COSMO · STUDY + TEACHING DECK

Algorithms

March 28, 2026

A question worth sharing

Greedy Algorithms

Make locally best choices only when an exchange argument proves they can form a global optimum.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use greedy choice, exchange argument, counterexample 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

Does choosing the largest coin always minimize coin count?

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.

greedy choice

locally preferred next step

exchange argument

swap proving an optimal solution can use the greedy step

counterexample

instance disproving a rule

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

The core idea

Keep the idea. Explain the conditions.

\text{choose }I\text{ with the smallest finish time among compatible intervals}

The idea

Make locally best choices only when an exchange argument proves they can form a global optimum.

What to remember

A greedy choice earns trust through a proof, not through good-looking examples.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Use earliest-finish-time greedy selection for intervals [1,4],[3,5],[0,6],[5,7],[8,9].

Predict each next step

Choose [1,4], the earliest finishing interval.
Discard overlapping intervals.
Choose [5,7].
Then choose [8,9], for three intervals.

Which step needs a reason, not just a calculation?

See the structure

The structure behind greedy algorithms

\text{choose }I\text{ with the smallest finish time among compatible intervals}

For maximizing the number of nonoverlapping intervals, an exchange argument makes this choice safe.

Read the representation

Make locally best choices only when an exchange argument proves they can form a global optimum.

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 greedy algorithms

Your shared task

Teams propose greedy rules, attack them with counterexamples, then repair one rule with an exchange proof.

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

A greedy choice earns trust through a proof, not through good-looking examples.
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

Explain the exchange argument for interval scheduling.

Predict each next step

Take an optimal schedule whose first interval is not the greedy one.
Replace its first interval by the earliest-finishing interval.
The replacement leaves at least as much room for later intervals.
Thus an optimal schedule exists with the greedy choice.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

A greedy algorithm is correct only when every local choice can be extended to an optimum.

Build the argument

Fast local improvement is not enough.
An exchange or cut argument connects the choice to a global solution.
A single counterexample defeats an unsupported rule.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

Construct the smallest target where largest-coin-first fails for coins 1,4,5.

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

Greedy methods solve scheduling, spanning trees, compression, and some resource-allocation problems.

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

Does choosing the largest coin always minimize coin count?

Question 2

Explain why this claim is true: A greedy algorithm is correct only when every local choice can be extended to an optimum.

Question 3

Construct the smallest target where largest-coin-first fails for coins 1,4,5.

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

A greedy choice earns trust through a proof, not through good-looking examples.

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

A good question belongs to everyone in the room.