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A small step before this topic
Let’s picture it Let’s start with three stops.
Home. Then the park. Then the café. Just a few things in a row.
One little change Point to the middle one.
The park is second. In Python, its position is 1, because counting starts at 0.
Now it has a name That’s a list.
A list keeps things in order. An index is just a position in that list.
No pressure. Just try it.Which place is at position 0?
1 / 4 A small step before this topic
Let’s picture it Now visit each stop.
You don’t need three different instructions. “Visit the next place” works each time.
One little change One, then the next.
Finish home. Move to the park. Then do the same at the café.
Now it has a name That repeat is a loop.
One small action, repeated for each item. Three stops mean three visits.
No pressure. Just try it.Add one more stop. How many visits now?
1 / 4 A small step before this topic
Let’s picture it What if it rains?
Let’s choose the café on rainy days, and the park on sunny days.
One little change Today is sunny.
The rule stays the same. The weather changed, so we choose the other place.
Now it has a name That’s an if / else.
Raining?
Yes: café
No: park
A condition is a yes-or-no question. Its answer chooses which step happens next.
No pressure. Just try it.It’s raining. Where does our rule send us?
1 / 4 A small step before this topic
Let’s picture it Name a tiny job.
Our job is “add one.” Give it 3, and it gives back 4.
One little change Try a new input.
Give the same job 8. It gives back 9. We can reuse the instructions.
Now it has a name That job is a function.
Input goes in. An answer comes back. In Python, return sends the answer back.
No pressure. Just try it.What does add-one return for 5?
1 / 4 A small step before this topic
Let’s picture it Picture a pile of plates.
Blue goes down first. Pink goes on top.
One little change Take the top one.
Pink comes off first, even though blue arrived first.
Now it has a name That’s a stack.
Last in, first out. Paused function calls work this way too: finish the newest call, then return.
No pressure. Just try it.Add green on top. Which plate comes off next?
1 / 4 A small step before this topic
Let’s picture it Now picture a line of people.
Ana arrives first. Bo waits behind her.
One little change Ana goes first.
Bo moves to the front. New arrivals join the back.
Now it has a name That’s a queue.
First in, first out. Unlike plates, we serve the oldest waiting item first.
No pressure. Just try it.Who goes before Cy?
1 / 4 A small step before this topic
Let’s picture it Have we visited the park?
Keep a little checklist of places we’ve seen.
One little change Write park again.
We still only need one check beside park. We’re remembering whether, not how many times.
Now it has a name That’s a set.
A set keeps distinct items. “Is park in here?” becomes a quick membership check.
No pressure. Just try it.Add park twice. How many copies in a set?
1 / 4 A small step before this topic
Let’s picture it Let’s remember a little more.
The park is five minutes away. Write that beside its name.
One little change Look it up later.
Ask for “park.” The notebook gives you 5. No need to measure the walk again.
Now it has a name That’s a dictionary.
A key finds a value. Here, the place name is the key and the time is the value.
No pressure. Just try it.We now write park: 6. What does park return?
1 / 4 A small step before this topic
Let’s picture it Draw our little walk.
A dot for each place. A line where we can walk directly.
One little change Two short walks can join.
Home to park, then park to café. We can reach the café without a direct road.
Now it has a name Dots and links have names.
A dot is a node. A link is an edge. Together, they make a graph.
No pressure. Just try it.Must two places share a direct road to be reachable?
1 / 4 A small step before this topic
Let’s picture it Where did we put the book?
Try “second shelf, first box.” We need two positions, not just one.
One little change Pick a row, then a column.
Python calls the second row 1. The first column is 0. So the book is at [1][0].
Now it has a name That’s a two-dimensional table.
Two labels find one cell. A third label could tell us which cupboard contains the table.
No pressure. Just try it.Two cupboards have the same shelves. What extra label helps?
1 / 4 A small step before this topic
Let’s picture it Same door. One small difference.
You have a key. Your friend doesn’t. The door is the same, but your next move isn’t.
One little change Remember the key.
If we only write down “at the door,” we lose something important.
Now it has a name That little note is a state.
A state remembers what we need to continue. Here: where we are, and whether we have the key.
No pressure. Just try it.Is “at the door” enough to decide if we can enter?
1 / 4 A small step before this topic
Let’s picture it How many little jobs?
Visit three places. Say hello once at each. That’s three hellos.
One little change Now try every pair.
Three people each greet all three people, including themselves for this toy count. That’s nine greetings.
Now it has a name Count how the work grows.
One pass grows with n. Trying every pair grows with n × n. Code length doesn’t tell us that.
No pressure. Just try it.With 4 items, how many ordered pairs, including self-pairs?
1 / 4 A small step before this topic
Let’s picture it Find 8 in a sorted row.
Start with 2, 4, 6, 8, 10. Look at the middle: 6.
One little change Keep the side that can work.
8 is bigger than 6. Everything on the left is too small, so we can leave it behind.
Now it has a name Keep one promise.
If 8 is present, it stays in the part we keep. That repeating promise is called an invariant.
No pressure. Just try it.Would this left-or-right shortcut work on an unsorted row?
1 / 4 A small step before this topic
Let’s picture it Open a box.
There’s a smaller box inside. We already know how to open a box.
One little change Do the same small job again.
Open the smaller box. Keep going until there isn’t another box inside.
Now it has a name That’s the idea of recursion.
A function asks itself to handle a smaller case. The empty box is our stopping point: the base case.
No pressure. Just try it.What if every box contains another box forever?
1 / 4 A small step before this topic
Let’s picture it A question can make two more.
Climb one or two stairs at a time. Four stairs can lead to a smaller question with three or two left.
One little change Look, the same question returns.
ways(3) also asks for ways(2). We’ve reached that smaller question twice.
Now it has a name Notice the repeated work.
Draw the calls to spot repeated questions. Later, we’ll save their answers instead of solving them again.
No pressure. Just try it.Do two calls with the same input always need separate work here?
1 / 4 A small step before this topic
Let’s picture it Count a tiny family of dots.
One parent has two children. Each child says, “There’s one of me.”
One little change Let the answers travel up.
The parent adds the two answers, then counts itself: 1 + 1 + 1.
Now it has a name That works for bigger trees too.
my answer
left answer
right answer
Ask each child about its little tree. Combine the answers. You don’t need to hold the whole tree in your head.
No pressure. Just try it.Do we also count the parent itself?
1 / 4 A small step before this topic
Let’s picture it Two trees can make different promises.
In a search tree, smaller keys go left and larger keys go right. We’ll use distinct keys here.
One little change A heap promises something else.
A min-heap keeps the smallest value at the top. Its left and right sides need not be sorted.
Now it has a name Choose the promise you need.
Looking for an ordered key? Think search tree. Repeatedly taking the smallest? Think min-heap.
No pressure. Just try it.Does a min-heap promise its left side is smaller than its right?
1 / 4 A small step before this topic
Let’s picture it Only the smallest waiting item matters next.
Tasks cost 5,1,4. We need the cheapest task now, then the next cheapest after adding new tasks.
One little change A heap orders one frontier, not everything.
minimum root
partly ordered below
The root is smallest. Removing it repairs only the tree paths needed to expose the next minimum.
Now it has a name Push and pop logarithmically.
Use a heap when priorities change incrementally and repeatedly sorting the entire collection would waste work.
No pressure. Just try it.Does iterating the internal heap array return sorted order?
1 / 4 A small step before this topic
Let’s picture it Come back to our little map.
Home, park, café. Each place is a dot; each allowed walk is a link.
One little change A dot can also mean a situation.
“At park with a key” may allow a move that “at park without a key” cannot.
Now it has a name Choose dots that tell the whole story.
situation
legal move
new situation
A graph organizes possible moves. Sometimes places are enough. Sometimes a place plus a small extra fact is needed.
No pressure. Just try it.Must graph nodes always represent physical places?
1 / 4 A small step before this topic
Let’s picture it Try the nearest places first.
From home, find everything one road away before looking two roads away.
One little change Put new places at the back.
A queue keeps the earlier discoveries in front. That lets us finish one layer before the next.
Now it has a name That’s breadth-first search: BFS.
With equal-cost roads, the first layer reaching a place uses the fewest roads. Unequal costs need more care.
No pressure. Just try it.One road costs 100 minutes; two others cost 1 each. Fewest roads means fastest?
1 / 4 A small step before this topic
Let’s picture it Follow one path for a while.
Instead of finishing a whole layer, keep walking down one branch. This is depth-first search, or DFS.
One little change Remember where you’ve been.
For reachability, a seen set helps avoid walking around the same cycle forever.
Now it has a name But ask what “seen” means.
this attempt
undo mark
next attempt
For routes that cannot repeat a place, a place used in one attempt may be free in another. Clear path-only marks when returning.
No pressure. Just try it.Does “used in this path” always mean “forbidden in every path”?
1 / 4 A small step before this topic
Let’s picture it Socks before shoes.
Some jobs must wait for other jobs. Draw an arrow from the earlier job to the later one.
One little change Start with a ready job.
Do a job with nothing unfinished before it. Remove it from the waiting list and look for the next ready job.
Now it has a name That makes a dependency order.
A waits for B
B waits for A
Its name is topological order. A cycle means some jobs keep waiting on one another, so no such order exists.
No pressure. Just try it.A waits for B, and B waits for A. Can either start under these rules?
1 / 4 A small step before this topic
Let’s picture it Try a snack combination.
Pick apple first. See what can go with it. We’re exploring, not committing forever.
One little change Put it back and try again.
After exploring the apple choices, undo that pick. Now start with banana.
Now it has a name That’s backtracking.
Choose, explore, undo. Undoing keeps the next attempt from accidentally inheriting the last attempt’s choices.
No pressure. Just try it.Before a sibling attempt, what happens to path-only choices?
1 / 4 A small step before this topic
Let’s picture it This bag is already too heavy.
The limit is 5 kg. We’ve packed 6 kg, and every remaining item has positive weight.
One little change More items cannot fix that.
6 kg
add more
still too heavy
We can stop exploring this branch. There’s no need to try every heavier version.
Now it has a name That early stop is pruning.
Skip a branch when you can explain why it cannot work. “It looks unpromising” is not enough.
No pressure. Just try it.If later choices can subtract weight, is “already too heavy” enough to stop?
1 / 4 A small step before this topic
Let’s picture it We’ve answered this before.
route A
3 stairs left
route B
Two routes leave us with three stairs left. We may climb one or two at a time.
One little change Save these three little routes.
The routes are 1+1+1, 1+2, and 2+1. Count them once; both arrivals can reuse the answer.
Now it has a name That sharing is the heart of DP.
same question
reuse answer
Dynamic programming saves answers to smaller questions. Share only when the remaining question really is the same.
No pressure. Just try it.One route has a key left and the other doesn’t. Always the same remaining question?
1 / 4 A small step before this topic
Let’s picture it Two ways to fill a notebook.
One way: when a question appears, solve it if needed, then write down the answer.
One little change Or prepare the small answers first.
Fill answers in an order where each needed earlier answer is already available.
Now it has a name Same idea, different order.
Saving answers on demand is memoization. Filling them in a planned order is tabulation.
No pressure. Just try it.When filling a table, should an answer be ready before another cell uses it?
1 / 4 A small step before this topic
Let’s picture it Pick snacks, but not neighbors.
The values are 4, 5, 4. Taking the middle snack blocks both neighbors.
One little change Taking less first can win.
Take the first 4 and the last 4. Their total is 8, which beats 5.
Now it has a name Ask about the row from here.
At position i, compare taking it and moving two places, or skipping it and moving one. One position names this smaller question.
No pressure. Just try it.After taking position i, where can we continue?
1 / 4 A small step before this topic
Let’s picture it Make 3 using coins 1 and 2.
We could use 1 then 2, or 2 then 1.
One little change What are we counting?
Those are two orders, but the same combination of coins. The question decides whether to count them separately.
Now it has a name Same ingredients, different tasks.
fewest?
orders?
combinations?
Fewest coins, number of orders, and number of combinations are different questions. Decide which answer one cell is storing.
No pressure. Just try it.If order doesn’t matter, are 1+2 and 2+1 two combinations?
1 / 4 A small step before this topic
Let’s picture it Walk across four squares.
Start at the top-left. Reach the bottom-right. For now, moves only go right or down.
One little change Two tiny routes.
Right, then down. Or down, then right. Both reach the same finish.
Now it has a name A square can share its answer.
Row and column name the remaining journey. With only right/down moves, we can fill answers without moving in circles.
No pressure. Just try it.If moves can go in circles, is this simple fill order automatically safe?
1 / 4 A small step before this topic
Let’s picture it Will this snack fit?
snack: 3
room: 4 ✓
room: 2 ×
The snack weighs 3. A bag with 4 spaces can take it; a bag with 2 cannot.
One little change Remember the room left.
Seeing the same next item doesn’t mean having the same choices. The remaining capacity matters too.
Now it has a name Give the question two labels.
For 0/1 knapsack, remember which item comes next and how much capacity remains. Each item can be chosen at most once.
No pressure. Just try it.For 0/1 knapsack, after taking an item, can we take that same item again?
1 / 4 A small step before this topic
Let’s picture it Take the biggest snack first?
Values are 4, 5, 4, and neighbors cannot both be taken. The biggest is 5.
One little change But the two 4s make 8.
The tempting first choice blocked a better total. Looking best now wasn’t enough.
Now it has a name Try to break a greedy rule.
Greedy means committing to a local choice. Before trusting it, ask what that choice might block later.
No pressure. Just try it.Does one failing example disprove “always take the biggest” here?
1 / 4 A small step before this topic
Let’s picture it Fit as many meetings as possible.
Meetings cannot overlap. Choosing the one that ends earliest leaves the most room afterward.
One little change Can we swap without losing?
later finish
earlier finish ✓
Replace another schedule’s first meeting with the earliest-ending one. Later meetings still fit because we finish no later.
Now it has a name That swap explains the choice.
This is an exchange argument. It shows why earliest finish is safe for maximizing meeting count, not for every meeting objective.
No pressure. Just try it.Would this explanation also prove maximum total reward when meetings pay different amounts?
1 / 4 A small step before this topic
Let’s picture it Now meetings have rewards.
One long meeting pays 100. Two short, compatible meetings pay 10 each.
One little change More meetings isn’t more reward.
The two short meetings total 20. The long one wins if we care about payment.
Now it has a name Bring back take or skip.
skip
take + compatible best
Compare skipping this meeting with taking it plus the best compatible earlier reward. This turns into DP.
No pressure. Just try it.Does earliest finish alone still solve the reward version?
1 / 4 A small step before this topic
Let’s picture it Each value extends reach.
At position 0 with value 2, we can reach positions 1 or 2. We do not need to choose the exact landing yet.
One little change Keep the farthest reachable boundary.
Every index at or before farthest can be processed. Its jump may extend the same reachable prefix.
Now it has a name A gap ends the proof.
If the scan reaches an index beyond farthest, no earlier reachable position could jump across that gap.
No pressure. Just try it.Must greedy choose one exact jump path?
1 / 4 A small step before this topic
Let’s picture it Deadlines create a prefix constraint.
A course taking 3 days with deadline 5 must fit before day 5 alongside all earlier chosen courses.
One little change Sort by deadline.
early deadline
later deadline
Processing earliest deadlines first makes each prefix the tightest set that must already fit.
Now it has a name If late, replace the longest course.
too late
remove longest
same count potential
Removing the largest chosen duration frees the most time while losing only one course. A max-heap finds it.
No pressure. Just try it.Why remove the longest chosen duration when a deadline is missed?
1 / 4 A small step before this topic
Let’s picture it Frequent symbols should stay shallow.
freq 10 → shallow
freq 1 → deeper
A symbol used 10 times should usually have a shorter code than one used once. Code cost is frequency times depth.
One little change Merge the two smallest weights.
Make the least frequent symbols siblings. Their combined weight represents one subtree in the remaining problem.
Now it has a name Repeat with a min-heap.
Each merge adds the two weights to total encoded length. The heap exposes the next two smallest subtrees.
No pressure. Just try it.What does a merged subtree’s weight represent?
1 / 4 Keep a connection for later.
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