class Node: def __init__(self, value: int = 0) -> None: self.value = value self.left = None self.right = None class PersistentSegmentTree: def __init__(self, arr: list[int]) -> None: self.n = len(arr) self.roots: list[Node] = [] self.roots.append(self._build(arr, 0, self.n - 1)) def _build(self, arr: list[int], start: int, end: int) -> Node: """ Builds a segment tree from the provided array. >>> pst = PersistentSegmentTree([1, 2, 3]) >>> root = pst._build([1, 2, 3], 0, 2) >>> root.value # Sum of the whole array 6 """ if start == end: return Node(arr[start]) mid = (start + end) // 2 node = Node() node.left = self._build(arr, start, mid) node.right = self._build(arr, mid + 1, end) node.value = node.left.value + node.right.value return node def update(self, version: int, index: int, value: int) -> int: """ Updates the segment tree with a new value at the specified index. >>> pst = PersistentSegmentTree([1, 2, 3]) >>> version_1 = pst.update(0, 1, 5) >>> pst.query(version_1, 0, 2) # Query sum from index 0 to 2 9 """ new_root = self._update(self.roots[version], 0, self.n - 1, index, value) self.roots.append(new_root) return len(self.roots) - 1 # return the index of the new version def _update(self, node: Node, start: int, end: int, index: int, value: int) -> Node: if start == end: new_node = Node(value) return new_node mid = (start + end) // 2 new_node = Node() if index <= mid: new_node.left = self._update(node.left, start, mid, index, value) new_node.right = node.right else: new_node.left = node.left new_node.right = self._update(node.right, mid + 1, end, index, value) new_node.value = new_node.left.value + new_node.right.value return new_node def query(self, version: int, left: int, right: int) -> int: """ Queries the sum in the given range for the specified version. >>> pst = PersistentSegmentTree([1, 2, 3]) >>> version_1 = pst.update(0, 1, 5) >>> pst.query(version_1, 0, 1) # Query sum from index 0 to 1 6 >>> pst.query(version_1, 0, 2) # Query sum from index 0 to 2 9 """ return self._query(self.roots[version], 0, self.n - 1, left, right) def _query(self, node: Node, start: int, end: int, left: int, right: int) -> int: if left > end or right < start: return 0 if left <= start and right >= end: return node.value mid = (start + end) // 2 return self._query(node.left, start, mid, left, right) + self._query( node.right, mid + 1, end, left, right )