from __future__ import annotations class Node: def __init__(self, value: int = 0) -> None: self.value: int = value self.left: Node | None = None self.right: Node | None = None class PersistentSegmentTree: def __init__(self, arr: list[int]) -> None: self.n: int = 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. """ 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 value at the given index and returns the new version. """ new_root = self._update(self.roots[version], 0, self.n - 1, index, value) self.roots.append(new_root) return len(self.roots) - 1 def _update( self, node: Node | None, start: int, end: int, index: int, value: int ) -> Node: """ Updates the node for the specified index and value and returns the new node. """ if node is None: # Handle the None case node = Node() # Create a new node if None if start == end: return Node(value) 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 # Ensure right node is the same as the original else: new_node.left = node.left # Ensure left node is the same as the original new_node.right = self._update(node.right, mid + 1, end, index, value) new_node.value = new_node.left.value + ( new_node.right.value if new_node.right else 0 ) return new_node def query(self, version: int, left: int, right: int) -> int: """ Queries the sum in the given range for the specified version. """ return self._query(self.roots[version], 0, self.n - 1, left, right) def _query( self, node: Node | None, start: int, end: int, left: int, right: int ) -> int: """ Queries the sum of values in the range [left, right] for the given node. """ if node is None or 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 ) # Running the doctests if __name__ == "__main__": import doctest print("Running doctests...") result = doctest.testmod() print(f"Ran {result.attempted} tests, {result.failed} failed.")