diff --git a/data_structures/persistent_segment_tree.py b/data_structures/persistent_segment_tree.py new file mode 100644 index 000000000000..05194fa20a4a --- /dev/null +++ b/data_structures/persistent_segment_tree.py @@ -0,0 +1,138 @@ +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.")