Your concept feed | DSA & Design, Connected

Start small

Let’s picture it

Let’s start with three stops.

🏠 Home

🌳 Park

☕ Café

Home. Then the park. Then the café. Just a few things in a row.

Swipe when you’re ready

One little change

Point to the middle one.

0 · Home

1 · Park

2 · Café

The park is second. In Python, its position is 1, because counting starts at 0.

Same idea. One more small step.

Now it has a name

That’s a list.

Home

Park

Café

A list keeps things in order. An index is just a position in that list.

Want to try a tiny question?

No pressure. Just try it.

Which place is at position 0?

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Start small

One small step at a time.

Nothing to learn first. Let’s replay this little example.

Let’s start with three stops.

🏠 Home

🌳 Park

☕ Café

Home. Then the park. Then the café. Just a few things in a row.

1 / 3

No need to read the code yet. It’s here when you want to connect the picture to Python.

Show the Python example

stops = ["home", "park", "cafe"]
assert stops[1] == "park"
assert len(stops) == 3

More detail, examples, and edge cases

The foundations

Let’s picture it

Same door. One small difference.

You + key

Locked door

You have a key. Your friend doesn’t. The door is the same, but your next move isn’t.

Swipe when you’re ready

One little change

Remember the key.

at door

has key?

If we only write down “at the door,” we lose something important.

Same idea. One more small step.

Now it has a name

That little note is a state.

place

key: yes

A state remembers what we need to continue. Here: where we are, and whether we have the key.

Want to try a tiny question?

No pressure. Just try it.

Is “at the door” enough to decide if we can enter?

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The foundations

One small step at a time.

This idea builds on these smaller ones. Pick any to revisit it here.

What if it rains?

☔ Rain

☕ Café

Let’s choose the café on rainy days, and the park on sunny days.

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No need to read the code yet. It’s here when you want to connect the picture to Python.

Show the Python example

def can_enter(cell, tokens_left):
    return cell == "." or tokens_left > 0

assert can_enter("#", 1)
assert not can_enter("#", 0)

More detail, examples, and edge cases

Recursion

Let’s picture it

Open a box.

Big box

Small box

Empty

There’s a smaller box inside. We already know how to open a box.

Swipe when you’re ready

One little change

Do the same small job again.

Big box

Small box

Empty

Open the smaller box. Keep going until there isn’t another box inside.

Same idea. One more small step.

Now it has a name

That’s the idea of recursion.

smaller

stop

A function asks itself to handle a smaller case. The empty box is our stopping point: the base case.

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What if every box contains another box forever?

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Recursion

One small step at a time.

This idea builds on these smaller ones. Pick any to revisit it here.

Same door. One small difference.

You + key

Locked door

You have a key. Your friend doesn’t. The door is the same, but your next move isn’t.

1 / 3

No need to read the code yet. It’s here when you want to connect the picture to Python.

Show the Python example

def suffix_sum(values):
    def total(i):
        if i == len(values):
            return 0
        return values[i] + total(i + 1)
    return total(0)

assert suffix_sum([4, 2, 7]) == 13
assert suffix_sum([]) == 0

More detail, examples, and edge cases

Arrays, search & sorting

Let’s picture it

Try every pair first.

1

2

4

7

In a sorted row [1,2,4,7], seek a pair totaling 9. Checking every pair repeats work.

Swipe when you’re ready

One little change

Look at both ends.

1 ↑

7 ↑

sum 8

1 + 7 is too small. Moving the right pointer left only makes the sum smaller, so move the left pointer right.

Same idea. One more small step.

Now it has a name

Discard with a reason.

too small → left

too large → right

equal ✓

Now 2 + 7 = 9. Each move removes an entire family of impossible pairs, so both pointers move only forward overall.

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When the sum is too small, which pointer can safely move?

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1 / 4

Arrays, search & sorting

One small step at a time.

This idea builds on these smaller ones. Pick any to revisit it here.

Let’s start with three stops.

🏠 Home

🌳 Park

☕ Café

Home. Then the park. Then the café. Just a few things in a row.

1 / 3

Python for Two pointers: shrink one search space. The complete example includes small checks you can run; the full notes state its scope.

Show the complete Python solution

def two_sum_sorted(values, target):
    left, right = 0, len(values) - 1
    while left < right:
        total = values[left] + values[right]
        if total == target:
            return left, right
        if total < target:
            left += 1
        else:
            right -= 1
    return None

assert two_sum_sorted([1, 2, 4, 7], 9) == (1, 3)
assert two_sum_sorted([1, 2, 4, 7], 20) is None

More detail, examples, and edge cases

Trees

Let’s picture it

Count a tiny family of dots.

Parent

Child: 1

One parent has two children. Each child says, “There’s one of me.”

Swipe when you’re ready

One little change

Let the answers travel up.

Total: 3

1

The parent adds the two answers, then counts itself: 1 + 1 + 1.

Same idea. One more small step.

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.

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No pressure. Just try it.

Do we also count the parent itself?

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Trees

One small step at a time.

This idea builds on these smaller ones. Pick any to revisit it here.

Open a box.

Big box

Small box

Empty

There’s a smaller box inside. We already know how to open a box.

1 / 3

No need to read the code yet. It’s here when you want to connect the picture to Python.

Show the Python example

def height(node):
    if node is None:
        return 0
    _, left, right = node
    return 1 + max(height(left), height(right))

tree = (8, (3, None, None), (10, None, (12, None, None)))
assert height(tree) == 3
assert height(None) == 0

More detail, examples, and edge cases

Graphs

Let’s picture it

Come back to our little map.

Home

Park

Café

Home, park, café. Each place is a dot; each allowed walk is a link.

Swipe when you’re ready

One little change

A dot can also mean a situation.

Park + key

Park, no key

“At park with a key” may allow a move that “at park without a key” cannot.

Same idea. One more small step.

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.

Want to try a tiny question?

No pressure. Just try it.

Must graph nodes always represent physical places?

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1 / 4

Graphs

One small step at a time.

This idea builds on these smaller ones. Pick any to revisit it here.

Same door. One small difference.

You + key

Locked door

You have a key. Your friend doesn’t. The door is the same, but your next move isn’t.

1 / 3

No need to read the code yet. It’s here when you want to connect the picture to Python.

Show the Python example

def neighbors(row, col, rows, cols):
    for dr, dc in [(1, 0), (-1, 0), (0, 1), (0, -1)]:
        nr, nc = row + dr, col + dc
        if 0 <= nr < rows and 0 <= nc < cols:
            yield nr, nc

assert set(neighbors(0, 0, 2, 3)) == {(1, 0), (0, 1)}

More detail, examples, and edge cases

Backtracking

Let’s picture it

Try a snack combination.

Apple

Banana

Pear

Pick apple first. See what can go with it. We’re exploring, not committing forever.

Swipe when you’re ready

One little change

Put it back and try again.

Apple

Banana

Pear

After exploring the apple choices, undo that pick. Now start with banana.

Same idea. One more small step.

Now it has a name

That’s backtracking.

choose

explore

undo

Choose, explore, undo. Undoing keeps the next attempt from accidentally inheriting the last attempt’s choices.

Want to try a tiny question?

No pressure. Just try it.

Before a sibling attempt, what happens to path-only choices?

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Backtracking

One small step at a time.

This idea builds on these smaller ones. Pick any to revisit it here.

A question can make two more.

ways(4)

ways(3)

ways(2)

Climb one or two stairs at a time. Four stairs can lead to a smaller question with three or two left.

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No need to read the code yet. It’s here when you want to connect the picture to Python.

Show the Python example

def subsets(values):
    answers, path = [], []
    def visit(i):
        if i == len(values):
            answers.append(path.copy())
            return
        visit(i + 1)               # Exclude this item.
        path.append(values[i])     # Include this item.
        visit(i + 1)
        path.pop()                 # Restore the caller's path.
    visit(0)
    return answers

assert subsets(["A", "B"]) == [[], ["B"], ["A"], ["A", "B"]]
assert subsets([]) == [[]]

More detail, examples, and edge cases

Tries

Let’s picture it

Car and cat start the same way.

c

a

r / t

Write c, then a, just once. Split where the words become different.

Swipe when you’re ready

One little change

A prefix isn’t always a word.

c

a

r

t ✓

If we only stored cart, reaching c-a-r is not enough to say car was stored too.

Same idea. One more small step.

Now it has a name

Mark where each word ends.

car ✓

cart ✓

cat ✓

This shared-prefix tree is a trie. The word-end mark tells a whole word apart from just its beginning.

Want to try a tiny question?

No pressure. Just try it.

Only cart is stored. Does car count as a stored word?

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Tries

One small step at a time.

This idea builds on these smaller ones. Pick any to revisit it here.

Count a tiny family of dots.

Parent

Child: 1

One parent has two children. Each child says, “There’s one of me.”

1 / 3

No need to read the code yet. It’s here when you want to connect the picture to Python.

Show the Python example

def insert(root, word):
    node = root
    for char in word:
        node = node.setdefault(char, {})
    node[None] = True

def follow(root, text):
    node = root
    for char in text:
        if char not in node:
            return None
        node = node[char]
    return node

trie = {}
insert(trie, "cart")
assert follow(trie, "car") is not None
assert None not in follow(trie, "car")
insert(trie, "car")
assert None in follow(trie, "car")
assert follow(trie, "dog") is None

More detail, examples, and edge cases

Thinking in DP

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.

Swipe when you’re ready

One little change

Save these three little routes.

1+1+1

1+2

2+1

The routes are 1+1+1, 1+2, and 2+1. Count them once; both arrivals can reuse the answer.

Same idea. One more small step.

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.

Want to try a tiny question?

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One route has a key left and the other doesn’t. Always the same remaining question?

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Thinking in DP

One small step at a time.

This idea builds on these smaller ones. Pick any to revisit it here.

Same door. One small difference.

You + key

Locked door

You have a key. Your friend doesn’t. The door is the same, but your next move isn’t.

1 / 3

No need to read the code yet. It’s here when you want to connect the picture to Python.

Show the Python example

from functools import cache

def count_climbs(n):
    @cache
    def ways(r):
        if r == 0:
            return 1
        if r < 0:
            return 0
        return ways(r - 1) + ways(r - 2)
    return ways(n)

assert count_climbs(5) == 8
assert count_climbs(0) == 1

More detail, examples, and edge cases

Greedy choices

Let’s picture it

Take the biggest snack first?

4

5

4

Values are 4, 5, 4, and neighbors cannot both be taken. The biggest is 5.

Swipe when you’re ready

One little change

But the two 4s make 8.

4 ✓

5 ×

4 ✓

The tempting first choice blocked a better total. Looking best now wasn’t enough.

Same idea. One more small step.

Now it has a name

Try to break a greedy rule.

best now?

best overall?

Greedy means committing to a local choice. Before trusting it, ask what that choice might block later.

Want to try a tiny question?

No pressure. Just try it.

Does one failing example disprove “always take the biggest” here?

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Greedy choices

One small step at a time.

This idea builds on these smaller ones. Pick any to revisit it here.

Find 8 in a sorted row.

2

4

6

8

10

Start with 2, 4, 6, 8, 10. Look at the middle: 6.

1 / 3

No need to read the code yet. It’s here when you want to connect the picture to Python.

Show the Python example

def largest_first(coins, target):
    used = 0
    for coin in sorted(coins, reverse=True):
        count, target = divmod(target, coin)
        used += count
    return used if target == 0 else -1

assert largest_first([1, 3, 4], 6) == 3  # Demonstrates the bad rule.
assert 3 + 3 == 6                      # A better two-coin construction.

More detail, examples, and edge cases

Connecting it all

Let’s picture it

You don’t have to guess the name.

remember?

next moves?

Start smaller: what do I need to remember, and what can I do next?

Swipe when you’re ready

One little change

Then look for one useful clue.

layers

saved answers

safe choices

Equal-cost moves suggest BFS. Repeated remaining questions suggest DP. A safe local choice suggests greedy.

Same idea. One more small step.

Now it has a name

Check the clue before choosing.

small example

reason

name

Change one rule and ask if the explanation still works. An algorithm’s name comes after the reason, not before it.

Want to try a tiny question?

No pressure. Just try it.

Should a familiar keyword replace checking the problem’s rules?

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Connecting it all

One small step at a time.

This idea builds on these smaller ones. Pick any to revisit it here.

Do we still need yesterday’s notes?

older rows

yesterday

today

Suppose today’s row only reads yesterday’s row. Rows before yesterday will never be used again.

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