Programming, problem solving, and algorithms

CPSC 203, 2026 W1

October 29, 2026

Announcements

• Examlet 3 today through Monday.
• Lab 7 (in-place list manipulation) and POTW 7 (numpy arrays) due Sunday.
• Project 2 (Game of Life) released, due Monday November 16.
  • One week grace period for projects at 50% credit.
• Final exam:
  • Booking opens Tuesday December 1.
  • Slots available Tuesday December 15 through Saturday December 19 (inclusive).
• Feeling overwhelmed? You are not alone!
  • Visit student health services’ mental health resources.

Graphs

Graphs: A Model for Representing Relationships between Entities

• Nodes or vertices represent entities.
• Edges denote a relationship.
• Nodes are neighbours if they share an edge.

• How many nodes does this graph have?
• How many edges does this graph have?
• What are the neighbours of B?
• There are many variations; for example:
  • Edges can be directional.
  • Nodes may be disconnected.
  • There may be multiple edges between two nodes.

Graphs: Representation for Humans

A graph is a pair of lists \(G = (V,E)\).

• The nodes / vertices \(V = \{ v_i \}\).
• The edges are pairs of nodes \(E = \{ (v_i, v_j) \}\).

Yes, in CS we almost always use the letter \(V\) even though we often call them nodes rather than vertices.

Work through the representation:

• \(V = \{ E, C, A, D, B, F \}\)
• \(E = \{ (E,C), (C,A), (C,D), (A,B), (D,B), (B,F) \}\)

Does the order of the nodes / edge-pairs matter? No.

Does the order of nodes within an edge pair matter? Not if edges are bi-directional / the graph is undirected.

Surprise: We Have Already Been Using a Graph

Consider the neighbourhood of a pixel in our Voronoi algorithm.

• Each pixel is a node in a graph.
• Edges of central pixel (denoted by “+”) shown in red.
• Edges of all other pixels shown in black.

Our original visualization

4 neighbours

8 neighbours

Example Graphs

The Internet, circa 2003

In this case:

• \(V\): The routers which communicate between local networks.
• \(E\): Identify which routers are connected together.

The degree of a vertex is the number of edges to which it connects.

Degree is the number of neighbours / adjacent vertices (in an undirected graph).

A Conflict Graph

The nodes of this graph are labelled by their shape and colour.

• \(V\): Courses at the university.
• \(E\): Connect courses with a common student enrolled in both.

Labels can be attached to nodes or edges to capture features of the underlying problem.

• What is special about this labelling?

All graphs give at least a unique label / name to each node so that we can tell the nodes apart; for example, the indexes of the pixels in our Voronoi art. Additional labels (not necessarily unique) can help to solve the underlying problem; for example, the colour of the pixels.

Hint: Don’t look at the three red boxes in the upper right. They look connected but they are actually overlapping edges.

Because no two courses with the same label are connected by an edge, this graph represents a feasible exam scheduling (each label is a different exam time).

Vertex labeling problem: In this case the labels are chosen by the algorithm. Unfortunately, there is no efficient algorithm known for creating an optimal labelling. We could try to loop through all possible labellings, but that would be infeasibly slow.

A Weighted Graph

• \(V\): Major characters in The Office.
• \(E\): Conversations. Weight denotes the number of conversations.

A graph is complete if it contains all possible edges.

• Is this graph complete?

How many edges in a complete graph of \(n\) vertices?

It is not complete: Look at Jan and Holly.

Every node connects to every other node: \(n(n-1)\). But that double counts each edge, so divide by 2. - Looks like \(O(n^2)\).

Rush Hour Game

• \(V\): State of the board / position of all cars / configuration.
• \(E\): Connect two configurations if there is a legal move between them.

A path is a sequence of vertices with an edge between every adjacent pair of vertices.

• Try to get the red car out through the exit. Vehicles can only move up/down or left/right.
• Green blobs are a path to a solution.

Social Network (Cinda’s Facebook in 2012)

• \(V\): Cinda’s friends.
• \(E\): Connect friends who are friends of each other.

A graph is connected if there is a path between any pair of vertices.

A connected component is a subgraph which is connected.

• Clusters are hometown, extended family, workplace, …
• All friends are actually connected through Cinda, but in this graph Cinda’s edges do not appear. There are a few disconnected nodes.
• When we refer to a connected component, we typically make them as big as possible.
• Draw an example graph with disconnected components.

State Machine: Is a Number Divisible by Seven?

• Start at the circle node at the top.
• Read the number from left to right.
• For each digit \(d\) in the given number, follow \(d\) blue / solid edges in succession.
• As you move from one digit to the next, follow 1 red / dashed edge.
• If you end up back at the circle node, your number is divisible by 7.

Consider the number 3703.

In a directed graph, edges connect nodes only in one direction.

• How do you check for divisible by 2 / 5? Check last digit.
• Divisible by 3? Sum the digits and check if that is divisible by 3. Use recursion.
• Divisible by 4? Check if last two digits are divisible by 4.
• Divisible by 6? Divisible by 3 and 2.

Note that even though all nodes are connected to two blue and two red edges, one is incoming and one is outgoing.

The Stanford Bunny

• \(V\): Points on the surface of the bunny.
• \(E\): Connect vertices into triangles.

In a planar graph, the vertices can be placed on the plane and the graph drawn without any crossing edges.

An early example of a 3D object represented in a computer. Used extensively in academic graphics research.

• More points were kept in regions with high curvature.
• Edges are chosen to try to minimize the distance between the triangles and the actual surface.
• It is then easy to colour / shade / add hair to little triangles.
• The whole bunny is not a planar graph, but local patches of it are because it is a triangularization.
• Note that (unlike our previous examples but like our Voronoi images) the vertices have positions in 3D space (a form of label).

Complexity of Graph Algorithms

Graph Explorations

A graph \(G\) is defined by a set of vertices \(V\) and a set of edges \(E\).

• We will often denote the size of these sets by \(|V|\) and \(|E|\).

Algorithm complexity is often described in terms of the number of vertices.

• Actual execution time usually depends on the number of edges.
• So how are \(|V|\) and \(|E|\) related?

Why describe complexity by \(|V|\) when it actually depends on \(|E|\)?

• For real problems, we often have an estimate or intuition about \(|V|\) but not \(|E|\).
• For many problems, we generate \(E\) (or even \(V\)) on the fly.

Demo

Open today’s activity. You’ll work through some graph size calculations.

• Launch Class Activity: Graph Explorations in PrairieLearn.

If possible, work through at least the “How many edges?” question.