code: 
  
  
  from __future__ import annotations
  
  from abc import abstractmethod
  from collections.abc import Iterable
  from typing import Generic, Protocol, TypeVar
  
  
  
  
  
  
  
  class 					__lowercase					(       a_     ):
  
  
  
  		"""simple docstring"""
  
  		@abstractmethod
  		def __lt__(   self		,			A  )     ->   bool:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					pass
  
  
  
  
  
  
  
  		@abstractmethod
  		def __gt__(   self		,			A  )     ->   bool:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					pass
  
  
  
  
  
  
  
  		@abstractmethod
  		def __eq__(   self		,			A  )     ->   bool:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					pass
  
  
  
  
  
  
  UpperCAmelCase      :						List[str]    =		TypeVar("T", bound=Comparable)
  
  
  
  
  
  
  
  class 					__lowercase					(       Generic[T]     ):
  
  
  
  		"""simple docstring"""
  
  		def __init__(   self  )     ->   None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					lowerCamelCase								=     []
  					lowerCamelCase								=     0
  
  
  
  
  
  
  
  		def __repr__(   self  )     ->   str:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					return str(self.h  )
  
  
  
  
  
  
  
  		def 				__A      (   self		,			A  )     ->   int | None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					if child_idx > 0:
  								return (child_idx - 1) // 2
  					return None
  
  
  
  
  
  
  
  		def 				__A      (   self		,			A  )     ->   int | None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					lowerCamelCase								=     2 * parent_idx + 1
  					if left_child_index < self.heap_size:
  								return left_child_index
  					return None
  
  
  
  
  
  
  
  		def 				__A      (   self		,			A  )     ->   int | None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					lowerCamelCase								=     2 * parent_idx + 2
  					if right_child_index < self.heap_size:
  								return right_child_index
  					return None
  
  
  
  
  
  
  
  		def 				__A      (   self		,			A  )     ->   None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					if index < self.heap_size:
  								lowerCamelCase								=     index
  								lowerCamelCase								=     self.left_child_idx(A  )
  								lowerCamelCase								=     self.right_child_idx(A  )
  								# check which child is larger than its parent
  								if left_child is not None and self.h[left_child] > self.h[violation]:
  											lowerCamelCase								=     left_child
  								if right_child is not None and self.h[right_child] > self.h[violation]:
  											lowerCamelCase								=     right_child
  								# if violation indeed exists
  								if violation != index:
  											# swap to fix the violation
  											lowerCamelCase		,		lowerCamelCase								=     self.h[index], self.h[violation]
  											# fix the subsequent violation recursively if any
  											self.max_heapify(A  )
  
  
  
  
  
  
  
  		def 				__A      (   self		,			A  )     ->   None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					lowerCamelCase								=     list(A  )
  					lowerCamelCase								=     len(self.h  )
  					if self.heap_size > 1:
  								# max_heapify from right to left but exclude leaves (last level)
  								for i in range(self.heap_size // 2 - 1		,			-1		,			-1  ):
  											self.max_heapify(A  )
  
  
  
  
  
  
  
  		def 				__A      (   self  )     ->   T:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					if self.heap_size >= 2:
  								lowerCamelCase								=     self.h[0]
  								lowerCamelCase								=     self.h.pop(-1  )
  								self.heap_size -= 1
  								self.max_heapify(0  )
  								return me
  					elif self.heap_size == 1:
  								self.heap_size -= 1
  								return self.h.pop(-1  )
  					else:
  								raise Exception("""Empty heap"""  )
  
  
  
  
  
  
  
  		def 				__A      (   self		,			A  )     ->   None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					self.h.append(A  )
  					lowerCamelCase								=     (self.heap_size - 1) // 2
  					self.heap_size += 1
  					while idx >= 0:
  								self.max_heapify(A  )
  								lowerCamelCase								=     (idx - 1) // 2
  
  
  
  
  
  
  
  		def 				__A      (   self  )     ->   None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					lowerCamelCase								=     self.heap_size
  					for j in range(size - 1		,			0		,			-1  ):
  								lowerCamelCase		,		lowerCamelCase								=     self.h[j], self.h[0]
  								self.heap_size -= 1
  								self.max_heapify(0  )
  					lowerCamelCase								=     size
  
  
  if __name__ == "__main__":
  		import doctest
  
  		# run doc test
  		doctest.testmod()
  
  		# demo
  		for unsorted in [
  		    [0],
  		    [2],
  		    [3, 5],
  		    [5, 3],
  		    [5, 5],
  		    [0, 0, 0, 0],
  		    [1, 1, 1, 1],
  		    [2, 2, 3, 5],
  		    [0, 2, 2, 3, 5],
  		    [2, 5, 3, 0, 2, 3, 0, 3],
  		    [6, 1, 2, 7, 9, 3, 4, 5, 10, 8],
  		    [1_03, 9, 1, 7, 11, 15, 25, 2_01, 2_09, 1_07, 5],
  		    [-45, -2, -5],
  		]:
  				print(f"""unsorted array: {unsorted}""")
  
  				UpperCAmelCase      :						Heap[int]    =		Heap()
  				heap.build_max_heap(unsorted)
  				print(f"""after build heap: {heap}""")
  
  				print(f"""max value: {heap.extract_max()}""")
  				print(f"""after max value removed: {heap}""")
  
  				heap.insert(1_00)
  				print(f"""after new value 100 inserted: {heap}""")
  
  				heap.heap_sort()
  				print(f"""heap-sorted array: {heap}\n""")
  
  
style_context: 
  
  
  from __future__ import annotations
  
  from collections.abc import Callable
  from typing import Generic, TypeVar
  
  UpperCAmelCase      :						List[str]    =		TypeVar("T")
  UpperCAmelCase      :						Dict    =		TypeVar("U")
  
  
  
  
  
  
  
  class 					__lowercase					(       Generic[T, U]     ):
  
  
  
  		"""simple docstring"""
  
  		def __init__(   self		,			A		,			A  )     ->   Any:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					lowerCamelCase								=     key
  					lowerCamelCase								=     val
  					lowerCamelCase								=     0
  					lowerCamelCase								=     None
  					lowerCamelCase								=     None
  
  
  
  
  
  
  
  		def __repr__(   self  )     ->   str:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					return "Node: key: {}, val: {}, freq: {}, has next: {}, has prev: {}".format(
  					    self.key		,			self.val		,			self.freq		,			self.next is not None		,			self.prev is not None  )
  
  
  
  
  
  
  
  class 					__lowercase					(       Generic[T, U]     ):
  
  
  
  		"""simple docstring"""
  
  		def __init__(   self  )     ->   None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					lowerCamelCase								=     DoubleLinkedListNode(A		,			A  )
  					lowerCamelCase								=     DoubleLinkedListNode(A		,			A  )
  					lowerCamelCase		,		lowerCamelCase								=     self.rear, self.head
  
  
  
  
  
  
  
  		def __repr__(   self  )     ->   str:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					lowerCamelCase								=     ["""DoubleLinkedList"""]
  					lowerCamelCase								=     self.head
  					while node.next is not None:
  								rep.append(str(A  )  )
  								lowerCamelCase								=     node.next
  					rep.append(str(self.rear  )  )
  					return ",\n    ".join(A  )
  
  
  
  
  
  
  
  		def 				__A      (   self		,			A  )     ->   None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					lowerCamelCase								=     self.rear.prev
  
  					# All nodes other than self.head are guaranteed to have non-None previous
  					assert previous is not None
  
  					lowerCamelCase								=     node
  					lowerCamelCase								=     previous
  					lowerCamelCase								=     node
  					lowerCamelCase								=     self.rear
  					node.freq += 1
  					self._position_node(A  )
  
  
  
  
  
  
  
  		def 				__A      (   self		,			A  )     ->   None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					while node.prev is not None and node.prev.freq > node.freq:
  								# swap node with previous node
  								lowerCamelCase								=     node.prev
  
  								lowerCamelCase								=     previous_node.prev
  								lowerCamelCase								=     node.prev
  								lowerCamelCase								=     previous_node
  								lowerCamelCase								=     node
  
  
  
  
  
  
  
  		def 				__A      (   self		,			A  )     ->   DoubleLinkedListNode[T, U] | None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					if node.prev is None or node.next is None:
  								return None
  
  					lowerCamelCase								=     node.next
  					lowerCamelCase								=     node.prev
  					lowerCamelCase								=     None
  					lowerCamelCase								=     None
  					return node
  
  
  
  
  
  
  
  class 					__lowercase					(       Generic[T, U]     ):
  
  
  
  		"""simple docstring"""
  
  		UpperCamelCase    :				dict[Callable[[T], U], LFUCache[T, U]] 					=		{}
  
  
  
  
  
  
  
  		def __init__(   self		,			A  )     ->   Union[str, Any]:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					lowerCamelCase								=     DoubleLinkedList()
  					lowerCamelCase								=     capacity
  					lowerCamelCase								=     0
  					lowerCamelCase								=     0
  					lowerCamelCase								=     0
  					lowerCamelCase								=     {}
  
  
  
  
  
  
  
  		def __repr__(   self  )     ->   str:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					return (
  					    F'CacheInfo(hits={self.hits}, misses={self.miss}, '
  					    F'capacity={self.capacity}, current_size={self.num_keys})'
  					)
  
  
  
  
  
  
  
  		def __contains__(   self		,			A  )     ->   bool:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					return key in self.cache
  
  
  
  
  
  
  
  		def 				__A      (   self		,			A  )     ->   U | None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					if key in self.cache:
  								self.hits += 1
  								lowerCamelCase								=     self.cache[key]
  								lowerCamelCase								=     self.list.remove(self.cache[key]  )
  								assert node == value_node
  
  								# node is guaranteed not None because it is in self.cache
  								assert node is not None
  								self.list.add(A  )
  								return node.val
  					self.miss += 1
  					return None
  
  
  
  
  
  
  
  		def 				__A      (   self		,			A		,			A  )     ->   None:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					if key not in self.cache:
  								if self.num_keys >= self.capacity:
  											# delete first node when over capacity
  											lowerCamelCase								=     self.list.head.next
  
  											# guaranteed to have a non-None first node when num_keys > 0
  											# explain to type checker via assertions
  											assert first_node is not None
  											assert first_node.key is not None
  											assert self.list.remove(A  ) is not None
  											# first_node guaranteed to be in list
  
  											del self.cache[first_node.key]
  											self.num_keys -= 1
  								lowerCamelCase								=     DoubleLinkedListNode(A		,			A  )
  								self.list.add(self.cache[key]  )
  								self.num_keys += 1
  
  					else:
  								lowerCamelCase								=     self.list.remove(self.cache[key]  )
  								assert node is not None  # node guaranteed to be in list
  								lowerCamelCase								=     value
  								self.list.add(A  )
  
  
  
  
  
  
  
  		@classmethod
  		def 				__A      (   cls		,			A = 1_28  )     ->   Callable[[Callable[[T], U]], Callable[..., U]]:
  
  
  
  
  
  
  
  					'''simple docstring'''
  
  
  
  
  
  
  
  					def cache_decorator_inner(A  ) -> Callable[..., U]:
  								def cache_decorator_wrapper(*A  ) -> U:
  											if func not in cls.decorator_function_to_instance_map:
  														lowerCamelCase								=     LFUCache(A  )
  
  											lowerCamelCase								=     cls.decorator_function_to_instance_map[func].get(args[0]  )
  											if result is None:
  														lowerCamelCase								=     func(*A  )
  														cls.decorator_function_to_instance_map[func].put(args[0]		,			A  )
  											return result
  
  								def cache_info() -> LFUCache[T, U]:
  											return cls.decorator_function_to_instance_map[func]
  
  								setattr(A		,			"""cache_info"""		,			A  )  # noqa: B010
  
  								return cache_decorator_wrapper
  
  					return cache_decorator_inner
  
  
  if __name__ == "__main__":
  		import doctest
  
  		doctest.testmod()
  
  