Optional Lesson: Algorithms Detour - Minimum Spanning Tree
In this and the subsequent lesson, we consider some of the strategies used to construct networks and find paths for data in them. While this has a connection to ideas about the Internet, the focus of these lessons is on algorithms, formal techniques, and processes for solving problems. Students will explore and solve the Minimum Spanning Tree (MST) problem, first, in an unplugged fashion on paper. The real challenge is not in solving a particular instance of the minimum spanning tree, but to develop an algorithm, a clear series of steps, that if followed properly, will solve any instance of the problem. There is a possible misconception to look out for: the MST has a definite, verifiable optimal solution, as opposed to the Text Compression problem (from Unit 1), which does not.
The MST is a problem in a field of study known as Graph Theory in mathematics and computer science. Problems involving graphs come up a lot in computer science, not only related to networking problems, but also in describing more sophisticated or interconnected relationships between data and information, for example, complicated scheduling problems, logistics, or even sociology problems, or interactions between molecules. Many real-world problems can be expressed or visualized as graphs.
The MST problem is interesting because it has an optimal best solution, and the algorithm for finding the MST on a graph is relatively straightforward to understand. The interesting thing about the MST is that a graph might have multiple “best” solutions, and there are several different algorithms for finding them.
Students will be able to:
- Use the terms algorithm, graph, node, edge correctly.
- Write an algorithm for solving the minimum spanning tree (MST) problem.
- Explain the benefits of developing an algorithm for solving a problem versus solving an instance of a problem.
- Identify the minimum spanning tree on a given graph.
- Copies of Activity Guide
For the Teacher
For the Students
- Algorithms, Graphs and the MST Problem - Activity Guide
Present this diagram on the board or projector.
Let’s say that you are in charge of not one router, but several, and your job is to connect them so that a) any packet can get from any router to any other router in your system, and b) you want to build these connections as cheaply as possible.
Say you are in charge of 4 routers placed at various locations in a region. The diagram shows the possible connections that could be made between any pair of routers and the associated cost of building a connection between them (in millions of dollars).
Question: Which connections would you choose to build to keep the total cost down?
Let students work it out in pairs or individually and share solutions. (See Key: Lesson Plan Activities for solutions.)
Question: Can you come up with a clear strategy, or process, for finding the minimal-cost connections you need, not only for this network, but any network, especially if the network were bigger?
Why do we care about coming up with a strategy or process for solving this problem, rather than just finding an answer?
Questions (also in Code Studio):
What does a minimum spanning tree tell you about a graph?
a) The shortest path from a particular point in the graph to another point in the graph.
b) The shortest path between any two points for all points in the graph
c) The fewest number and smallest total distance of connections necessary to connect all points in a graph
d) Whether or not the graph represents a network
e) The fewest number and smallest total distance of connections necessary to travel from one point to all the other points without having to visit a point twice
The images below all show the same map (or graph) as the one depicted above, but have different paths between the points highlighted. Please choose the image that is highlighting a Minimum Spanning Tree for the map. NOTE: The “distance” between points is depicted by the number of line segments connecting any two points.
Go over the answers and discuss (See Key: Lesson Plan Activities for solutions.)
Establish that the minimum spanning tree for the first graph on the worksheet had a total cost of 25, and there were two possible solutions to the minimum spanning tree.
If necessary, act out the algorithm, or ask students to act out the algorithm on the original graph, or a new one that they just make up.
Ask students to compare and contrast their strategies with the one above.
Was your algorithm clearly written and easy to understand? Were there two different algorithms that still solved the problem correctly?
Here is an informal proof of the algorithm we wrote above. Note: you do not need to go over this with students, but many students might want to know why this works.
1) First you need 9 edges to make a spanning tree. Think about any configuration of 10 nodes with no edges connecting them. Say they are all in line, or one in the middle with others surrounding it. You’d need a minimum of 9 edges to string them all together. If there are n nodes, you always need one less (n-1) to minimally connect them.
2) If you take all of the edge weights from the graph and line them up in order:
2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6
We know that we need 9 of these edges to connect the 10 nodes. So let’s look at the 9 edges that we chose as a result of the algorithm:
2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6
These are the 9 lowest possible values we could select out of the entire list except for a 3. Why not one of those 3s? If you look at the graph there is a triangular set of edges with weights of 2, 2, and 3. Adding the 3 would make a cycle, so we can’t use it. Other than that 3, the 9 edges we’ve chosen are actually the 9 smallest numbers in the entire graph. There is no set of 9 edges that could possibly give us a lower total cost.
Transition to Activity Guide.
The Activity Guide - Algorithms, Graphs and the MST Problem - Activity Guide
- Provides some initial background reading that covers:
- What are algorithms and why do we want them?
- Defining terms for graph problems: Graph, Node, Edge, Cost, Cycle, Tree
- Explaining the Minimum Spanning Tree problem
- Gives a few examples of graphs to experiment with finding the MST
- Asks students to write an algorithm, or series of steps, that would find the MST for any graph.
It’s common for students to struggle at first and give a strategy that is essentially a restatement of the problem. For example: “My strategy is to randomly try edges until I find the set that connects all the nodes with minimal cost.” Don’t let them do this. While it’s fine to choose edges at random, what’s missing from this strategy is how or why an edge is kept as part of the set or discarded.
Some questions that might help students get started or make progress:
- How do you know when to stop? i.e. How do you know you’ve found the minimum?
- Which edge should you start with?
- Can you define a strategy for considering an edge, then either keeping it or discarding it?
It’s possible students might make the wrong connections between this and the Text Compression activity from Unit 1, which also asked them to write out a process for solving that problem.
The difference between the two is that the Text Compression problem actually has no known efficient way for finding the best or optimal solution. To know the best solution you’d have to produce every possible compression for a piece of text and see which one came out the best. The only algorithms that exist do a decent job of compressing the text but they are just heuristics.
The minimum spanning tree has a definite best solution. The algorithm will always find the optimal solution. The interesting part is that since it’s possible that in choosing the next edge you have to pick randomly from a set of choices that are equally good, it means that the MSTs produced by the same algorithm might be different trees but they will have the same total cost.
- You might check out some real pseudocode and a visualization of the minimum spanning tree in action on VisuAlgo.
- Go here: https://visualgo.net/en/mst?slide=1
- From the list of algorithms, choose “Kruskal’s Algorithm.”
- You can see some of the other algorithms for solving the MST problem as well.
- Reflection: starting out in computer science
- (click tabs to see student view)
Starting out in Computer Science
Computer science has changed the way we communicate with each other, make art and movies, grow food, and even treat illnesses. Everyone can learn computer science and make a difference.
Quotes from students
Still, we understand that taking a computer science course can be difficult at first. Here are a few student quotes describing their strategies and tips for taking this course. Please read the quotes carefully and respond to the prompt below.
In the first week of this class I was falling behind quickly. There was a lot of new information to learn. To keep up, I had to find a better way to study. I tried to find connections between the material and what I already know. That really helped me remember things. I also tried to not overdo it. I started taking small breaks in-between lessons and when I came back I checked if I still remembered what I was studying before. It helped a lot
Sofia P. (age 16)
Some days I felt tired and would drift away in my thoughts. It was a real problem because I would miss so much of what we were learning. So I started going to bed a bit earlier and I tried my best to pay attention. At the end of every class our teacher summarized what we learned that day and that was really helpful. I started taking more notes because that also kept my mind from wandering. These little tricks got me through the class and I learned more.
Jasmin D. (age 17)
I can be pretty forgetful sometimes and it was a problem in this class. I think it's because we did so much on the computer. For my other classes I take notes on paper and read through them again at home. So the trick that I found helpful in this class was to take notes on paper anyway and to test myself about the concepts. I wasn't sure if it would work at first, but I think it ended up being a big help.
Sam J. (age 17)
Now consider the strategies and insights for how to learn best that you just read.
Reflect and Summarize:
What are your own strategies and insights about how to learn best? And, how are they similar or different to the ones that you just heard about from other students?
Please write a short paragraph. Don't worry about spelling, grammar, or how well written it is.
CSTA K-12 Computer Science Standards (2011)
CT - Computational Thinking
- CT.L3B:3 - Critically examine classical algorithms and implement an original algorithm.
- CT.L3B:4 - Evaluate algorithms by their efficiency, correctness, and clarity.
- CT.L3B:6 - Compare and contrast simple data structures and their uses (e.g., arrays and lists).
Computer Science Principles
4.1 - Algorithms are precise sequences of instructions for processes that can be executed by a computer and are implemented using programming languages.
4.1.1 - Develop an algorithm for implementation in a program. [P2]
- 4.1.1B - Sequencing is the application of each step of an algorithm in the order in which the statements are given.
- 4.1.1H - Different algorithms can be developed to solve the same problem.
- 4.1.1I - Developing a new algorithm to solve a problem can yield insight into the problem.
4.1.2 - Express an algorithm in a language. [P5]
- 4.1.2A - Languages for algorithms include natural language, pseudocode, and visual and textual programming languages.
- 4.1.2B - Natural language and pseudocode describe algorithms so that humans can understand them.
- 4.1.2C - Algorithms described in programming languages can be executed on a computer.
- 4.1.2F - The language used to express an algorithm can affect characteristics such as clarity or readability but not whether an algorithmic solution exists.
- 4.1.2I - Clarity and readability are important considerations when expressing an algorithm in a language.