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"""
A non-recursive Segment Tree implementation with range query and single element update,
works virtually with any list of the same type of elements with a "commutative"
combiner.
Explanation:
https://www.geeksforgeeks.org/iterative-segment-tree-range-minimum-query/
https://www.geeksforgeeks.org/segment-tree-efficient-implementation/
>>> SegmentTree([1, 2, 3], lambda a, b: a + b).query(0, 2)
6
>>> SegmentTree([3, 1, 2], min).query(0, 2)
1
>>> SegmentTree([2, 3, 1], max).query(0, 2)
3
>>> st = SegmentTree([1, 5, 7, -1, 6], lambda a, b: a + b)
>>> st.update(1, -1)
>>> st.update(2, 3)
>>> st.query(1, 2)
2
>>> st.query(1, 1)
-1
>>> st.update(4, 1)
>>> st.query(3, 4)
0
>>> st = SegmentTree([[1, 2, 3], [3, 2, 1], [1, 1, 1]], lambda a, b: [a[i] + b[i] for i
... in range(len(a))])
>>> st.query(0, 1)
[4, 4, 4]
>>> st.query(1, 2)
[4, 3, 2]
>>> st.update(1, [-1, -1, -1])
>>> st.query(1, 2)
[0, 0, 0]
>>> st.query(0, 2)
[1, 2, 3]
"""
from __future__ importannotations
fromcollections.abcimportCallable
fromtypingimportAny, Generic, TypeVar
T=TypeVar("T")
classSegmentTree(Generic[T]):
def__init__(self, arr: list[T], fnc: Callable[[T, T], T]) ->None:
"""
Segment Tree constructor, it works just with commutative combiner.
:param arr: list of elements for the segment tree
:param fnc: commutative function for combine two elements
>>> SegmentTree(['a', 'b', 'c'], lambda a, b: f'{a}{b}').query(0, 2)
'abc'
>>> SegmentTree([(1, 2), (2, 3), (3, 4)],
... lambda a, b: (a[0] + b[0], a[1] + b[1])).query(0, 2)
(6, 9)
"""
any_type: Any|T=None
self.N: int=len(arr)
self.st: list[T] = [any_typefor_inrange(self.N)] +arr
self.fn=fnc
self.build()
defbuild(self) ->None:
forpinrange(self.N-1, 0, -1):
self.st[p] =self.fn(self.st[p*2], self.st[p*2+1])
defupdate(self, p: int, v: T) ->None:
"""
Update an element in log(N) time
:param p: position to be update
:param v: new value
>>> st = SegmentTree([3, 1, 2, 4], min)
>>> st.query(0, 3)
1
>>> st.update(2, -1)
>>> st.query(0, 3)
-1
"""
p+=self.N
self.st[p] =v
whilep>1:
p=p//2
self.st[p] =self.fn(self.st[p*2], self.st[p*2+1])
defquery(self, l: int, r: int) ->T|None: # noqa: E741
"""
Get range query value in log(N) time
:param l: left element index
:param r: right element index
:return: element combined in the range [l, r]
>>> st = SegmentTree([1, 2, 3, 4], lambda a, b: a + b)
>>> st.query(0, 2)
6
>>> st.query(1, 2)
5
>>> st.query(0, 3)
10
>>> st.query(2, 3)
7
"""
l, r=l+self.N, r+self.N
res: T|None=None
whilel<=r:
ifl%2==1:
res=self.st[l] ifresisNoneelseself.fn(res, self.st[l])
ifr%2==0:
res=self.st[r] ifresisNoneelseself.fn(res, self.st[r])
l, r= (l+1) //2, (r-1) //2
returnres
if__name__=="__main__":
fromfunctoolsimportreduce
test_array= [1, 10, -2, 9, -3, 8, 4, -7, 5, 6, 11, -12]
test_updates= {
0: 7,
1: 2,
2: 6,
3: -14,
4: 5,
5: 4,
6: 7,
7: -10,
8: 9,
9: 10,
10: 12,
11: 1,
}
min_segment_tree=SegmentTree(test_array, min)
max_segment_tree=SegmentTree(test_array, max)
sum_segment_tree=SegmentTree(test_array, lambdaa, b: a+b)
deftest_all_segments() ->None:
"""
Test all possible segments
"""
foriinrange(len(test_array)):
forjinrange(i, len(test_array)):
min_range=reduce(min, test_array[i : j+1])
max_range=reduce(max, test_array[i : j+1])
sum_range=reduce(lambdaa, b: a+b, test_array[i : j+1])
assertmin_range==min_segment_tree.query(i, j)
assertmax_range==max_segment_tree.query(i, j)
assertsum_range==sum_segment_tree.query(i, j)
test_all_segments()
forindex, valueintest_updates.items():
test_array[index] =value
min_segment_tree.update(index, value)
max_segment_tree.update(index, value)
sum_segment_tree.update(index, value)
test_all_segments()