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

@@ -1,159 +0,0 @@
----
-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 keeps 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 likelihook 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 `left`/`right` nodes
-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)
-```