blog/content/posts/2024-07-27-treap-revisited/index.md

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---
title: "Treap, revisited"
date: 2024-07-27T14:12:27+01:00
draft: false # I don't care for draft mode, git has branches for that
description: "An even simpler BST"
tags:
- algorithms
- data structures
- python
categories:
- programming
series:
- Cool algorithms
favorite: false
disable_feed: false
---
My [last post]({{< relref "../2024-07-20-treap/index.md" >}}) about the _Treap_
showed an implementation using tree rotations, as is commonly done with [AVL
Trees][avl] and [Red Black Trees][rb].
But the _Treap_ lends itself well to a simple and elegant implementation with no
tree rotations. This makes it especially easy to implement the removal of a key,
rather than the fiddly process of deletion using tree rotations.
[avl]: https://en.wikipedia.org/wiki/AVL_tree
[rb]: https://en.wikipedia.org/wiki/Red%E2%80%93black_tree
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## Implementation
All operations on the tree will be implemented in terms of two fundamental
operations: `split` and `merge`.
We'll be reusing the same structures as in the last post, so let's skip straight
to implementing those fundaments, and building on them for `insert` and
`delete`.
### Split
Splitting a tree means taking a key, and getting the following output:
* a `left` node, root of the tree of all keys lower than the input.
* an extracted `node` which corresponds to the input `key`.
* a `right` node, root of the tree of all keys higher than the input.
```python
type OptionalNode[K, V] = Node[K, V] | None
class SplitResult(NamedTuple):
left: OptionalNode
node: OptionalNode
right: OptionalNode
def split(root: OptionalNode[K, V], key: K) -> SplitResult:
# Base case, empty tree
if root is None:
return SplitResult(None, None, None)
# If we found the key, simply extract left and right
if root.key == key:
left, right = root.left, root.right
root.left, root.right = None, None
return SplitResult(left, root, right)
# Otherwise, recurse on the corresponding side of the tree
if root.key < key:
left, node, right = split(root.right, key)
root.right = left
return SplitResult(root, node, right)
if key < root.key:
left, node, right = split(root.left, key)
root.left = right
return SplitResult(left, node, root)
raise RuntimeError("Unreachable")
```
### Merge
Merging a `left` and `right` tree means (cheaply) building a new tree containing
both of them. A pre-condition for merging is that the `left` tree is composed
entirely of nodes that are lower than any key in `right` (i.e: as in `left` and
`right` after a `split`).
```python
def merge(
left: OptionalNode[K, V],
right: OptionalNode[K, V],
) -> OptionalNode[K, V]:
# Base cases, left or right being empty
if left is None:
return right
if right is None:
return left
# Left has higher priority, it must become the root node
if left.priority >= right.priority:
# We recursively reconstruct its right sub-tree
left.right = merge(left.right, right)
return left
# Right has higher priority, it must become the root node
if left.priority < right.priority:
# We recursively reconstruct its left sub-tree
right.left = merge(left, right.left)
return right
raise RuntimeError("Unreachable")
```
### Insertion
Inserting a node into the tree is done in two steps:
1. `split` the tree to isolate the middle insertion point
2. `merge` it back up to form a full tree with the inserted key
```python
def insert(self, key: K, value: V) -> bool:
# `left` and `right` come before/after the key
left, node, right = split(self._root, key)
was_updated: bool
# Create the node, or update its value, if the key was already in the tree
if node is None:
node = Node(key, value)
was_updated = False
else:
node.value = value
was_updated = True
# Rebuild the tree with a couple of merge operations
self._root = merge(left, merge(node, right))
# Signal whether the key was already in the key
return was_updated
```
### Removal
Removing a key from the tree is similar to inserting a new key, and forgetting
to insert it back: simply `split` the tree and `merge` it back without the
extracted middle node.
```python
def remove(self, key: K) -> bool:
# `node` contains the key, or `None` if the key wasn't in the tree
left, node, right = split(self._root, key)
# Put the tree back together, without the extract node
self._root = merge(left, right)
# Signal whether `key` was mapped in the tree
return node is not None
```