


"""simple docstring"""


import sys
from random import randint
class snake_case   :







		def __init__(       self   :							Optional[int]     , a__   :							str     , a__   :							Optional[Any]		)							->    int:





						'''simple docstring'''







						_A       	=      key
						_A       	=      freq







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





						'''simple docstring'''







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






def   a__			(					__lowercase  ,      __lowercase  ,      __lowercase  ,      __lowercase  ,      __lowercase  ,      __lowercase )						->						Union[str, Any]:






				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   a__			(					__lowercase )						->						Optional[int]:
				nodes.sort(key=lambda __lowercase : 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   a__			(					)						->						Dict:
				_A       	=      [Node(_UpperCAmelCase  ,      randint(1  ,      50 ) ) for i in range(10  ,      0  ,      -1 )]
				find_optimal_binary_search_tree(_UpperCAmelCase )


if __name__ == "__main__":
		main()





"""simple docstring"""

class 							lowercase   :







				def __init__(				self :       Union[str, Any]   ,  _lowerCamelCase :       int							):



					"""simple docstring"""

					A_	:					Optional[int]						 =		data
					A_	:					int
					A_	:					Node

def 		lowercase_   (		_UpperCAmelCase						):




	"""simple docstring"""






	A_	:					List[str]						 =		0
	A_	:					List[str]						 =		x

def 		lowercase_   (		_UpperCAmelCase	,   _UpperCAmelCase						):




	"""simple docstring"""






	A_ ,			A_	:					Any						 =		find_set(_UpperCAmelCase						), find_set(_UpperCAmelCase						)
	if x == y:
		return

	elif x.rank > y.rank:
		A_	:					Optional[int]						 =		x
	else:
		A_	:					Optional[int]						 =		y
		if x.rank == y.rank:
			y.rank += 1

def 		lowercase_   (		_UpperCAmelCase						):




	"""simple docstring"""






	if x != x.parent:
		A_	:					Union[str, Any]						 =		find_set(x.parent						)
	return x.parent

def 		lowercase_   (		_UpperCAmelCase						):




	"""simple docstring"""






	A_	:					str						 =		({0, 1, 2}, {3, 4, 5})
	for s in sets:
		if node.data in s:
			return s
	A_	:					Optional[int]						 =		f"""{node.data} is not in {sets}"""
	raise ValueError(_UpperCAmelCase						)

def 		lowercase_   (		):




	"""simple docstring"""






	A_	:					Dict						 =		[Node(_UpperCAmelCase						) for i in range(6						)]
	for v in vertex:
		make_set(_UpperCAmelCase						)

	union_set(vertex[0]	,   vertex[1]						)
	union_set(vertex[1]	,   vertex[2]						)
	union_set(vertex[3]	,   vertex[4]						)
	union_set(vertex[3]	,   vertex[5]						)

	for nodea in vertex:
		for nodea in vertex:
			if find_python_set(_UpperCAmelCase						).isdisjoint(find_python_set(_UpperCAmelCase						)						):
				assert find_set(_UpperCAmelCase						) != find_set(_UpperCAmelCase						)
			else:
				assert find_set(_UpperCAmelCase						) == find_set(_UpperCAmelCase						)


if __name__ == "__main__":
		test_disjoint_set()





