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from copy import deepcopy

class FenwickTree:

"""

Fenwick Tree

More info: https://en.wikipedia.org/wiki/Fenwick_tree

"""

def __init__(self, arr: list[int] = None, size: int = None) -> None:

"""

Constructor for the Fenwick tree

Parameters:

arr (list): list of elements to initialize the tree with (optional)

size (int): size of the Fenwick tree (if arr is None)

"""

if arr is None and size is not None:

self.size = size

self.tree = [0] * size

elif arr is not None:

self.init(arr)

else:

raise ValueError("Either arr or size must be specified")

def init(self, arr: list[int]) -> None:

"""

Initialize the Fenwick tree with arr in O(N)

Parameters:

arr (list): list of elements to initialize the tree with

Returns:

None

>>> a = [1, 2, 3, 4, 5]

>>> f1 = FenwickTree(a)

>>> f2 = FenwickTree(size=len(a))

>>> for index, value in enumerate(a):

... f2.add(index, value)

>>> f1.tree == f2.tree

True

"""

self.size = len(arr)

self.tree = deepcopy(arr)

for i in range(1, self.size):

j = self.next(i)

if j < self.size:

self.tree[j] += self.tree[i]

def get_array(self) -> list[int]:

"""

Get the Normal Array of the Fenwick tree in O(N)

Returns:

list: Normal Array of the Fenwick tree

>>> a = [i for i in range(128)]

>>> f = FenwickTree(a)

>>> f.get_array() == a

True

"""

arr = self.tree[:]

for i in range(self.size - 1, 0, -1):

j = self.next(i)

if j < self.size:

arr[j] -= arr[i]

return arr

@staticmethod

def next(index: int) -> int:

return index + (index & (-index))

@staticmethod

def prev(index: int) -> int:

return index - (index & (-index))

def add(self, index: int, value: int) -> None:

"""

Add a value to index in O(lg N)

Parameters:

index (int): index to add value to

value (int): value to add to index

Returns:

None

>>> f = FenwickTree([1, 2, 3, 4, 5])

>>> f.add(0, 1)

>>> f.add(1, 2)

>>> f.add(2, 3)

>>> f.add(3, 4)

>>> f.add(4, 5)

>>> f.get_array()

[2, 4, 6, 8, 10]

"""

if index == 0:

self.tree[0] += value

return

while index < self.size:

self.tree[index] += value

index = self.next(index)

def update(self, index: int, value: int) -> None:

"""

Set the value of index in O(lg N)

Parameters:

index (int): index to set value to

value (int): value to set in index

Returns:

None

>>> f = FenwickTree([5, 4, 3, 2, 1])

>>> f.update(0, 1)

>>> f.update(1, 2)

>>> f.update(2, 3)

>>> f.update(3, 4)

>>> f.update(4, 5)

>>> f.get_array()

[1, 2, 3, 4, 5]

"""

self.add(index, value - self.get(index))

def prefix(self, right: int) -> int:

"""

Prefix sum of all elements in [0, right) in O(lg N)

Parameters:

right (int): right bound of the query (exclusive)

Returns:

int: sum of all elements in [0, right)

>>> a = [i for i in range(128)]

>>> f = FenwickTree(a)

>>> res = True

>>> for i in range(len(a)):

... res = res and f.prefix(i) == sum(a[:i])

>>> res

True

"""

if right == 0:

return 0

result = self.tree[0]

right -= 1 # make right inclusive

while right > 0:

result += self.tree[right]

right = self.prev(right)

return result

def query(self, left: int, right: int) -> int:

"""

Query the sum of all elements in [left, right) in O(lg N)

Parameters:

left (int): left bound of the query (inclusive)

right (int): right bound of the query (exclusive)

Returns:

int: sum of all elements in [left, right)

>>> a = [i for i in range(128)]

>>> f = FenwickTree(a)

>>> res = True

>>> for i in range(len(a)):

... for j in range(i + 1, len(a)):

... res = res and f.query(i, j) == sum(a[i:j])

>>> res

True

"""

return self.prefix(right) - self.prefix(left)

def get(self, index: int) -> int:

"""

Get value at index in O(lg N)

Parameters:

index (int): index to get the value

Returns:

int: Value of element at index

>>> a = [i for i in range(128)]

>>> f = FenwickTree(a)

>>> res = True

>>> for i in range(len(a)):

... res = res and f.get(i) == a[i]

>>> res

True

"""

return self.query(index, index + 1)

def rank_query(self, value: int) -> int:

"""

Find the largest index with prefix(i) <= value in O(lg N)

NOTE: Requires that all values are non-negative!

Parameters:

value (int): value to find the largest index of

Returns:

-1: if value is smaller than all elements in prefix sum

int: largest index with prefix(i) <= value

>>> f = FenwickTree([1, 2, 0, 3, 0, 5])

>>> f.rank_query(0)

-1

>>> f.rank_query(2)

0

>>> f.rank_query(1)

0

>>> f.rank_query(3)

2

>>> f.rank_query(5)

2

>>> f.rank_query(6)

4

>>> f.rank_query(11)

5

"""

value -= self.tree[0]

if value < 0:

return -1

j = 1 # Largest power of 2 <= size

while j * 2 < self.size:

j *= 2

i = 0

while j > 0:

if i + j < self.size and self.tree[i + j] <= value:

value -= self.tree[i + j]

i += j

j //= 2

return i

if __name__ == "__main__":

import doctest

doctest.testmod()