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. >>> pst = PersistentSegmentTree([1, 2, 3, 4]) >>> root = pst._build([1, 2, 3, 4], 0, 3) >>> root.value # Sum of the whole array 10 >>> root.left.value # Sum of the left half 3 >>> root.right.value # Sum of the right half 7 """ 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. >>> pst = PersistentSegmentTree([1, 2, 3, 4]) >>> version_1 = pst.update(0, 1, 5) # Update index 1 to 5 >>> pst.query(version_1, 0, 3) # Query sum of all elements in new version 13 >>> pst.query(0, 0, 3) # Original version remains unchanged 10 >>> version_2 = pst.update(version_1, 3, 6) # Update index 3 to 6 in version_1 >>> pst.query(version_2, 0, 3) # Query sum of all elements in newest version 15 """ 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, start: int, end: int, index: int, value: int) -> Node: """ Updates the node for the specified index and value and returns the new node. >>> pst = PersistentSegmentTree([1, 2, 3, 4]) >>> old_root = pst.roots[0] >>> new_root = pst._update(old_root, 0, 3, 1, 5) # Update index 1 to 5 >>> new_root.value # New sum after update 13 >>> old_root.value # Old root remains unchanged 10 >>> new_root.left.value # Updated left child 6 >>> new_root.right.value # Right child remains the same 7 """ 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. >>> pst = PersistentSegmentTree([1, 2, 3, 4]) >>> pst.query(0, 0, 3) # Sum of all elements in original version 10 >>> pst.query(0, 1, 2) # Sum of elements at index 1 and 2 in original version 5 >>> version_1 = pst.update(0, 1, 5) # Update index 1 to 5 >>> pst.query(version_1, 0, 3) # Sum of all elements in new version 13 >>> pst.query(version_1, 1, 2) # Sum of elements at index 1 and 2 8 """ 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: """ Queries the sum of values in the range [left, right] for the given node. >>> pst = PersistentSegmentTree([1, 2, 3, 4]) >>> root = pst.roots[0] >>> pst._query(root, 0, 3, 1, 2) # Sum of elements at index 1 and 2 5 >>> pst._query(root, 0, 3, 0, 3) # Sum of all elements 10 >>> pst._query(root, 0, 3, 2, 3) # Sum of elements at index 2 and 3 7 """ 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.")