

from __future__ import annotations

from abc import abstractmethod
from collections.abc import Iterable
from typing import Generic, Protocol, TypeVar







class     a__    (				UpperCAmelCase    ):







  """simple docstring"""




  @abstractmethod
  def __lt__(     self      :	T			,			UpperCAmelCase__      :	T						)   ->bool:
        """simple docstring"""







        pass


  @abstractmethod
  def __gt__(     self      :	T			,			UpperCAmelCase__      :	T						)   ->bool:
        """simple docstring"""







        pass




  @abstractmethod
  def __eq__(     self      :	T			,			UpperCAmelCase__      :	object						)   ->bool:
        """simple docstring"""







        pass


UpperCAmelCase__	:       Tuple			      =      TypeVar("""T""", bound=Comparable)







class     a__    (				Generic[T]    ):







  """simple docstring"""




  def __init__(     self      :	Optional[Any]						)   ->None:
        """simple docstring"""







        SCREAMING_SNAKE_CASE	: list[T]					       =				[]
        SCREAMING_SNAKE_CASE	: int					       =				0


  def __repr__(     self      :	Any						)   ->str:
        """simple docstring"""







        return str(self.h						)


  def 	_lowercase (     self      :	Optional[int]			,			UpperCAmelCase__      :	int						)   ->int | None:
        """simple docstring"""







        if child_idx > 0:
              return (child_idx - 1) // 2
        return None


  def 	_lowercase (     self      :	List[str]			,			UpperCAmelCase__      :	int						)   ->int | None:
        """simple docstring"""







        SCREAMING_SNAKE_CASE	: str					       =				2 * parent_idx + 1
        if left_child_index < self.heap_size:
              return left_child_index
        return None


  def 	_lowercase (     self      :	str			,			UpperCAmelCase__      :	int						)   ->int | None:
        """simple docstring"""







        SCREAMING_SNAKE_CASE	: str					       =				2 * parent_idx + 2
        if right_child_index < self.heap_size:
              return right_child_index
        return None


  def 	_lowercase (     self      :	Optional[int]			,			UpperCAmelCase__      :	int						)   ->None:
        """simple docstring"""







        if index < self.heap_size:
              SCREAMING_SNAKE_CASE	: int					       =				index
              SCREAMING_SNAKE_CASE	: List[Any]					       =				self.left_child_idx(UpperCAmelCase__						)
              SCREAMING_SNAKE_CASE	: Optional[Any]					       =				self.right_child_idx(UpperCAmelCase__						)
              # check which child is larger than its parent
              if left_child is not None and self.h[left_child] > self.h[violation]:
                    SCREAMING_SNAKE_CASE	: Union[str, Any]					       =				left_child
              if right_child is not None and self.h[right_child] > self.h[violation]:
                    SCREAMING_SNAKE_CASE	: Optional[int]					       =				right_child
              # if violation indeed exists
              if violation != index:
                    # swap to fix the violation
                    SCREAMING_SNAKE_CASE       ,		SCREAMING_SNAKE_CASE	: Tuple					       =				self.h[index], self.h[violation]
                    # fix the subsequent violation recursively if any
                    self.max_heapify(UpperCAmelCase__						)


  def 	_lowercase (     self      :	int			,			UpperCAmelCase__      :	Iterable[T]						)   ->None:
        """simple docstring"""







        SCREAMING_SNAKE_CASE	: Dict					       =				list(UpperCAmelCase__						)
        SCREAMING_SNAKE_CASE	: str					       =				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(UpperCAmelCase__						)


  def 	_lowercase (     self      :	Tuple						)   ->T:
        """simple docstring"""







        if self.heap_size >= 2:
              SCREAMING_SNAKE_CASE	: List[str]					       =				self.h[0]
              SCREAMING_SNAKE_CASE	: str					       =				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 	_lowercase (     self      :	Tuple			,			UpperCAmelCase__      :	T						)   ->None:
        """simple docstring"""







        self.h.append(UpperCAmelCase__						)
        SCREAMING_SNAKE_CASE	: Dict					       =				(self.heap_size - 1) // 2
        self.heap_size += 1
        while idx >= 0:
              self.max_heapify(UpperCAmelCase__						)
              SCREAMING_SNAKE_CASE	: int					       =				(idx - 1) // 2




  def 	_lowercase (     self      :	List[Any]						)   ->None:
        """simple docstring"""







        SCREAMING_SNAKE_CASE	: Tuple					       =				self.heap_size
        for j in range(size - 1			,			0			,			-1						):
              SCREAMING_SNAKE_CASE       ,		SCREAMING_SNAKE_CASE	: List[Any]					       =				self.h[j], self.h[0]
              self.heap_size -= 1
              self.max_heapify(0						)
        SCREAMING_SNAKE_CASE	: Optional[int]					       =				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],
      [103, 9, 1, 7, 11, 15, 25, 201, 209, 107, 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(100)
    print(F"""after new value 100 inserted: {heap}""")

    heap.heap_sort()
    print(F"""heap-sorted array: {heap}\n""")







from __future__ import annotations

UpperCAmelCase__	:       Optional[int]			      =      [2, 1, 4, 5, 6, 0, 8, 9, 1, 2, 0, 6, 4, 2, 0, 6, 5, 3, 2, 7]







class     a__    :







  """simple docstring"""




  def __init__(     self      :	Optional[Any]			,			UpperCAmelCase__      :	int						)   ->None:
        """simple docstring"""







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




  def __repr__(     self      :	Dict						)   ->str:
        """simple docstring"""







        return f"Node(min_value={self.minn} max_value={self.maxx})"






def    __lowercase (       _A     )       ->  Node | None:
      SCREAMING_SNAKE_CASE	: Tuple					       =				Node(len(_A     )     )
      SCREAMING_SNAKE_CASE       ,		SCREAMING_SNAKE_CASE	: int					       =				min(_A     ), max(_A     )
      # Leaf node case where the node contains only one unique value
      if root.minn == root.maxx:
            return root
      SCREAMING_SNAKE_CASE	: int					       =				(root.minn + root.maxx) // 2

      SCREAMING_SNAKE_CASE	: list[int]					       =				[]
      SCREAMING_SNAKE_CASE	: list[int]					       =				[]

      for index, num in enumerate(_A     ):
            if num <= pivot:
                  left_arr.append(_A     )
            else:
                  right_arr.append(_A     )
            SCREAMING_SNAKE_CASE	: int					       =				len(_A     )
      SCREAMING_SNAKE_CASE	: int					       =				build_tree(_A     )
      SCREAMING_SNAKE_CASE	: Tuple					       =				build_tree(_A     )
      return root






def    __lowercase (       _A  ,      _A  ,      _A     )       ->  int:
      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
      SCREAMING_SNAKE_CASE	: List[str]					       =				(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  ,      _A  ,      node.map_left[index] - 1     )
      else:
            # go to the right subtree and map index to the right subtree
            return rank_till_index(node.right  ,      _A  ,      index - node.map_left[index]     )






def    __lowercase (       _A  ,      _A  ,      _A  ,      _A     )       ->  int:
      if start > end:
            return 0
      SCREAMING_SNAKE_CASE	: Any					       =				rank_till_index(_A  ,      _A  ,      _A     )
      SCREAMING_SNAKE_CASE	: Optional[int]					       =				rank_till_index(_A  ,      _A  ,      start - 1     )
      return rank_till_end - rank_before_start






def    __lowercase (       _A  ,      _A  ,      _A  ,      _A     )       ->  int:
      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]
      SCREAMING_SNAKE_CASE	: 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  ,      _A  ,      (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    __lowercase (       _A  ,      _A  ,      _A  ,      _A  ,      _A     )       ->  int:
      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
      SCREAMING_SNAKE_CASE	: Dict					       =				range_counting(
          node.left  ,      (node.map_left[start - 1] if start else 0)  ,      node.map_left[end] - 1  ,      _A  ,      _A  ,      )
      SCREAMING_SNAKE_CASE	: List[str]					       =				range_counting(
          node.right  ,      start - (node.map_left[start - 1] if start else 0)  ,      end - node.map_left[end]  ,      _A  ,      _A  ,      )
      return left + right


if __name__ == "__main__":
  import doctest

  doctest.testmod()




