



# 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      :							Optional[Any]				, __lowerCAmelCase      :							List[Any]				, __lowerCAmelCase      :							Union[str, Any]				)							->					Optional[int]:


    """simple docstring"""






    a							=		key
    a							=		freq




   def __str__(   self      :							Optional[Any]				)							->					Dict:


    """simple docstring"""






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


def 						UpperCAmelCase__   (   UpperCAmelCase__			:Optional[int]			,       UpperCAmelCase__			:List[str]			,       UpperCAmelCase__			:int			,       UpperCAmelCase__			:Union[str, Any]			,       UpperCAmelCase__			:List[Any]			,       UpperCAmelCase__			:Optional[int]       ):






 '''simple docstring'''

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

 a							=		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 						UpperCAmelCase__   (   UpperCAmelCase__			:str       ):






 '''simple docstring'''

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

 a							=		len(UpperCAmelCase__       )

 a							=		[nodes[i].key for i in range(UpperCAmelCase__       )]
 a							=		[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.
 a							=		[[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
 a							=		[[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
 a							=		[[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       ):
   a							=		i + interval_length - 1

   a							=		sys.maxsize  # set the value to "infinity"
   a							=		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
    a							=		dp[i][r - 1] if r != i else 0  # optimal cost for left subtree
    a							=		dp[r + 1][j] if r != j else 0  # optimal cost for right subtree
    a							=		left + total[i][j] + right

    if dp[i][j] > cost:
     a							=		cost
     a							=		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 						UpperCAmelCase__   (   ):






 '''simple docstring'''

 a							=		[Node(UpperCAmelCase__			,       randint(1			,       50       )       ) for i in range(10			,       0			,       -1       )]
 find_optimal_binary_search_tree(UpperCAmelCase__       )


if __name__ == "__main__":
     main()





from typing import TYPE_CHECKING

from ...utils import (
    OptionalDependencyNotAvailable,
    _LazyModule,
    is_flax_available,
    is_tf_available,
    is_tokenizers_available,
    is_torch_available,
)


__UpperCamelCase							:							List[Any]    							=					{
    "configuration_mpnet": ["MPNET_PRETRAINED_CONFIG_ARCHIVE_MAP", "MPNetConfig"],
    "tokenization_mpnet": ["MPNetTokenizer"],
}

try:
       if not is_tokenizers_available():
              raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
       pass
else:
       __UpperCamelCase							:							Union[str, Any]    							=					["MPNetTokenizerFast"]

try:
       if not is_torch_available():
              raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
       pass
else:
       __UpperCamelCase							:							Union[str, Any]    							=					[
           "MPNET_PRETRAINED_MODEL_ARCHIVE_LIST",
           "MPNetForMaskedLM",
           "MPNetForMultipleChoice",
           "MPNetForQuestionAnswering",
           "MPNetForSequenceClassification",
           "MPNetForTokenClassification",
           "MPNetLayer",
           "MPNetModel",
           "MPNetPreTrainedModel",
       ]

try:
       if not is_tf_available():
              raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
       pass
else:
       __UpperCamelCase							:							List[Any]    							=					[
           "TF_MPNET_PRETRAINED_MODEL_ARCHIVE_LIST",
           "TFMPNetEmbeddings",
           "TFMPNetForMaskedLM",
           "TFMPNetForMultipleChoice",
           "TFMPNetForQuestionAnswering",
           "TFMPNetForSequenceClassification",
           "TFMPNetForTokenClassification",
           "TFMPNetMainLayer",
           "TFMPNetModel",
           "TFMPNetPreTrainedModel",
       ]


if TYPE_CHECKING:
       from .configuration_mpnet import MPNET_PRETRAINED_CONFIG_ARCHIVE_MAP, MPNetConfig
       from .tokenization_mpnet import MPNetTokenizer

       try:
              if not is_tokenizers_available():
                     raise OptionalDependencyNotAvailable()
       except OptionalDependencyNotAvailable:
              pass
       else:
              from .tokenization_mpnet_fast import MPNetTokenizerFast

       try:
              if not is_torch_available():
                     raise OptionalDependencyNotAvailable()
       except OptionalDependencyNotAvailable:
              pass
       else:
              from .modeling_mpnet import (
                  MPNET_PRETRAINED_MODEL_ARCHIVE_LIST,
                  MPNetForMaskedLM,
                  MPNetForMultipleChoice,
                  MPNetForQuestionAnswering,
                  MPNetForSequenceClassification,
                  MPNetForTokenClassification,
                  MPNetLayer,
                  MPNetModel,
                  MPNetPreTrainedModel,
              )

       try:
              if not is_tf_available():
                     raise OptionalDependencyNotAvailable()
       except OptionalDependencyNotAvailable:
              pass
       else:
              from .modeling_tf_mpnet import (
                  TF_MPNET_PRETRAINED_MODEL_ARCHIVE_LIST,
                  TFMPNetEmbeddings,
                  TFMPNetForMaskedLM,
                  TFMPNetForMultipleChoice,
                  TFMPNetForQuestionAnswering,
                  TFMPNetForSequenceClassification,
                  TFMPNetForTokenClassification,
                  TFMPNetMainLayer,
                  TFMPNetModel,
                  TFMPNetPreTrainedModel,
              )

else:
       import sys

       __UpperCamelCase							:							Tuple    							=					_LazyModule(__name__, globals()["__file__"], _import_structure, module_spec=__spec__)


