Asymptotic Notation Big o omega theta – Big o Notation in Data Structure – Asymptotic Bounds

The line in question:

“Example: If an algorithm takes at most 3n² + 2n + 1 operations, then it’s O(n²).”

What it means:

An algorithm might have a time complexity that looks like this:

T(n) = 3n² + 2n + 1

This means:

• When the input size n increases, the number of operations grows like:
  • 3n² → dominates for large n
  • 2n and 1 → smaller contributions

Why it becomes O(n²):

When using Big O (O) notation, we care only about how the algorithm grows for large input sizes, so:

• We ignore constants like 3.
• We ignore smaller terms like 2n2n2n and 111, because they grow slower than n2n^2n2.

So, we say:

T(n) = 3n² + 2n + 1 ∈ O(n²)

That means:

“This algorithm’s time won’t grow faster than n² in the worst case.”

Real-life analogy:

Imagine someone says:

“It’ll take 3 hours, plus a bit more time for traffic and rest stops.”

We say:

“Okay, we can assume it takes about 3 hours, and ignore the small extras.”

Summary:

• 3n² + 2n + 1 means the number of steps depends mostly on n².
• In Big O, we write just the dominant term: O(n²).

The Symbol: ∈

In math, the symbol: ∈

means “belongs to” or “is in”.

So, this expression:

T(n) = 3n² + 2n + 1 ∈ O(n²)

Is read as:

“T(n) belongs to the set of functions that grow like O(n²)“

Or more simply:

“T(n) is in Big O of n²“

Alternative (More Common) Way to Write It:

Instead of using the ∈ symbol, most people write:

T(n) = O(n²)

This means the same thing — that the function grows at most like n², ignoring constants and smaller terms.

Final Explanation:

• ✅ ∈ O(n²) → Correct but more formal
• ✅ = O(n²) → Common and easier to understand

📌 The line:

“Real-world analogy: Like saying a delivery takes at most 2 hours, even if it’s usually faster.”

✅ What it means:

This is an analogy (a comparison) to explain Big O Notation (O).

💡 Big Idea Behind Big O:

• Big O gives the maximum time an algorithm could take.
• Even if it usually runs faster, Big O tells us the worst-case time.

🛵 Real-World Example:

Imagine you’re ordering food online.

• The app says:
  “Delivery will take at most 2 hours.”

Even though:

• The delivery usually takes only 40 minutes.
• Sometimes it’s even faster.

But the app guarantees:

“No matter what, it will never take more than 2 hours.”

🎯 Why this analogy is useful:

It helps you understand that Big O is about:

• The maximum possible time (just like the delivery time limit)
• Not the usual or average time

So even if an algorithm is often fast, if it could take a long time in the worst case, we use Big O to describe that upper limit.