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def bubble_sort(arr):
    n = len(arr)
    for i in range(n):
        # Repeat, excluding the already sorted section at the end of the list
        for j in range(0, n-i-1):
            # Compare adjacent elements and swap if necessary
            if arr[j] > arr[j+1]:
                arr[j], arr[j+1] = arr[j+1], arr[j]
    return arr


example_list = [64, 34, 25, 12, 22, 11, 90]

sorted_list = bubble_sort(example_list)

print('Sorted array', sorted_list) # [11, 12, 22, 25, 34, 64, 90]

# Understanding Bubble Sort in Detail

## 1. Utilizing Nested For Loops

Bubble Sort employs two nested loops. The outer loop determines the number of passes through the array, while the inner loop compares each element of the array.

```python title="Utilizing Nested For Loops"
for i in range(n):
  for j in range(0, n-i-1):
```

<br />

## 2. Element Comparison and Swap

Within the inner loop, adjacent elements are compared and swapped if necessary.

```python title="Element Comparison and Swap Example"
if arr[j] > arr[j+1]:
  arr[j], arr[j+1] = arr[j+1], arr[j]
```

<br />

## Bubble Sort Time Complexity

1. `Worst-case Time Complexity`: O(n^2)

   - This occurs when the array is sorted in reverse order compared to the desired sort. In this case, every element must be compared with every other element. If the size of the array is n, it requires (n x n) comparisons.

2. `Average Time Complexity`: O(n^2)

   - Even in typical situations where array elements are randomly arranged, approximately (n x n) comparisons are needed.

Understanding Bubble Sort in Detail