





import sys
from random import randint






class 							__snake_case    :





	def __init__(		self			,					UpperCamelCase_			,					UpperCamelCase_		)			->		Optional[Any]:
		snake_case__  					=      key
		snake_case__  					=      freq





	def __str__(		self		)			->		str:
		return F'''Node(key={self.key}, freq={self.freq})'''





def 	__lowerCamelCase (  UpperCAmelCase_				,							UpperCAmelCase_				,							UpperCAmelCase_				,							UpperCAmelCase_				,							UpperCAmelCase_				,							UpperCAmelCase_       )      ->Any:
	if i > j or i < 0 or j > len(UpperCAmelCase_       ) - 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(UpperCAmelCase_				,							UpperCAmelCase_				,							UpperCAmelCase_				,							node - 1				,							key[node]				,							UpperCAmelCase_       )
	print_binary_search_tree(UpperCAmelCase_				,							UpperCAmelCase_				,							node + 1				,							UpperCAmelCase_				,							key[node]				,							UpperCAmelCase_       )




def 	__lowerCamelCase (  UpperCAmelCase_       )      ->Union[str, Any]:
	nodes.sort(key=lambda UpperCAmelCase_       : node.key       )

	snake_case__  					=      len(UpperCAmelCase_       )

	snake_case__  					=      [nodes[i].key for i in range(UpperCAmelCase_       )]
	snake_case__  					=      [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.
	snake_case__  					=      [[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
	snake_case__  					=      [[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
	snake_case__  					=      [[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       ):
			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(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 	__lowerCamelCase (  )      ->Union[str, Any]:
	# A sample binary search tree
	snake_case__  					=      [Node(UpperCAmelCase_				,							randint(1				,							50       )       ) for i in range(10				,							0				,							-1       )]
	find_optimal_binary_search_tree(UpperCAmelCase_       )


if __name__ == "__main__":
			main()










'''simple docstring'''




a__				: Any									=    {
    0: '''underwear''',
    1: '''pants''',
    2: '''belt''',
    3: '''suit''',
    4: '''shoe''',
    5: '''socks''',
    6: '''shirt''',
    7: '''tie''',
    8: '''watch''',
}

a__				: Optional[Any]									=    [[1, 4], [2, 4], [3], [], [], [4], [2, 7], [3], []]

a__				: List[str]									=    [0 for x in range(len(graph))]
a__				: List[Any]									=    []




def 	__lowerCamelCase (  UpperCAmelCase_				,							UpperCAmelCase_       )      ->Dict:
	snake_case__  					=      1
	while stack:
		snake_case__  					=      stack.pop()
		print(UpperCAmelCase_				,							clothes[current_clothing]       )
		order += 1




def 	__lowerCamelCase (  UpperCAmelCase_				,							UpperCAmelCase_				,							UpperCAmelCase_       )      ->Dict:
	snake_case__  					=      1
	for v in graph[u]:
		if not visited[v]:
			depth_first_search(UpperCAmelCase_				,							UpperCAmelCase_				,							UpperCAmelCase_       )

	stack.append(UpperCAmelCase_       )




def 	__lowerCamelCase (  UpperCAmelCase_				,							UpperCAmelCase_       )      ->Union[str, Any]:
	for v in range(len(UpperCAmelCase_       )       ):
		if not visited[v]:
			depth_first_search(UpperCAmelCase_				,							UpperCAmelCase_				,							UpperCAmelCase_       )


if __name__ == "__main__":
			topological_sort(graph, visited)
			print(stack)
			print_stack(stack, clothes)



