diff --git a/notes-and-examples/2026.07.15/graphs-intro.md b/notes-and-examples/2026.07.15/graphs-intro.md new file mode 100644 index 0000000..1ad0bfa --- /dev/null +++ b/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. + diff --git a/notes-and-examples/2026.07.15/graphs.excalidraw b/notes-and-examples/2026.07.15/graphs.excalidraw new file mode 100644 index 0000000..10da892 --- /dev/null +++ b/notes-and-examples/2026.07.15/graphs.excalidraw @@ -0,0 +1,17688 @@ +{ + "type": "excalidraw", + "version": 2, + "source": "https://excalidraw.com", + "elements": [ + { + "id": "CdpR1bm5ny6F9dMrtBB7C", + "type": "freedraw", + "x": 229.89301986535696, + "y": 308.13964635181947, + "width": 32.94397164654953, + "height": 21.14314598211388, + "angle": 0, + "strokeColor": "#1e1e1e", + "backgroundColor": "transparent", + "fillStyle": "solid", + "strokeWidth": 0.5, + "strokeStyle": "solid", + "roughness": 1, + "opacity": 100, + "groupIds": [], + "frameId": null, + "index": "a0", + 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--git a/notes-and-examples/2026.07.15/graphs.png b/notes-and-examples/2026.07.15/graphs.png new file mode 100644 index 0000000..3a6999e Binary files /dev/null and b/notes-and-examples/2026.07.15/graphs.png differ diff --git a/notes-and-examples/2026.07.15/overview.md b/notes-and-examples/2026.07.15/overview.md new file mode 100644 index 0000000..f308d0a --- /dev/null +++ b/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) +