"""simple docstring"""







import sys
from random import randint


class       _lowercase		:

				def __init__(  self							:     List[str]							,     _lowercase							:     Optional[int]							,     _lowercase							:     Tuple  ):


										'''simple docstring'''
										UpperCAmelCase__       						=		key
										UpperCAmelCase__       						=		freq

				def __str__(  self							:     int  ):


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


def 	a__   (		UpperCAmelCase__							:      str		,      UpperCAmelCase__							:      List[str]		,      UpperCAmelCase__							:      Dict		,      UpperCAmelCase__							:      List[Any]		,      UpperCAmelCase__							:      int		,      UpperCAmelCase__							:      Optional[int]      )->						str:





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

						UpperCAmelCase__       						=		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 	a__   (		UpperCAmelCase__							:      Optional[int]      )->						Union[str, Any]:
						nodes.sort(key=lambda UpperCAmelCase__      : node.key      )

						UpperCAmelCase__       						=		len(lowerCamelCase_      )

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

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

																								if dp[i][j] > cost:
																														UpperCAmelCase__       						=		cost
																														UpperCAmelCase__       						=		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 	a__   (		)->						Optional[Any]:
						UpperCAmelCase__       						=		[Node(lowerCamelCase_		,      randint(1		,      5_0      )      ) for i in range(1_0		,      0		,      -1      )]
						find_optimal_binary_search_tree(lowerCamelCase_      )


if __name__ == "__main__":
						main()


'''simple docstring'''


from __future__ import annotations

snake_case__	:     str      =    [2, 1, 4, 5, 6, 0, 8, 9, 1, 2, 0, 6, 4, 2, 0, 6, 5, 3, 2, 7]






class 	__SCREAMING_SNAKE_CASE		:







  '''simple docstring'''






  def __init__(					self ,						snake_case_		):




       '''simple docstring'''

       UpperCAmelCase_				:  int  = -1
       UpperCAmelCase_				:  int  = -1
       UpperCAmelCase_				:  list[int]  = [-1] * length
       UpperCAmelCase_				:  Node | None  = None
       UpperCAmelCase_				:  Node | None  = None






  def __repr__(					self		):




       '''simple docstring'''

       return F'''Node(min_value={self.minn} max_value={self.maxx})'''







def    _lowerCamelCase       ( lowerCamelCase_   :				list[int]  ):



     """simple docstring"""





     UpperCAmelCase_				:  List[str]  = Node(len(lowerCamelCase_  )  )
     UpperCAmelCase_  ,     UpperCAmelCase_				:  Any  = min(lowerCamelCase_  ), max(lowerCamelCase_  )
     # Leaf node case where the node contains only one unique value
     if root.minn == root.maxx:
          return root
     UpperCAmelCase_				:  str  = (root.minn + root.maxx) // 2

     UpperCAmelCase_				:  list[int]  = []
     UpperCAmelCase_				:  list[int]  = []

     for index, num in enumerate(lowerCamelCase_  ):
          if num <= pivot:
               left_arr.append(lowerCamelCase_  )
          else:
               right_arr.append(lowerCamelCase_  )
          UpperCAmelCase_				:  Tuple  = len(lowerCamelCase_  )
     UpperCAmelCase_				:  Optional[Any]  = build_tree(lowerCamelCase_  )
     UpperCAmelCase_				:  Optional[Any]  = build_tree(lowerCamelCase_  )
     return root


def    _lowerCamelCase       ( lowerCamelCase_   :				Node | None      ,	lowerCamelCase_   :				int      ,	lowerCamelCase_   :				int  ):



     """simple docstring"""





     if index < 0 or node is None:
          return 0
     # Leaf node cases
     if node.minn == node.maxx:
          return index + 1 if node.minn == num else 0
     UpperCAmelCase_				:  Any  = (node.minn + node.maxx) // 2
     if num <= pivot:
          # go the left subtree and map index to the left subtree
          return rank_till_index(node.left      ,	lowerCamelCase_      ,	node.map_left[index] - 1  )
     else:
          # go to the right subtree and map index to the right subtree
          return rank_till_index(node.right      ,	lowerCamelCase_      ,	index - node.map_left[index]  )


def    _lowerCamelCase       ( lowerCamelCase_   :				Node | None      ,	lowerCamelCase_   :				int      ,	lowerCamelCase_   :				int      ,	lowerCamelCase_   :				int  ):



     """simple docstring"""





     if start > end:
          return 0
     UpperCAmelCase_				:  int  = rank_till_index(lowerCamelCase_      ,	lowerCamelCase_      ,	lowerCamelCase_  )
     UpperCAmelCase_				:  List[Any]  = rank_till_index(lowerCamelCase_      ,	lowerCamelCase_      ,	start - 1  )
     return rank_till_end - rank_before_start


def    _lowerCamelCase       ( lowerCamelCase_   :				Node | None      ,	lowerCamelCase_   :				int      ,	lowerCamelCase_   :				int      ,	lowerCamelCase_   :				int  ):



     """simple docstring"""





     if index > (end - start) or start > end or node is None:
          return -1
     # Leaf node case
     if node.minn == node.maxx:
          return node.minn
     # Number of elements in the left subtree in interval [start, end]
     UpperCAmelCase_				:  Tuple  = node.map_left[end] - (
         node.map_left[start - 1] if start else 0
     )
     if num_elements_in_left_tree > index:
          return quantile(
              node.left      ,	lowerCamelCase_      ,	(node.map_left[start - 1] if start else 0)      ,	node.map_left[end] - 1      ,	)
     else:
          return quantile(
              node.right      ,	index - num_elements_in_left_tree      ,	start - (node.map_left[start - 1] if start else 0)      ,	end - node.map_left[end]      ,	)


def    _lowerCamelCase       ( lowerCamelCase_   :				Node | None      ,	lowerCamelCase_   :				int      ,	lowerCamelCase_   :				int      ,	lowerCamelCase_   :				int      ,	lowerCamelCase_   :				int  ):



     """simple docstring"""





     if (
         start > end
         or node is None
         or start_num > end_num
         or node.minn > end_num
         or node.maxx < start_num
     ):
          return 0
     if start_num <= node.minn and node.maxx <= end_num:
          return end - start + 1
     UpperCAmelCase_				:  Any  = range_counting(
         node.left      ,	(node.map_left[start - 1] if start else 0)      ,	node.map_left[end] - 1      ,	lowerCamelCase_      ,	lowerCamelCase_      ,	)
     UpperCAmelCase_				:  List[str]  = range_counting(
         node.right      ,	start - (node.map_left[start - 1] if start else 0)      ,	end - node.map_left[end]      ,	lowerCamelCase_      ,	lowerCamelCase_      ,	)
     return left + right


if __name__ == "__main__":
 import doctest

 doctest.testmod()






