   ,       snake_case__:	bool = True   ,       snake_case__:	bool = True   ,       snake_case__:	Optional[Union[float, List[float]]] = None   ,       snake_case__:	Optional[Union[float, List[float]]] = None   ,       snake_case__:	bool = True   ,       **snake_case__:	Optional[Any]   ,       )  ->		None:
  
  
         """simple docstring"""
  
  
  
         super().__init__(**snake_case__	)
         _lowercase     =	size if size is not None else {'height': 346, 'width': 346}
         _lowercase     =	get_size_dict(snake_case__	)
         _lowercase     =	crop_size if crop_size is not None else {'height': 289, 'width': 289}
         _lowercase     =	get_size_dict(snake_case__   ,       param_name='crop_size'	)
  
         _lowercase     =	do_resize
         _lowercase     =	size
         _lowercase     =	resample
         _lowercase     =	do_center_crop
         _lowercase     =	crop_size
         _lowercase     =	do_rescale
         _lowercase     =	rescale_factor
         _lowercase     =	rescale_offset
         _lowercase     =	do_normalize
         _lowercase     =	image_mean if image_mean is not None else IMAGENET_STANDARD_MEAN
         _lowercase     =	image_std if image_std is not None else IMAGENET_STANDARD_STD
         _lowercase     =	include_top
  
     def 			SCREAMING_SNAKE_CASE__				(    self:	str   ,       snake_case__:	np.ndarray   ,       snake_case__:	Dict[str, int]   ,       snake_case__:	PILImageResampling = PIL.Image.NEAREST   ,       snake_case__:	Optional[Union[str, ChannelDimension]] = None   ,       **snake_case__:	Optional[int]   ,       )  ->		np.ndarray:
  
  
         """simple docstring"""
  
  
  
         _lowercase     =	get_size_dict(snake_case__	)
         if "height" not in size or "width" not in size:
             raise ValueError(f'''The size dictionary must have keys \'height\' and \'width\'. Got {size.keys()}'''	)
         return resize(
             snake_case__   ,       size=(size['height'], size['width'])   ,       resample=snake_case__   ,       data_format=snake_case__   ,       **snake_case__	)
  
     def 			SCREAMING_SNAKE_CASE__				(    self:	Optional[Any]   ,       snake_case__:	np.ndarray   ,       snake_case__:	Dict[str, int]   ,       snake_case__:	Optional[Union[str, ChannelDimension]] = None   ,       **snake_case__:	str   ,       )  ->		np.ndarray:
  
  
         """simple docstring"""
  
  
  
         _lowercase     =	get_size_dict(snake_case__	)
         if "height" not in size or "width" not in size:
             raise ValueError(f'''The size dictionary must have keys \'height\' and \'width\'. Got {size.keys()}'''	)
         return center_crop(snake_case__   ,       size=(size['height'], size['width'])   ,       data_format=snake_case__   ,       **snake_case__	)
  
     def 			SCREAMING_SNAKE_CASE__				(    self:	Optional[Any]   ,       snake_case__:	np.ndarray   ,       snake_case__:	Union[int, float]   ,       snake_case__:	bool = True   ,       snake_case__:	Optional[Union[str, ChannelDimension]] = None   ,       **snake_case__:	List[str]   ,       )  ->		Optional[int]:
  
  
         """simple docstring"""
  
  
  
         if offset:
             _lowercase     =	(image - 1_27.5) * scale
             if data_format is not None:
                 _lowercase     =	to_channel_dimension_format(snake_case__   ,       snake_case__	)
             _lowercase     =	rescaled_image.astype(np.floataa	)
         else:
             _lowercase     =	rescale(snake_case__   ,       scale=snake_case__   ,       data_format=snake_case__   ,       **snake_case__	)
         return rescaled_image
  
     def 			SCREAMING_SNAKE_CASE__				(    self:	Dict   ,       snake_case__:	np.ndarray   ,       snake_case__:	Union[float, List[float]]   ,       snake_case__:	Union[float, List[float]]   ,       snake_case__:	Optional[Union[str, ChannelDimension]] = None   ,       **snake_case__:	Any   ,       )  ->		np.ndarray:
  
  
         """simple docstring"""
  
  
  
         return normalize(snake_case__   ,       mean=snake_case__   ,       std=snake_case__   ,       data_format=snake_case__   ,       **snake_case__	)
  
     def 			SCREAMING_SNAKE_CASE__				(    self:	Optional[int]   ,       snake_case__:	ImageInput   ,       snake_case__:	bool = None   ,       snake_case__:	Dict[str, int] = None   ,       snake_case__:	Union[str, Any]=None   ,       snake_case__:	bool = None   ,       snake_case__:	Dict[str, int] = None   ,       snake_case__:	bool = None   ,       snake_case__:	float = None   ,       snake_case__:	bool = None   ,       snake_case__:	bool = None   ,       snake_case__:	Optional[Union[float, List[float]]] = None   ,       snake_case__:	Optional[Union[float, List[float]]] = None   ,       snake_case__:	bool = None   ,       snake_case__:	Optional[Union[str, TensorType]] = None   ,       snake_case__:	ChannelDimension = ChannelDimension.FIRST   ,       **snake_case__:	int   ,       )  ->		PIL.Image.Image:
  
  
         """simple docstring"""
  
  
  
         _lowercase     =	do_resize if do_resize is not None else self.do_resize
         _lowercase     =	resample if resample is not None else self.resample
         _lowercase     =	do_center_crop if do_center_crop is not None else self.do_center_crop
         _lowercase     =	do_rescale if do_rescale is not None else self.do_rescale
         _lowercase     =	rescale_factor if rescale_factor is not None else self.rescale_factor
         _lowercase     =	rescale_offset if rescale_offset is not None else self.rescale_offset
         _lowercase     =	do_normalize if do_normalize is not None else self.do_normalize
         _lowercase     =	image_mean if image_mean is not None else self.image_mean
         _lowercase     =	image_std if image_std is not None else self.image_std
         _lowercase     =	include_top if include_top is not None else self.include_top
  
         _lowercase     =	size if size is not None else self.size
         _lowercase     =	get_size_dict(snake_case__	)
         _lowercase     =	crop_size if crop_size is not None else self.crop_size
         _lowercase     =	get_size_dict(snake_case__   ,       param_name='crop_size'	)
  
         _lowercase     =	make_list_of_images(snake_case__	)
  
         if not valid_images(snake_case__	):
             raise ValueError(
                 'Invalid image type. Must be of type PIL.Image.Image, numpy.ndarray, '
                 'torch.Tensor, tf.Tensor or jax.ndarray.'	)
  
         if do_resize and size is None or resample is None:
             raise ValueError('Size and resample must be specified if do_resize is True.'	)
  
         if do_center_crop and crop_size is None:
             raise ValueError('Crop size must be specified if do_center_crop is True.'	)
  
         if do_rescale and rescale_factor is None:
             raise ValueError('Rescale factor must be specified if do_rescale is True.'	)
  
         if do_normalize and (image_mean is None or image_std is None):
             raise ValueError('Image mean and std must be specified if do_normalize is True.'	)
  
         # All transformations expect numpy arrays.
         _lowercase     =	[to_numpy_array(snake_case__	) for image in images]
  
         if do_resize:
             _lowercase     =	[self.resize(image=snake_case__   ,       size=snake_case__   ,       resample=snake_case__	) for image in images]
  
         if do_center_crop:
             _lowercase     =	[self.center_crop(image=snake_case__   ,       size=snake_case__	) for image in images]
  
         if do_rescale:
             _lowercase     =	[self.rescale(image=snake_case__   ,       scale=snake_case__   ,       offset=snake_case__	) for image in images]
  
         if do_normalize:
             _lowercase     =	[self.normalize(image=snake_case__   ,       mean=snake_case__   ,       std=snake_case__	) for image in images]
  
         if include_top:
             _lowercase     =	[self.normalize(image=snake_case__   ,       mean=[0, 0, 0]   ,       std=snake_case__	) for image in images]
  
         _lowercase     =	[to_channel_dimension_format(snake_case__   ,       snake_case__	) for image in images]
  
         _lowercase     =	{'pixel_values': images}
         return BatchFeature(data=snake_case__   ,       tensor_type=snake_case__	)
  