Lazy Segment Tree

A
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```from __future__ import annotations

import math

class SegmentTree:
def __init__(self, size: int) -> None:
self.size = size
# approximate the overall size of segment tree with given value
self.segment_tree = [0 for i in range(4 * size)]
# create array to store lazy update
self.lazy = [0 for i in range(4 * size)]
self.flag = [0 for i in range(4 * size)]  # flag for lazy update

def left(self, idx: int) -> int:
"""
>>> segment_tree = SegmentTree(15)
>>> segment_tree.left(1)
2
>>> segment_tree.left(2)
4
>>> segment_tree.left(12)
24
"""
return idx * 2

def right(self, idx: int) -> int:
"""
>>> segment_tree = SegmentTree(15)
>>> segment_tree.right(1)
3
>>> segment_tree.right(2)
5
>>> segment_tree.right(12)
25
"""
return idx * 2 + 1

def build(
self, idx: int, left_element: int, right_element: int, a: list[int]
) -> None:
if left_element == right_element:
self.segment_tree[idx] = a[left_element - 1]
else:
mid = (left_element + right_element) // 2
self.build(self.left(idx), left_element, mid, a)
self.build(self.right(idx), mid + 1, right_element, a)
self.segment_tree[idx] = max(
self.segment_tree[self.left(idx)], self.segment_tree[self.right(idx)]
)

def update(
self, idx: int, left_element: int, right_element: int, a: int, b: int, val: int
) -> bool:
"""
update with O(lg n) (Normal segment tree without lazy update will take O(nlg n)
for each update)

update(1, 1, size, a, b, v) for update val v to [a,b]
"""
if self.flag[idx] is True:
self.segment_tree[idx] = self.lazy[idx]
self.flag[idx] = False
if left_element != right_element:
self.lazy[self.left(idx)] = self.lazy[idx]
self.lazy[self.right(idx)] = self.lazy[idx]
self.flag[self.left(idx)] = True
self.flag[self.right(idx)] = True

if right_element < a or left_element > b:
return True
if left_element >= a and right_element <= b:
self.segment_tree[idx] = val
if left_element != right_element:
self.lazy[self.left(idx)] = val
self.lazy[self.right(idx)] = val
self.flag[self.left(idx)] = True
self.flag[self.right(idx)] = True
return True
mid = (left_element + right_element) // 2
self.update(self.left(idx), left_element, mid, a, b, val)
self.update(self.right(idx), mid + 1, right_element, a, b, val)
self.segment_tree[idx] = max(
self.segment_tree[self.left(idx)], self.segment_tree[self.right(idx)]
)
return True

# query with O(lg n)
def query(
self, idx: int, left_element: int, right_element: int, a: int, b: int
) -> int | float:
"""
query(1, 1, size, a, b) for query max of [a,b]
>>> A = [1, 2, -4, 7, 3, -5, 6, 11, -20, 9, 14, 15, 5, 2, -8]
>>> segment_tree = SegmentTree(15)
>>> segment_tree.build(1, 1, 15, A)
>>> segment_tree.query(1, 1, 15, 4, 6)
7
>>> segment_tree.query(1, 1, 15, 7, 11)
14
>>> segment_tree.query(1, 1, 15, 7, 12)
15
"""
if self.flag[idx] is True:
self.segment_tree[idx] = self.lazy[idx]
self.flag[idx] = False
if left_element != right_element:
self.lazy[self.left(idx)] = self.lazy[idx]
self.lazy[self.right(idx)] = self.lazy[idx]
self.flag[self.left(idx)] = True
self.flag[self.right(idx)] = True
if right_element < a or left_element > b:
return -math.inf
if left_element >= a and right_element <= b:
return self.segment_tree[idx]
mid = (left_element + right_element) // 2
q1 = self.query(self.left(idx), left_element, mid, a, b)
q2 = self.query(self.right(idx), mid + 1, right_element, a, b)
return max(q1, q2)

def __str__(self) -> str:
return str([self.query(1, 1, self.size, i, i) for i in range(1, self.size + 1)])

if __name__ == "__main__":
A = [1, 2, -4, 7, 3, -5, 6, 11, -20, 9, 14, 15, 5, 2, -8]
size = 15
segt = SegmentTree(size)
segt.build(1, 1, size, A)
print(segt.query(1, 1, size, 4, 6))
print(segt.query(1, 1, size, 7, 11))
print(segt.query(1, 1, size, 7, 12))
segt.update(1, 1, size, 1, 3, 111)
print(segt.query(1, 1, size, 1, 15))
segt.update(1, 1, size, 7, 8, 235)
print(segt)
```