# 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()






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()





