Add Treap post

content/posts/2024-07-20-treap/index.md

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+---
+title: "Treap"
+date: 2024-07-20T14:12:27+01:00
+draft: false # I don't care for draft mode, git has branches for that
+description: "A simpler BST"
+tags:
+  - algorithms
+  - data structures
+  - python
+categories:
+  - programming
+series:
+  - Cool algorithms
+favorite: false
+disable_feed: false
+graphviz: true
+---
+
+The [_Treap_][wiki] is a mix between a _Binary Search Tree_ and a _Heap_.
+
+Like a _Binary Search Tree_, it keeps an ordered set of keys in the shape of a
+tree, allowing for binary search traversal.
+
+Like a _Heap_, it associates each node with a priority, making sure that a
+parent's priority is always higher than any of its children.
+
+[wiki]: https://en.wikipedia.org/wiki/Treap
+
+<!--more-->
+
+## What does it do?
+
+By randomizing the priority value of each key at insertion time, we ensure a
+high likelihood that the tree stays _roughly_ balanced, avoiding degenerating to
+unbalanced O(N) height.
+
+Here's a sample tree created by inserting integers from 0 to 250 into the tree:
+
+{{< graphviz file="treap.gv" />}}
+
+## Implementation
+
+I'll be keeping the theme for this [series] by using Python to implement the
+_Treap_. This leads to somewhat annoying code to handle the rotation process,
+which is easier to do in C using pointers.
+
+[series]: {{< ref "/series/cool-algorithms/" >}}
+
+### Representation
+
+Creating a new `Treap` is easy: the tree starts off empty, waiting for new nodes
+to insert.
+
+Each `Node` must keep track of the `key`, the mapped `value`, and the node's
+`priority` (which is assigned randomly). Finally it must also allow for storing
+two children (`left` and `right`).
+
+```python
+class Node[K, V]:
+    key: K
+    value: V
+    priority: float
+    left: Node[K, V] | None
+    righg: Node[K, V] | None
+
+    def __init__(self, key: K, value: V):
+        # Store key and value, like a normal BST node
+        self.key = key
+        self.value = value
+        # Priority is derived randomly
+        self.priority = random()
+        self.left = None
+        self.right = None
+
+class Treap[K, V]:
+    _root: Node[K, V] | None
+
+    def __init__(self):
+        # The tree starts out empty
+        self._root = None
+```
+
+### Search
+
+Searching the tree is the same as in any other _Binary Search Tree_.
+
+```python
+def get(self, key: K) -> T | None:
+    node = self._root
+    # The usual BST traversal
+    while node is not None:
+        if node.key == key:
+            return node.value
+        elif node.key < key:
+            node = node.right
+        else:
+            node = node.left
+    return None
+```
+
+### Insertion
+
+To insert a new `key` into the tree, we identify which leaf position it should
+be inserted at. We then generate the node's priority, insert it at this
+position, and rotate the node upwards until the heap property is respected.
+
+```python
+type ChildField = Literal["left, right"]
+
+def insert(self, key: K, value: V) -> bool:
+    # Empty treap base-case
+    if self._root is None:
+        self._root = Node(key, value)
+        # Signal that we're not overwriting the value
+        return False
+    # Keep track of the parent chain for rotation after insertion
+    parents = []
+    node = self._root
+    while node is not None:
+        # Insert a pre-existing key
+        if node.key == key:
+            node.value = value
+            return True
+        #  Go down the tree, keep track of the path through the tree
+        field = "left" if key < node.key else "right"
+        parents.append((node, field))
+        node = getattr(node, field)
+    #  Key wasn't found, we're inserting a new node
+    child = Node(key, value)
+    parent, field = parents[-1]
+    setattr(parent, field, child)
+    # Rotate the new node up until we respect the decreasing priority property
+    self._rotate_up(child, parents)
+    # Key wasn't found, signal that we inserted a new node
+    return False
+
+def _rotate_up(
+    self,
+    node: Node[K, V],
+    parents: list[tuple[Node[K, V], ChildField]],
+) -> None:
+    while parents:
+        parent, field = parents.pop()
+        # If the parent has higher priority, we're done rotating
+        if parent.priority >= node.priority:
+            break
+        # Check for grand-parent/root of tree edge-case
+        if parents:
+            # Update grand-parent to point to the new rotated node
+            grand_parent, field = parents[-1]
+            setattr(grand_parent, field, node)
+        else:
+            # Point the root to the new rotated node
+            self._root = node
+        other_field = "left" if field == "right" else "right"
+        # Rotate the node up
+        setattr(parent, field, getattr(node, other_field))
+        setattr(node, other_field, parent)
+```