





# 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				,							A					:       Union[str, Any]				,							A					:       Any						):
         __snake_case:		Tuple					    =		key
         __snake_case:		Dict					    =		freq






       def __str__(    self					:       Tuple						):
         return f'''Node(key={self.key}, freq={self.freq})'''







def      A__				(       SCREAMING_SNAKE_CASE__   ,   SCREAMING_SNAKE_CASE__   ,   SCREAMING_SNAKE_CASE__   ,   SCREAMING_SNAKE_CASE__   ,   SCREAMING_SNAKE_CASE__   ,   SCREAMING_SNAKE_CASE__)  ->       Any:
  if i > j or i < 0 or j > len(SCREAMING_SNAKE_CASE__) - 1:
    return

  __snake_case:		int					    =		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      A__				(       SCREAMING_SNAKE_CASE__)  ->       int:
  nodes.sort(key=lambda SCREAMING_SNAKE_CASE__: node.key)

  __snake_case:		Optional[int]					    =		len(SCREAMING_SNAKE_CASE__)

  __snake_case:		List[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:		Tuple					    =		[[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:		Union[str, Any]					    =		[[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:		List[str]					    =		i + interval_length - 1

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

        if dp[i][j] > cost:
          __snake_case:		Tuple					    =		cost
          __snake_case:		int					    =		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      A__				(       )  ->       Dict:
  # A sample binary search tree
  __snake_case:		Optional[Any]					    =		[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


__UpperCAmelCase			:  List[str]										=				{"configuration_levit": ["LEVIT_PRETRAINED_CONFIG_ARCHIVE_MAP", "LevitConfig", "LevitOnnxConfig"]}

try:
      if not is_vision_available():
            raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
      pass
else:
      __UpperCAmelCase			:  Any										=				["LevitFeatureExtractor"]
      __UpperCAmelCase			:  Tuple										=				["LevitImageProcessor"]

try:
      if not is_torch_available():
            raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
      pass
else:
      __UpperCAmelCase			:  str										=				[
          "LEVIT_PRETRAINED_MODEL_ARCHIVE_LIST",
          "LevitForImageClassification",
          "LevitForImageClassificationWithTeacher",
          "LevitModel",
          "LevitPreTrainedModel",
      ]


if TYPE_CHECKING:
      from .configuration_levit import LEVIT_PRETRAINED_CONFIG_ARCHIVE_MAP, LevitConfig, LevitOnnxConfig

      try:
            if not is_vision_available():
                  raise OptionalDependencyNotAvailable()
      except OptionalDependencyNotAvailable:
            pass
      else:
            from .feature_extraction_levit import LevitFeatureExtractor
            from .image_processing_levit import LevitImageProcessor

      try:
            if not is_torch_available():
                  raise OptionalDependencyNotAvailable()
      except OptionalDependencyNotAvailable:
            pass
      else:
            from .modeling_levit import (
                LEVIT_PRETRAINED_MODEL_ARCHIVE_LIST,
                LevitForImageClassification,
                LevitForImageClassificationWithTeacher,
                LevitModel,
                LevitPreTrainedModel,
            )
else:
      import sys

      __UpperCAmelCase			:  Union[str, Any]										=				_LazyModule(__name__, globals()["__file__"], _import_structure, module_spec=__spec__)






