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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
+
+<!--more-->
+
+## 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
+```