Add 2026.07.15 session

notes-and-examples/2026.07.01/red_black_tree.py

@@ -84,17 +84,95 @@ class RedBlackTree:
         elif root.value < value:
             self.delete(value, root.right, root)
 
-    def rebalance_from_just_inserted(self, node: Node):
+    def rebalance_from_just_inserted(self, node: Node | None):
-        # This is what you'll be implementing.
+        if node is None:
-        #
+            return
-        # By the time this is called, `node` has already been inserted like a
+
-        # normal BST node, colored red, and had its `parent` pointer set (see
+        parent = node.parent
-        # insert()). Your job is to restore the red-black properties by
+        if parent is not None and not parent.is_red:
-        # rebalancing the subtree around `node`.
+            return
-        #
+        elif parent is None:
-        # Rebalancing is only needed when node.parent is red. Remember to keep
+            node.is_red = False
-        # the root black at the end.
+            return
-        pass
+
+        grandparent = parent.parent
+        if grandparent is None:
+            return
+
+        uncle = grandparent.right if grandparent.left == parent else grandparent.left
+
+        if uncle is not None and uncle.is_red:
+            uncle.is_red = False
+            parent.is_red = False
+            grandparent.is_red = True
+
+            self.rebalance_from_just_inserted(grandparent)
+            return
+
+        if grandparent.left == parent and parent.right == node:
+            node, parent = parent, self.rotate_left(parent)
+        elif grandparent.right == parent and parent.left == node:
+            node, parent = parent, self.rotate_right(parent)
+
+        new_grandparent: Node | None = None
+        if grandparent.left == parent and parent.left == node:
+            new_grandparent = self.rotate_right(grandparent)
+        elif grandparent.right == parent and parent.right == node:
+            new_grandparent = self.rotate_left(grandparent)
+
+        if grandparent == self.root and new_grandparent is not None:
+            self.root = new_grandparent
+            self.root.is_red = False
+            grandparent.is_red = True
+
+
+    def rotate_left(self, node: Node) -> Node:
+        if node.right is None:
+            return node
+
+        original_parent = node.parent
+        new_parent = node.right
+        new_right_child = node.right.left
+
+        node.right = new_right_child
+        if new_right_child is not None:
+            new_right_child.parent = node
+
+        new_parent.left = node
+        node.parent = new_parent
+        new_parent.parent = original_parent
+
+        if original_parent is not None:
+            if original_parent.left == node:
+                original_parent.left = new_parent
+            elif original_parent.right == node:
+                original_parent.right = new_parent
+
+        return new_parent
+
+    def rotate_right(self, node: Node) -> Node:
+        if node.left is None:
+            return node
+
+        original_parent = node.parent
+        new_parent = node.left
+        new_left_child = node.left.right
+
+        node.left = new_left_child
+        if new_left_child is not None:
+            new_left_child.parent = node
+
+        new_parent.right = node
+        node.parent = new_parent
+        new_parent.parent = original_parent
+
+        if original_parent is not None:
+            if original_parent.left == node:
+                original_parent.left = new_parent
+            elif original_parent.right == node:
+                original_parent.right = new_parent
+
+        return new_parent
 
     def visualize(self) -> str:
         # renders the tree top-down with the root on top and branches

notes-and-examples/2026.07.15/graphs-intro.md

@@ -0,0 +1,70 @@
+## Fundamentals
+
+- A *node* or *vertex* contains a data point
+- An *edge* is what connects one node to another
+
+A graph is just a collection of vertices and edges.
+
+> [!QUESTION] What is a real-life example of a simple graph with only vertices and edges?
+
+Some additional properties we can put on a graph:
+
+- A *weight* is a numerical value that can be assigned to an edge
+- We can also assign a *direction* to an edge, such that it only points from one node to another, not the other way around
+
+We can, of course, combine both of these properties too.
+
+> [!QUESTION] What's something we can model with weights in a graph? What about with directional edges?
+
+Going forward, we'll use $V$ to represent the number of vertices in a graph, and $E$ to represent the number of edges.
+
+## Representing a graph
+
+The *adjacency list* stores a list of connected vertices for each node, and it can fit in a dictionary or hash map structure.
+
+```
+1 -> 2, 4
+2 -> 1, 3
+3 -> 2
+4 -> 1
+```
+
+> [!QUESTION] How would you draw out this graph?
+
+> [!QUESTION] What would this mapping look like if we wanted to add weights? What about directional edges?
+
+The *adjacency matrix* is a 2D array where each position `arr[x][y]` represents an edge, and `x` and `y` each represent a node.
+
+```
+[
+    [0, 1, 0, 1],
+    [1, 0, 1, 0],
+    [0, 1, 0, 0],
+    [1, 0, 0, 0]
+]
+```
+
+> [!QUESTION] How would you draw out this graph? What would it look like as an adjacency list?
+
+> [!QUESTION] How do we add weights and/or directional edges to this graph?
+
+## Categorizing graphs
+
+We say a graph is *directed and acyclic*, or a *directed acyclic graph (DAG)*, if there are no cycles formed using the directional edges.
+
+> [!QUESTION] What's something we can model with a DAG?
+
+> [!QUESTION] What's another data structure we went over that also classifies as a DAG?
+
+A graph is *dense* if $E$ is closer to $V^2$, and *sparse* if $E$ is closer to $V$.
+
+> [!QUESTION] What's the implication for the graph if $E$ is closer to $V^2$, i.e. how is it different compared to a sparse graph?
+
+> [!QUESTION] What's the maximum number of edges we can have in a graph with no self-loops, relative to the number of vertices $V$?
+
+A graph can contain *self-loops* (a loop from a vertex to itself). A graph can also have *multiple edges* going from one vertex to another.
+
+> [!QUESTION] How do we represent a self-loop using an adjacency matrix?
+
+Finally, a graph is *connected* if every vertex is reachable from every other vertex.
+

notes-and-examples/2026.07.15/overview.md

@@ -0,0 +1,18 @@
+## Outline
+
+- Go over the homework
+- Go over the basics of graphs, see [Introduction to graphs](./graphs-intro.md)
+
+![Graphs introduction whiteboard](./graphs.png)
+
+To open the `.excalidraw` file:
+
+- Launch https://excalidraw.com
+- Click the top left menu, then select "Open"
+- Select the `.excalidraw` file from the file picker
+
+## Assignment
+
+- Try to complete the red-black implementation from last week
+- Complete the exercise for the 2D array (see the picture)
+