Understanding the Complexity Class of Bubble Sort

[Rajeev Bagra]

Bubble Sort is one of the most beginner-friendly sorting algorithms.

It helps learners understand:

• loops
• comparisons
• swapping
• iteration
• algorithmic thinking
• time complexity

Even though Bubble Sort is simple, it also teaches an important computer science concept:

Small optimizations do not always change the complexity class of an algorithm.

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Basic Idea of Bubble Sort

Bubble Sort repeatedly compares neighboring elements.

Example:

[5, 1, 4, 2]

First pass:

• Compare 5 and 1 → swap
• Compare 5 and 4 → swap
• Compare 5 and 2 → swap

Largest value moves to the end.

This is why it is called:

“Bubble” Sort

because large elements bubble upward toward the end.

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Why Bubble Sort Is Slow

Bubble Sort only fixes problems locally.

It compares:

neighbor ↔ neighbor

Even if a small value belongs at the front of the list, it can move only one position at a time.

That means many repeated passes are needed.

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Plain English View of Complexity

Suppose we have:

n elements

Bubble Sort may need:

• almost n comparisons in first pass
• almost n comparisons again in second pass
• again in third pass
• and so on

Example for 1000 items:

Pass	Comparisons
1	999
2	998
3	997

The total work grows very quickly.

That is why Bubble Sort becomes inefficient for large datasets.

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Mathematical Complexity of Bubble Sort

The number of comparisons approximately becomes:

(n-1)+(n-2)+(n-3)+\dots+1

This sum equals:

 

Expanding:

\frac{n(n-1)}{2}
=
\frac{n^2-n}{2}

Ignoring constants and smaller terms:

O(n^2)

This is called:

Quadratic Time Complexity

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What Does Really Mean?

If input size doubles:

n → 2n

then work becomes approximately:

 

So doubling input roughly quadruples work.

Example:

Elements	Approximate Work
10	100
100	10,000
1000	1,000,000

This rapid growth is why Bubble Sort is considered inefficient for large inputs.

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Bubble Sort Optimization Using a Flag

Many implementations include:

didswap = False

If no swaps happen during a pass, the algorithm stops early.

Example:

[1,2,3,4,5]

The list is already sorted.

Only one pass is needed.

---

Does the Flag Change Complexity Class?

No.

The flag improves:

• practical performance
• best-case runtime

Best case becomes:

O(n)

because only one scan occurs.

But complexity class usually refers to:

Worst-Case Complexity

Worst case example:

[5,4,3,2,1]

The algorithm still performs approximately:

\frac{n(n-1)}{2}

operations.

So worst-case complexity remains:

O(n^2)

---

Optimization by Reducing Range

Another optimization:

for j in range(1, len(listofnum) - i):

Reason:

After each pass, the largest element is already fixed at the end.

So there is no need to compare it again.

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Why This Still Does NOT Change Complexity Class

The total comparisons become:

(n-1)+(n-2)+(n-3)+\dots+1

which equals:

 

Big-O notation ignores:

• constants
• smaller-order terms

So:

\frac{1}{2}n^2-\frac{1}{2}n

still becomes:

O(n^2)

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Important Insight About Big-O Notation

Big-O measures:

Growth Trend

not exact operation count.

These are treated as equivalent growth classes:

n^2

10n^2

\frac{1}{2}n^2

because all grow quadratically.

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Core Structural Limitation of Bubble Sort

The optimizations help somewhat, but Bubble Sort still has a structural limitation:

• elements move slowly
• swaps are local
• many repeated comparisons occur

A small element may require many swaps to reach the beginning.

That fundamental behavior remains unchanged.

So the algorithm remains:

O(n^2)

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Final Takeaway

Bubble Sort optimizations:

✅ improve practical runtime
✅ improve best-case behavior
✅ reduce unnecessary comparisons

But they do NOT change the fundamental scaling law.

Bubble Sort still grows quadratically because the algorithm fundamentally relies on repeated local swaps.

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