"""simple docstring"""







from __future__ import annotations

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






class 							__snake_case(					A_	):


    '''simple docstring'''





    @abstractmethod
    def __lt__(  self					,       __lowerCamelCase	):




         '''simple docstring'''



         pass
    @abstractmethod
    def __gt__(  self					,       __lowerCamelCase	):




         '''simple docstring'''



         pass

    @abstractmethod
    def __eq__(  self					,       __lowerCamelCase	):




         '''simple docstring'''



         pass







lowerCamelCase      : Dict       =TypeVar('''T''', bound=Comparable)






class 							__snake_case(					Generic[T]	):


    '''simple docstring'''





    def __init__(  self	):




         '''simple docstring'''



         __A					:					list[T]								=		[]
         __A					:					int								=		0
    def __repr__(  self	):




         '''simple docstring'''



         return str(self.h	)
    def 						_a							(  self					,       __lowerCamelCase	):




         '''simple docstring'''



         if child_idx > 0:
              return (child_idx - 1) // 2
         return None
    def 						_a							(  self					,       __lowerCamelCase	):




         '''simple docstring'''



         __A					:					Union[str, Any]								=		2 * parent_idx + 1
         if left_child_index < self.heap_size:
              return left_child_index
         return None
    def 						_a							(  self					,       __lowerCamelCase	):




         '''simple docstring'''



         __A					:					Any								=		2 * parent_idx + 2
         if right_child_index < self.heap_size:
              return right_child_index
         return None
    def 						_a							(  self					,       __lowerCamelCase	):




         '''simple docstring'''



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




         '''simple docstring'''



         __A					:					List[str]								=		list(__lowerCamelCase	)
         __A					:					Tuple								=		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(__lowerCamelCase	)
    def 						_a							(  self	):




         '''simple docstring'''



         if self.heap_size >= 2:
              __A					:					Union[str, Any]								=		self.h[0]
              __A					:					int								=		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					,       __lowerCamelCase	):




         '''simple docstring'''



         self.h.append(__lowerCamelCase	)
         __A					:					Union[str, Any]								=		(self.heap_size - 1) // 2
         self.heap_size += 1
         while idx >= 0:
              self.max_heapify(__lowerCamelCase	)
              __A					:					str								=		(idx - 1) // 2

    def 						_a							(  self	):




         '''simple docstring'''



         __A					:					Any								=		self.heap_size
         for j in range(size - 1					,       0					,       -1	):
              __A						,		__A					:					Union[str, Any]								=		self.h[j], self.h[0]
              self.heap_size -= 1
              self.max_heapify(0	)
         __A					:					str								=		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}')

              lowerCamelCase      : 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')




"""simple docstring"""







def 						_lowercase				(  _SCREAMING_SNAKE_CASE				:				list[int]						)    ->     int:







     '''simple docstring'''


     if not isinstance(_SCREAMING_SNAKE_CASE  ,       _SCREAMING_SNAKE_CASE						) or not all(
         isinstance(_SCREAMING_SNAKE_CASE  ,       _SCREAMING_SNAKE_CASE						) and item >= 0 for item in citations						):
          raise ValueError('The citations should be a list of non negative integers.'						)

     citations.sort()
     __A					:					Any								=		len(_SCREAMING_SNAKE_CASE						)

     for i in range(_SCREAMING_SNAKE_CASE						):
          if citations[len_citations - 1 - i] <= i:
               return i

     return len_citations


if __name__ == "__main__":
       import doctest

       doctest.testmod()



