Programming, problem solving, and algorithms

CPSC 203, 2026 W1

October 20, 2026

Announcements

• Practice examlet 3 available.
• Book your slot for examlet 3.
• Lab 6 (stacks) and POTW 6 (summations and stacks) due Sunday.
• Project 1 due Monday, October 26 (next week).

Let’s Take Stock

What Have We Done?

• Measuring complexity / efficiency.
• Representing information with computer data.
• Lists: Mutation, aliasing and comprehensions.
• Dataclasses for compound data (and methods).
• Pandas and dataframes for tabular data.
• Dictionaries for key-value lookup.
• Sprinkled throughout: Various bits of Python syntax and idiom.

We considered complexity as the size of the problem / amount of data increased. Focus on broad classes of complexity \(\log n\), \(n\), \(n \log n\), \(n^2\), \(n^3\), \(2^n\).

Computers represent everything as a binary number. But we can use numbers to represent anything else (at least approximately). Example was colour.

Dictionaries: Our previous data structures were chosen primarily for functionality (it was convenient that Pandas is fast, but mostly it is nice because it is easy to write filters and maps). We could implement a correct key-value store using lists, but we don’t. The dictionary is our first data structure we use primarily because of its efficiency for a particular operation.

Where Are We Going?

• Stacks & queues: Data structures of interest for their efficiency.
  • Application: Computing Voronoi diagrams.
• Graphs: Data structure of interest for representational power and (sometimes) efficiency.
  • Applications: Markov chains for random processes, solving Sudoku, shortest path on maps.
• Fundamentally complex problems: Traveling Salesperson.
• More graphs: textual analysis, social networks, gerrymandering election boundaries.
• All with sprinkling of yet more bits of Python syntax and idiom.

In CPSC 103, you learned about data types such as bool, int, str, range, enumeration, optional, compound and arbitrary-sized. To that CPSC 203 adds tables, dictionaries, stacks, queues and graphs.

In CPSC 103, you had:

• Simple rules mapping problem information into data type.
• A single code template for each type of data.
• Correct implementation as your singular focus.

In CPSC 203, we are breaking all of these simplifying features. You may choose different data types and code templates in order to achieve better efficiency when solving your problem.

Voronoi Diagrams

Voronoi diagrams

Given a (finite) set of “centres” \(c_{1}, c_{2}, \ldots, c_{k}\), a Voronoi region, \(R_{j}\) consists of the set of points nearer to centre \(c_{j}\), than to any other centre.

Together, the \(R_{j}\) regions compose the Voronoi diagram of a plane.

Every point in plane is in some region and no point is in 2 regions.

Applications

• Biology, Ecology, Forestry: Model and analyze plant competition.
• Zoology: Model and analyze the territories of animals.
• Robotics: Path planning in the presence of obstacles.
• Astronomy: Identify clusters of stars and clusters of galaxies.
• Cartography: Piece together satellite photographs into large “mosaic” maps.
• Geography: Analyzing patterns of urban settlements.
• Marketing: Model market of US metropolitan areas.
• Metallurgy: Modelling “grain growth” in metal films.

More Applications

• Physiology: Analysis of capillary distribution in cross-sections of muscle tissue to compute oxygen transport (“Capillary domains”).
• Meteorology: Estimate regional rainfall averages, given data at discrete rain gauges.
• Anthropology and Archeology: Identify regions under the influence of different neolithic clans, chiefdoms, ceremonial centers, or hill forts.
• Crystallography and Chemistry: Modelling alloy structures as sphere packings (“Domain of an atom”).
• Geology: Estimation of ore reserves in a deposit using info obtained from bore holes.

Motivational Aside: Pointillism

Un dimanche après-midi à l’Île de la Grande Jatte

Ferris Bueller’s Day Off: A Visit to the Art Institute of Chicago

A photo doesn’t communicate the scale of the painting. It took Seurat 18 months over two years to complete.

Algorithmic Art

1. Select a subset of points from the original image.
2. Use those points, with their colors, as centers in a new image of the same size.
3. Build the Voronoi diagram in the new image, using the center colors from the original image.

We are going to focus on calculating Voronoi regions over discrete domains, specifically images.

Choose some pixels as centers. For every other pixel, figure out which center is closest. For that operation, we need to compute distance between pixels (or some monotonically related measure).

Distance Between Pixels

Draw in the right angle triangle.

Give the pixels coordinates and write out the formula \(\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\).

Note that the location of the origin doesn’t matter.

Nearest Center

Note that we don’t need the exact distance, just which center is closer to yellow.

In particular, we don’t need to compute the square root, which is computationally expensive (compared to other arithmetic operations) and results in floating point numbers (ugly).

Design an Algorithm

Consider a pixel which is not a center (not coloured). What will we do?

What will we do for the whole image?

Planning the Data:

What data structures do we need?

• Point:
• Color:
• Center:
• Centers:
• Image:

• tuple[int, int]
• tuple[int, int, int]
• Point and Colour
• list[Center]
• PIL’s Image and PixelAccess

Planning the Algorithm:

How will data flow through our algorithm?

• Read input image.
• …

• Write output image.

1. Read image
2. Choose centers (now finished with original image!!)
3. Build new image – for each point, for each center find distance. Pick nearest center and use its colour.
4. Write out new image.

Let’s Practice!

Open today’s activity. We’ll start with the loop-only version of the algorithm.

• Launch Class Activity: Voronoi Art in PrairieLearn.
• Open voronoi_loop_nb.py.

Remember to demo doubling the density –> double the runtime.

Analysis

How much work is done?

1. Read image:
2. Choose centers:
3. Build new image:
4. Write out new image:

Size of the image: \(n\)
(\(n =\) width \(\times\) height)

Density of centers: \(d\)

Number of centers: \(dn\)

1. \(O(n)\).
2. Hopefully \(O(dn)\).
3. \(O(dn^2)\).
4. \(O(n)\).

Can we do better? Do we need to “visit” (do something for) every pixel? Yes: At a minimum we need to write out a colour for that pixel. So, we probably can’t avoid \(O(n)\). But what about the dependence on number of centers? Can we avoid doing something for each pixel-center combination?