
'''simple docstring'''




def 		lowerCAmelCase_				(   snake_case__						):




    '''simple docstring'''



    A       :   List[Any]					=						0
    A       :   Any					=						len(snake_case__						)

    for i in range(n - 1						):
        for j in range(i + 1		,				snake_case__						):
            if arr[i] > arr[j]:
                num_inversions += 1

    return num_inversions







def 		lowerCAmelCase_				(   snake_case__						):




    '''simple docstring'''



    if len(snake_case__						) <= 1:
        return arr, 0
    A       :   Any					=						len(snake_case__						) // 2
    A       :   List[Any]					=						arr[0:mid]
    A       :   Union[str, Any]					=						arr[mid:]

    A,  A       :   List[str]					=						count_inversions_recursive(snake_case__						)
    A,  A       :   Dict					=						count_inversions_recursive(snake_case__						)
    A,  A       :   Dict					=						_count_cross_inversions(snake_case__		,				snake_case__						)

    A       :   Optional[int]					=						inversion_p + inversions_q + cross_inversions
    return c, num_inversions







def 		lowerCAmelCase_				(   snake_case__		,				snake_case__						):




    '''simple docstring'''



    A       :   Tuple					=						[]
    A       :   Optional[Any]					=						0
    while i < len(snake_case__						) and j < len(snake_case__						):
        if p[i] > q[j]:
            # if P[1] > Q[j], then P[k] > Q[k] for all  i < k <= len(P)
            # These are all inversions. The claim emerges from the
            # property that P is sorted.
            num_inversion += len(snake_case__						) - i
            r.append(q[j]						)
            j += 1
        else:
            r.append(p[i]						)
            i += 1

    if i < len(snake_case__						):
        r.extend(p[i:]						)
    else:
        r.extend(q[j:]						)

    return r, num_inversion







def 		lowerCAmelCase_				(   ):




    '''simple docstring'''



    A       :   Tuple					=						[10, 2, 1, 5, 5, 2, 11]

    # this arr has 8 inversions:
    # (10, 2), (10, 1), (10, 5), (10, 5), (10, 2), (2, 1), (5, 2), (5, 2)

    A       :   List[Any]					=						count_inversions_bf(snake_case__						)
    A,  A       :   int					=						count_inversions_recursive(snake_case__						)

    assert num_inversions_bf == num_inversions_recursive == 8

    print('''number of inversions = '''		,				snake_case__						)

    # testing an array with zero inversion (a sorted arr_1)

    arr_a.sort()
    A       :   Tuple					=						count_inversions_bf(snake_case__						)
    A,  A       :   Optional[int]					=						count_inversions_recursive(snake_case__						)

    assert num_inversions_bf == num_inversions_recursive == 0
    print('''number of inversions = '''		,				snake_case__						)

    # an empty list should also have zero inversions
    A       :   int					=						[]
    A       :   List[Any]					=						count_inversions_bf(snake_case__						)
    A,  A       :   Optional[int]					=						count_inversions_recursive(snake_case__						)

    assert num_inversions_bf == num_inversions_recursive == 0
    print('''number of inversions = '''		,				snake_case__						)


if __name__ == "__main__":
      main()











"""simple docstring"""
class 					_UpperCAmelCase   :


					def __init__(					self			:List[str]     ,       __UpperCamelCase			:Union[str, Any]     ,       __UpperCamelCase			:Tuple		):
												A		=  name
												A		=  val


					def __str__(					self			:str		):
												return f"{self.__class__.__name__}({self.name}, {self.val})"


					def __lt__(					self			:List[Any]     ,       __UpperCamelCase			:Union[str, Any]		):
												return self.val < other.val







class 					_UpperCAmelCase   :


					def __init__(					self			:List[str]     ,       __UpperCamelCase			:Optional[Any]		):
												A		=  {}
												A		=  {}
												A		=  self.build_heap(__UpperCamelCase		)


					def __getitem__(					self			:int     ,       __UpperCamelCase			:Optional[int]		):
												return self.get_value(__UpperCamelCase		)


					def 	lowerCamelCase       (					self			:List[Any]     ,       __UpperCamelCase			:str		):
												return (idx - 1) // 2


					def 	lowerCamelCase       (					self			:int     ,       __UpperCamelCase			:Optional[Any]		):
												return idx * 2 + 1


					def 	lowerCamelCase       (					self			:Union[str, Any]     ,       __UpperCamelCase			:Optional[int]		):
												return idx * 2 + 2


					def 	lowerCamelCase       (					self			:Optional[Any]     ,       __UpperCamelCase			:str		):
												return self.heap_dict[key]


					def 	lowerCamelCase       (					self			:int     ,       __UpperCamelCase			:Optional[Any]		):
												A		=  len(__UpperCamelCase		) - 1
												A		=  self.get_parent_idx(__UpperCamelCase		)

												for idx, i in enumerate(__UpperCamelCase		):
																			A		=  idx
																			A		=  i.val

												for i in range(__UpperCamelCase     ,       -1     ,       -1		):
																			self.sift_down(__UpperCamelCase     ,       __UpperCamelCase		)
												return array


					def 	lowerCamelCase       (					self			:str     ,       __UpperCamelCase			:Optional[Any]     ,       __UpperCamelCase			:Dict		):
												while True:
																			A		=  self.get_left_child_idx(__UpperCamelCase		)  # noqa: E741
																			A		=  self.get_right_child_idx(__UpperCamelCase		)

																			A		=  idx
																			if l < len(__UpperCamelCase		) and array[l] < array[idx]:
																										A		=  l
																			if r < len(__UpperCamelCase		) and array[r] < array[smallest]:
																										A		=  r

																			if smallest != idx:
																										A,	A		=  array[smallest], array[idx]
																										(
																										    (
																										    A
																										),	(
																										    A
																										),	
																										)		=  (
																										    self.idx_of_element[array[smallest]],
																										    self.idx_of_element[array[idx]],
																										)
																										A		=  smallest
																			else:
																										break


					def 	lowerCamelCase       (					self			:Optional[Any]     ,       __UpperCamelCase			:Optional[int]		):
												A		=  self.get_parent_idx(__UpperCamelCase		)
												while p >= 0 and self.heap[p] > self.heap[idx]:
																			A,	A		=  self.heap[idx], self.heap[p]
																			A,	A		=  (
																			    self.idx_of_element[self.heap[idx]],
																			    self.idx_of_element[self.heap[p]],
																			)
																			A		=  p
																			A		=  self.get_parent_idx(__UpperCamelCase		)


					def 	lowerCamelCase       (					self			:Any		):
												return self.heap[0]


					def 	lowerCamelCase       (					self			:Tuple		):
												A,	A		=  self.heap[-1], self.heap[0]
												A,	A		=  (
												    self.idx_of_element[self.heap[-1]],
												    self.idx_of_element[self.heap[0]],
												)

												A		=  self.heap.pop()
												del self.idx_of_element[x]
												self.sift_down(0     ,       self.heap		)
												return x


					def 	lowerCamelCase       (					self			:Optional[int]     ,       __UpperCamelCase			:Optional[int]		):
												self.heap.append(__UpperCamelCase		)
												A		=  len(self.heap		) - 1
												A		=  node.val
												self.sift_up(len(self.heap		) - 1		)


					def 	lowerCamelCase       (					self			:Tuple		):
												return len(self.heap		) == 0


					def 	lowerCamelCase       (					self			:Any     ,       __UpperCamelCase			:str     ,       __UpperCamelCase			:Dict		):
												assert (
												    self.heap[self.idx_of_element[node]].val > new_value
												), "newValue must be less that current value"
												A		=  new_value
												A		=  new_value
												self.sift_up(self.idx_of_element[node]		)







_snake_case     :		Optional[int]       						= Node('R', -1)
_snake_case     :		Tuple       						= Node('B', 6)
_snake_case     :		Tuple       						= Node('A', 3)
_snake_case     :		Optional[int]       						= Node('X', 1)
_snake_case     :		List[Any]       						= Node('E', 4)

# Use one of these two ways to generate Min-Heap

# Generating Min-Heap from array
_snake_case     :		Tuple       						= MinHeap([r, b, a, x, e])

# Generating Min-Heap by Insert method
# myMinHeap.insert(a)
# myMinHeap.insert(b)
# myMinHeap.insert(x)
# myMinHeap.insert(r)
# myMinHeap.insert(e)

# Before
print('Min Heap - before decrease key')
for i in my_min_heap.heap:
		print(i)

print('Min Heap - After decrease key of node [B -> -17]')
my_min_heap.decrease_key(b, -17)

# After
for i in my_min_heap.heap:
		print(i)

if __name__ == "__main__":
		import doctest

		doctest.testmod()


