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 					UpperCamelCase	:
  
  
  
  
       def __init__( self :					List[str]      ,_lowerCAmelCase :					Optional[int]      ,_lowerCAmelCase :					str						):
  
  
         """simple docstring"""
  
  
  
  
         __snake_case		       =     key
         __snake_case		       =     freq
  
  
  
  
  
  
  
       def __str__( self :					Union[str, Any]						):
  
  
         """simple docstring"""
  
  
  
  
         return F"""Node(key={self.key}, freq={self.freq})"""
  
  
  
  
  def 							_lowerCamelCase(				__snake_case						,						__snake_case						,						__snake_case						,						__snake_case						,						__snake_case						,						__snake_case	)  ->   List[str]:
    if i > j or i < 0 or j > len(lowerCAmelCase__	) - 1:
      return
  
    __snake_case		       =     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(lowerCAmelCase__						,						lowerCAmelCase__						,						lowerCAmelCase__						,						node - 1						,						key[node]						,						lowerCAmelCase__	)
    print_binary_search_tree(lowerCAmelCase__						,						lowerCAmelCase__						,						node + 1						,						lowerCAmelCase__						,						key[node]						,						lowerCAmelCase__	)
  
  
  
  def 							_lowerCamelCase(				__snake_case	)  ->   List[str]:
    nodes.sort(key=lambda __snake_case	: node.key	)
  
    __snake_case		       =     len(lowerCAmelCase__	)
  
    __snake_case		       =     [nodes[i].key for i in range(lowerCAmelCase__	)]
    __snake_case		       =     [nodes[i].freq for i in range(lowerCAmelCase__	)]
  
    # 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		       =     [[freqs[i] if i == j else 0 for j in range(lowerCAmelCase__	)] for i in range(lowerCAmelCase__	)]
    # sum[i][j] stores the sum of key frequencies between i and j inclusive in nodes
    # array
    __snake_case		       =     [[freqs[i] if i == j else 0 for j in range(lowerCAmelCase__	)] for i in range(lowerCAmelCase__	)]
    # stores tree roots that will be used later for constructing binary search tree
    __snake_case		       =     [[i if i == j else 0 for j in range(lowerCAmelCase__	)] for i in range(lowerCAmelCase__	)]
  
    for interval_length in range(2						,						n + 1	):
      for i in range(n - interval_length + 1	):
        __snake_case		       =     i + interval_length - 1
  
        __snake_case		       =     sys.maxsize  # set the value to "infinity"
        __snake_case		       =     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		       =     dp[i][r - 1] if r != i else 0  # optimal cost for left subtree
          __snake_case		       =     dp[r + 1][j] if r != j else 0  # optimal cost for right subtree
          __snake_case		       =     left + total[i][j] + right
  
          if dp[i][j] > cost:
            __snake_case		       =     cost
            __snake_case		       =     r
  
    print("Binary search tree nodes:"	)
    for node in nodes:
      print(lowerCAmelCase__	)
  
    print(f"""\nThe cost of optimal BST for given tree nodes is {dp[0][n - 1]}."""	)
    print_binary_search_tree(lowerCAmelCase__						,						lowerCAmelCase__						,						0						,						n - 1						,						-1						,						lowerCAmelCase__	)
  
  
  
  def 							_lowerCamelCase(				)  ->   Union[str, Any]:
    # A sample binary search tree
    __snake_case		       =     [Node(lowerCAmelCase__						,						randint(1						,						50	)	) for i in range(10						,						0						,						-1	)]
    find_optimal_binary_search_tree(lowerCAmelCase__	)
  
  
  if __name__ == "__main__":
    main()
  
  
  
  
  
  
style_context: import posixpath
  from pathlib import Path
  from unittest.mock import patch
  
  import pytest
  from fsspec.implementations.local import AbstractFileSystem, LocalFileSystem, stringify_path
  from fsspec.registry import _registry as _fsspec_registry
  
  
  
  
  class 					UpperCamelCase	(     snake_case__      ):
  					__UpperCamelCase						=   """mock"""
  
  
  
  
  					def __init__( self :					List[Any]      ,*_lowerCAmelCase :					Optional[int]      ,_lowerCAmelCase :					Union[str, Any]      ,**_lowerCAmelCase :					Union[str, Any]						):
  
  
  							"""simple docstring"""
  
  
  
  
  							super().__init__()
  							__snake_case		       =     LocalFileSystem(*_lowerCAmelCase      ,**_lowerCAmelCase						)
  							__snake_case		       =     Path(_lowerCAmelCase						).resolve().as_posix() + "/"
  
  
  
  
  					def 							UpperCamelCase_						( self :					List[Any]      ,_lowerCAmelCase :					Optional[Any]      ,*_lowerCAmelCase :					Union[str, Any]      ,**_lowerCAmelCase :					Dict						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     posixpath.join(self.local_root_dir      ,self._strip_protocol(_lowerCAmelCase						)						)
  							return self._fs.mkdir(_lowerCAmelCase      ,*_lowerCAmelCase      ,**_lowerCAmelCase						)
  
  
  
  
  					def 							UpperCamelCase_						( self :					int      ,_lowerCAmelCase :					Tuple      ,*_lowerCAmelCase :					Optional[Any]      ,**_lowerCAmelCase :					Optional[Any]						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     posixpath.join(self.local_root_dir      ,self._strip_protocol(_lowerCAmelCase						)						)
  							return self._fs.makedirs(_lowerCAmelCase      ,*_lowerCAmelCase      ,**_lowerCAmelCase						)
  
  
  
  
  					def 							UpperCamelCase_						( self :					Tuple      ,_lowerCAmelCase :					Any						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     posixpath.join(self.local_root_dir      ,self._strip_protocol(_lowerCAmelCase						)						)
  							return self._fs.rmdir(_lowerCAmelCase						)
  
  
  
  
  					def 							UpperCamelCase_						( self :					List[Any]      ,_lowerCAmelCase :					List[Any]      ,_lowerCAmelCase :					str=True      ,*_lowerCAmelCase :					int      ,**_lowerCAmelCase :					List[str]						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     posixpath.join(self.local_root_dir      ,self._strip_protocol(_lowerCAmelCase						)						)
  							__snake_case		       =     self._fs.ls(_lowerCAmelCase      ,detail=_lowerCAmelCase      ,*_lowerCAmelCase      ,**_lowerCAmelCase						)
  							if detail:
  									return [{**info, "name": info["name"][len(self.local_root_dir						) :]} for info in out]
  							else:
  									return [name[len(self.local_root_dir						) :] for name in out]
  
  
  
  
  					def 							UpperCamelCase_						( self :					List[str]      ,_lowerCAmelCase :					Any      ,*_lowerCAmelCase :					Any      ,**_lowerCAmelCase :					str						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     posixpath.join(self.local_root_dir      ,self._strip_protocol(_lowerCAmelCase						)						)
  							__snake_case		       =     dict(self._fs.info(_lowerCAmelCase      ,*_lowerCAmelCase      ,**_lowerCAmelCase						)						)
  							__snake_case		       =     out["name"][len(self.local_root_dir						) :]
  							return out
  
  
  
  
  					def 							UpperCamelCase_						( self :					Tuple      ,_lowerCAmelCase :					Optional[int]      ,_lowerCAmelCase :					Optional[Any]      ,*_lowerCAmelCase :					int      ,**_lowerCAmelCase :					Optional[int]						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     posixpath.join(self.local_root_dir      ,self._strip_protocol(_lowerCAmelCase						)						)
  							__snake_case		       =     posixpath.join(self.local_root_dir      ,self._strip_protocol(_lowerCAmelCase						)						)
  							return self._fs.cp_file(_lowerCAmelCase      ,_lowerCAmelCase      ,*_lowerCAmelCase      ,**_lowerCAmelCase						)
  
  
  
  
  					def 							UpperCamelCase_						( self :					Union[str, Any]      ,_lowerCAmelCase :					int      ,*_lowerCAmelCase :					List[Any]      ,**_lowerCAmelCase :					int						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     posixpath.join(self.local_root_dir      ,self._strip_protocol(_lowerCAmelCase						)						)
  							return self._fs.rm_file(_lowerCAmelCase      ,*_lowerCAmelCase      ,**_lowerCAmelCase						)
  
  
  
  
  					def 							UpperCamelCase_						( self :					Optional[Any]      ,_lowerCAmelCase :					str      ,*_lowerCAmelCase :					Optional[int]      ,**_lowerCAmelCase :					int						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     posixpath.join(self.local_root_dir      ,self._strip_protocol(_lowerCAmelCase						)						)
  							return self._fs.rm(_lowerCAmelCase      ,*_lowerCAmelCase      ,**_lowerCAmelCase						)
  
  
  
  
  					def 							UpperCamelCase_						( self :					int      ,_lowerCAmelCase :					Optional[int]      ,*_lowerCAmelCase :					str      ,**_lowerCAmelCase :					Optional[Any]						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     posixpath.join(self.local_root_dir      ,self._strip_protocol(_lowerCAmelCase						)						)
  							return self._fs._open(_lowerCAmelCase      ,*_lowerCAmelCase      ,**_lowerCAmelCase						)
  
  
  
  
  					def 							UpperCamelCase_						( self :					Union[str, Any]      ,_lowerCAmelCase :					Tuple						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     posixpath.join(self.local_root_dir      ,self._strip_protocol(_lowerCAmelCase						)						)
  							return self._fs.created(_lowerCAmelCase						)
  
  
  
  
  					def 							UpperCamelCase_						( self :					Union[str, Any]      ,_lowerCAmelCase :					int						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     posixpath.join(self.local_root_dir      ,self._strip_protocol(_lowerCAmelCase						)						)
  							return self._fs.modified(_lowerCAmelCase						)
  
  
  
  
  
  
  
  					@classmethod
  					def 							UpperCamelCase_						( cls :					Dict      ,_lowerCAmelCase :					Dict						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     stringify_path(_lowerCAmelCase						)
  							if path.startswith("mock://"						):
  									__snake_case		       =     path[7:]
  							return path
  
  
  
  
  class 					UpperCamelCase	(     snake_case__      ):
  					__UpperCamelCase						=   """tmp"""
  					__UpperCamelCase						=   None
  
  
  
  
  					def __init__( self :					int      ,*_lowerCAmelCase :					Tuple      ,**_lowerCAmelCase :					Union[str, Any]						):
  
  
  							"""simple docstring"""
  
  
  
  
  							assert self.tmp_dir is not None, "TmpDirFileSystem.tmp_dir is not set"
  							super().__init__(*_lowerCAmelCase      ,**_lowerCAmelCase      ,local_root_dir=self.tmp_dir      ,auto_mkdir=_lowerCAmelCase						)
  
  
  
  
  
  
  
  					@classmethod
  					def 							UpperCamelCase_						( cls :					int      ,_lowerCAmelCase :					str						):
  
  
  							"""simple docstring"""
  
  
  
  
  							__snake_case		       =     stringify_path(_lowerCAmelCase						)
  							if path.startswith("tmp://"						):
  									__snake_case		       =     path[6:]
  							return path
  
  
  
  
  @pytest.fixture
  def 							_lowerCamelCase(				)  ->   str:
  		__snake_case		       =     MockFileSystem
  		__snake_case		       =     TmpDirFileSystem
  		yield
  		del _fsspec_registry["mock"]
  		del _fsspec_registry["tmp"]
  
  
  
  @pytest.fixture
  def 							_lowerCamelCase(				__snake_case						,						__snake_case	)  ->   Dict:
  		__snake_case		       =     tmp_path_factory.mktemp("mockfs"	)
  		return MockFileSystem(local_root_dir=__snake_case						,						auto_mkdir=__snake_case	)
  
  
  
  
  
  @pytest.fixture
  def 							_lowerCamelCase(				__snake_case						,						__snake_case	)  ->   List[Any]:
  		__snake_case		       =     tmp_path_factory.mktemp("tmpfs"	)
  		with patch.object(__snake_case						,						"tmp_dir"						,						__snake_case	):
  				yield TmpDirFileSystem()
  
  
  
  
  
  