
'''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      snake_case__    :







		"""simple docstring"""

		def __init__(	self							:       Tuple   ,     UpperCamelCase__							:       Tuple   ,     UpperCamelCase__							:       Optional[int]	)		->       str:






				"""simple docstring"""







				snake_case   :    Union[str, Any]			       =		key
				snake_case   :    str			       =		freq



		def __str__(	self							:       int	)		->       Dict:






				"""simple docstring"""







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







def 					_UpperCamelCase		(    SCREAMING_SNAKE_CASE__      ,    SCREAMING_SNAKE_CASE__      ,    SCREAMING_SNAKE_CASE__      ,    SCREAMING_SNAKE_CASE__      ,    SCREAMING_SNAKE_CASE__      ,    SCREAMING_SNAKE_CASE__       )   ->				int:


		'''simple docstring'''


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

		snake_case   :    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(SCREAMING_SNAKE_CASE__      ,    SCREAMING_SNAKE_CASE__      ,    SCREAMING_SNAKE_CASE__      ,    node - 1      ,    key[node]      ,    SCREAMING_SNAKE_CASE__       )
		print_binary_search_tree(SCREAMING_SNAKE_CASE__      ,    SCREAMING_SNAKE_CASE__      ,    node + 1      ,    SCREAMING_SNAKE_CASE__      ,    key[node]      ,    SCREAMING_SNAKE_CASE__       )




def 					_UpperCamelCase		(    SCREAMING_SNAKE_CASE__       )   ->				Optional[Any]:


		'''simple docstring'''


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

		snake_case   :    Union[str, Any]			       =		len(SCREAMING_SNAKE_CASE__       )

		snake_case   :    str			       =		[nodes[i].key for i in range(SCREAMING_SNAKE_CASE__       )]
		snake_case   :    int			       =		[nodes[i].freq for i in range(SCREAMING_SNAKE_CASE__       )]

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

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

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

								if dp[i][j] > cost:
										snake_case   :    Tuple			       =		cost
										snake_case   :    Tuple			       =		r

		print('''Binary search tree nodes:'''       )
		for node in nodes:
				print(SCREAMING_SNAKE_CASE__       )

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




def 					_UpperCamelCase		(    )   ->				int:


		'''simple docstring'''


		snake_case   :    Dict			       =		[Node(SCREAMING_SNAKE_CASE__      ,    randint(1      ,    50       )       ) for i in range(10      ,    0      ,    -1       )]
		find_optimal_binary_search_tree(SCREAMING_SNAKE_CASE__       )


if __name__ == "__main__":
						main()









from typing import TYPE_CHECKING

from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_torch_available, is_vision_available


_snake_case				= {"""configuration_bit""": ["""BIT_PRETRAINED_CONFIG_ARCHIVE_MAP""", """BitConfig""", """BitOnnxConfig"""]}

try:
  if not is_torch_available():
    raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
  pass
else:
  _snake_case				= [
      """BIT_PRETRAINED_MODEL_ARCHIVE_LIST""",
      """BitForImageClassification""",
      """BitModel""",
      """BitPreTrainedModel""",
      """BitBackbone""",
  ]


try:
  if not is_vision_available():
    raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
  pass
else:
  _snake_case				= ["""BitImageProcessor"""]


if TYPE_CHECKING:
  from .configuration_bit import BIT_PRETRAINED_CONFIG_ARCHIVE_MAP, BitConfig, BitOnnxConfig

  try:
    if not is_torch_available():
      raise OptionalDependencyNotAvailable()
  except OptionalDependencyNotAvailable:
    pass
  else:
    from .modeling_bit import (
        BIT_PRETRAINED_MODEL_ARCHIVE_LIST,
        BitBackbone,
        BitForImageClassification,
        BitModel,
        BitPreTrainedModel,
    )

  try:
    if not is_vision_available():
      raise OptionalDependencyNotAvailable()
  except OptionalDependencyNotAvailable:
    pass
  else:
    from .image_processing_bit import BitImageProcessor

else:
  import sys

  _snake_case				= _LazyModule(__name__, globals()["""__file__"""], _import_structure)






