COSMO · STUDY + TEACHING DECK

Algorithms

February 22, 2026

A question worth sharing

Shortest Paths

Build optimal routes by proving when a partial distance is final.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use weighted graph, relaxation, shortest-path tree to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

Why can breadth-first search find shortest paths in an unweighted graph?

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

Try a small case and record every step.
Separate what changes from what stays fixed.
State the pattern in a sentence before naming the theorem.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

weighted graph

network with edge costs

relaxation

improving a tentative distance

shortest-path tree

predecessor edges for optimal routes

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

d(v)\leftarrow\min\left(d(v),d(u)+w(u,v)\right)

The idea

Build optimal routes by proving when a partial distance is final.

What to remember

An optimal route algorithm needs a proof that local commitments cannot be regretted.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Run Dijkstra's algorithm on edges AB=2, AC=5, BC=1, BD=4, CD=1 from A.

Predict each next step

Start d(A)=0, others infinity.
Finalize B with distance 2 and update C=3, D=6.
Finalize C with distance 3 and update D=4.
Finalize D with distance 4.

Which step needs a reason, not just a calculation?

See the structure

The structure behind shortest paths

d(v)\leftarrow\min\left(d(v),d(u)+w(u,v)\right)

Relax an edge to improve a tentative distance; Dijkstra's finalization requires nonnegative weights.

Read the representation

Build optimal routes by proving when a partial distance is final.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

Structural method

Name the objects and conditions.
Use a reusable relationship.
Explain why the method must work.

Brute force

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Investigate shortest paths

Your shared task

Students compare BFS and Dijkstra on the same map after weights are added.

Explore

4 min - test cases

Compare

5 min - form a rule

Defend

4 min - defend the rule

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

An optimal route algorithm needs a proof that local commitments cannot be regretted.
A complete solution names the condition that makes the method valid.
A counterexample is useful when it exposes a missing condition.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Why does Dijkstra fail with negative edges?

Predict each next step

A finalized vertex may later receive a cheaper route.
The proof relies on every extension adding nonnegative cost.
A negative edge breaks that monotonicity.
Bellman-Ford is designed for such cases.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

With nonnegative edge weights, the smallest tentative Dijkstra distance is final.

Build the argument

Any unseen route must first cross from finalized to unfinalized vertices.
That crossing begins with distance at least the chosen tentative value.
Nonnegative later edges cannot make the route cheaper.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

Construct a four-vertex example where the fewest-edge path is not the minimum-weight path.

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

Shortest-path reasoning supports navigation, routing, scheduling, and dependency planning.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

Why can breadth-first search find shortest paths in an unweighted graph?

Question 2

Explain why this claim is true: With nonnegative edge weights, the smallest tentative Dijkstra distance is final.

Question 3

Construct a four-vertex example where the fewest-edge path is not the minimum-weight path.

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

An optimal route algorithm needs a proof that local commitments cannot be regretted.

Use the matching practice sets to move from direct examples to proof and transfer.

A good question belongs to everyone in the room.