
"""simple docstring"""


from __future__ import annotations

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

class 					A_			(  _a     ):



							"""simple docstring"""






							@abstractmethod
							def __lt__(						self						:T  ,							lowerCamelCase_						:T ):





										"""simple docstring"""

										pass





							@abstractmethod
							def __gt__(						self						:T  ,							lowerCamelCase_						:T ):





										"""simple docstring"""

										pass





							@abstractmethod
							def __eq__(						self						:T  ,							lowerCamelCase_						:object ):





										"""simple docstring"""

										pass






lowerCAmelCase			=		TypeVar("""T""", bound=Comparable)

class 					A_			(  Generic[T]     ):



							"""simple docstring"""






							def __init__(						self						:List[str] ):





										"""simple docstring"""

										lowerCamelCase__			:  Optional[Any]  =[]
										lowerCamelCase__			:  int  =0





							def __repr__(						self						:Tuple ):





										"""simple docstring"""

										return str(self.h )





							def    UpperCAmelCase__							(						self						:Any  ,							lowerCamelCase_						:int ):





										"""simple docstring"""

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





							def    UpperCAmelCase__							(						self						:List[str]  ,							lowerCamelCase_						:int ):





										"""simple docstring"""

										lowerCamelCase__			:  List[Any]  =2 * parent_idx + 1
										if left_child_index < self.heap_size:
													return left_child_index
										return None





							def    UpperCAmelCase__							(						self						:Tuple  ,							lowerCamelCase_						:int ):





										"""simple docstring"""

										lowerCamelCase__			:  Dict  =2 * parent_idx + 2
										if right_child_index < self.heap_size:
													return right_child_index
										return None





							def    UpperCAmelCase__							(						self						:str  ,							lowerCamelCase_						:int ):





										"""simple docstring"""

										if index < self.heap_size:
													lowerCamelCase__			:  Any  =index
													lowerCamelCase__			:  List[Any]  =self.left_child_idx(lowerCamelCase_ )
													lowerCamelCase__			:  Union[str, Any]  =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]:
																lowerCamelCase__			:  Optional[int]  =left_child
													if right_child is not None and self.h[right_child] > self.h[violation]:
																lowerCamelCase__			:  Optional[Any]  =right_child
													# if violation indeed exists
													if violation != index:
																# swap to fix the violation
																lowerCamelCase__ ,       lowerCamelCase__			:  Any  =self.h[index], self.h[violation]
																# fix the subsequent violation recursively if any
																self.max_heapify(lowerCamelCase_ )





							def    UpperCAmelCase__							(						self						:int  ,							lowerCamelCase_						:Iterable[T] ):





										"""simple docstring"""

										lowerCamelCase__			:  Dict  =list(lowerCamelCase_ )
										lowerCamelCase__			:  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    UpperCAmelCase__							(						self						:Any ):





										"""simple docstring"""

										if self.heap_size >= 2:
													lowerCamelCase__			:  List[str]  =self.h[0]
													lowerCamelCase__			:  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    UpperCAmelCase__							(						self						:Dict  ,							lowerCamelCase_						:T ):





										"""simple docstring"""

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





							def    UpperCAmelCase__							(						self						:int ):





										"""simple docstring"""

										lowerCamelCase__			:  List[str]  =self.heap_size
										for j in range(size - 1  ,							0  ,							-1 ):
													lowerCamelCase__ ,       lowerCamelCase__			:  List[Any]  =self.h[j], self.h[0]
													self.heap_size -= 1
													self.max_heapify(0 )
										lowerCamelCase__			:  Any  =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()
														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"""


# A Bipartite Graph is a graph whose vertices can be divided into two independent sets,
# U and V such that every edge (u, v) either connects a vertex from U to V or a vertex
# from V to U. In other words, for every edge (u, v), either u belongs to U and v to V,
# or u belongs to V and v to U. We can also say that there is no edge that connects
# vertices of same set.
from queue import Queue






def      lowerCAmelCase_     (	snake_case_	:    List[Any]    )     ->Optional[int]:
			lowerCamelCase__			:  Optional[int]  =Queue()
			lowerCamelCase__			:  Any  =[False] * len(snake_case_    )
			lowerCamelCase__			:  Optional[Any]  =[-1] * len(snake_case_    )

			def bfs():
						while not queue.empty():
									lowerCamelCase__			:  List[Any]  =queue.get()
									lowerCamelCase__			:  int  =True

									for neighbour in graph[u]:
												if neighbour == u:
															return False

												if color[neighbour] == -1:
															lowerCamelCase__			:  Union[str, Any]  =1 - color[u]
															queue.put(snake_case_    )

												elif color[neighbour] == color[u]:
															return False

						return True

			for i in range(len(snake_case_    )    ):
						if not visited[i]:
									queue.put(snake_case_    )
									lowerCamelCase__			:  Tuple  =0
									if bfs() is False:
												return False

			return True


if __name__ == "__main__":
							# Adjacency List of graph
							print(check_bipartite({0: [1, 3], 1: [0, 2], 2: [1, 3], 3: [0, 2]}))