Bubble sort is one of the simplest sorting algorithms that works by repeatedly swapping adjacent elements if they are in the wrong order. This process is repeated until the list is sorted. Despite its simplicity, bubble sort is not the most efficient algorithm for large datasets, but it provides a great introduction to the concept of sorting in computer science.

## How Bubble Sort Works

The algorithm gets its name from the way larger elements “bubble” to the top of the list (end of the array), while the smaller elements sink to the bottom (beginning of the array). The algorithm “bubbles up” the largest non-sorted element on each iteration. it’s crucial to understand that as we progress with each pass through the array, the right side of the array begins to hold sorted elements.

Here’s a step-by-step explanation:

**Start with the first element in the array.****Compare the current element with the next element.****If the current element is greater than the next element, swap them. This ensures that in each comparison, the larger of the two elements moves closer to its correct position on the right.****Move to the next element and repeat the comparison until the end of the array is reached.****After completing one full pass through the array, the largest unsorted element will have “bubbled up” to its correct position at the end of the array. This element is now considered sorted and will no longer be involved in subsequent passes.****Repeat the process for the remaining unsorted portion of the array. With each pass, the unsorted portion of the array reduces by one element, as a new element is correctly positioned at the end of the array and added to the sorted section.****This process continues until no swaps are needed, indicating that the list is sorted. At this point, the right side of the array contains all the elements in their sorted order, securely positioned after each iteration.**

By understanding that the right side of the array progressively becomes populated with sorted elements after each pass, we can visualize how bubble sort segments the array into sorted and unsorted portions, methodically expanding the sorted section until the entire array is ordered.

## Implementation

Here’s a simple Python implementation of bubble sort:

```
def bubble_sort(arr):
n = len(arr)
for i in range(n):
swapped = False
for j in range(0, n-i-1):
if arr[j] > arr[j+1]:
arr[j], arr[j+1] = arr[j+1], arr[j]
swapped = True
if not swapped:
break
return arr
```

Let’s break down the implementation so we can understand it:

controls the number of passes through the array, aiming for`i`

`n`

iterations, where`n`

is the array’s length. Each pass potentially reduces the unsorted portion by positioning the largest element at the end of it.iterates over the unsorted portion of the array, comparing and, if necessary, swapping adjacent elements (`j`

`j`

and`j+1`

). This moves the larger element closer to its final position with each comparison., a boolean flag, is used to track if any swaps were made during a pass. If no swaps occur, it indicates the array is sorted, allowing for early termination of the algorithm to save time.`swapped`

Notice that the swapped `variable`

is not mandatory, and the implementation will also work without it. It also does not actually improve the runtime complexity (since the worst-case scenario does not change) but it might cause the code to run faster in some cases.

## Time and Space Complexity

Let’s break down the runtime and space complexity for bubble sort:

Case | Time Complexity | Memory (Space) Complexity |
---|---|---|

Best Case | O(n) | O(1) |

Worst Case | O(n^2) | O(1) |

### Worst-case scenario

Let’s denote n as the number of elements in the array. In the worst-case scenario:

**First Pass:**We need to do n-1 comparisons and potentially n-1 swaps to move the largest element to the end of the array.**Second Pass:**We need to do n-2 comparisons and potentially n-2 swaps, as the last element is already sorted.**Third Pass:**This involves n-3 comparisons and potentially n-3 swaps, and so on.**Last Pass:**Only a single comparison (and potentially a swap) is needed.

To find the total number of comparisons (and the maximum number of swaps), we sum up the comparisons needed for each pass. We can ignore the swaps as these multiply our complexity by 2 and is just a constant.

Total Comparisons/Swaps = (n−1)+(n−2)+(n−3)+…+2+1

The sum of this series can be calculated using the formula for the sum of the first n natural numbers: \frac{n(n+1)}{2}

Substituting the actual count of elements to n-1, we get:

\frac{(n−1)n}{2}=O(n^2)

### Best-case scenario

This occurs when the array is already sorted. The algorithm only needs to make one pass through the array to confirm that no swaps are needed, resulting in a linear time complexity O(n).

### Space Complexity

Regardless of the case (best, or worst), bubble sort only requires a constant amount of additional memory for variables used in swapping elements, making its space complexity O(1). This indicates that bubble sort is an in-place sorting algorithm.

## The Reverse Variation

The reverse variation of bubble sort is essentially the same algorithm as the traditional bubble sort, but it operates in the opposite direction. Instead of larger elements “bubbling” up to the end of the array, smaller elements “bubble down” to the beginning of the array with each iteration.

In this variation, the algorithm starts comparisons from the end of the array and moves towards the beginning, swapping adjacent elements if they are in the wrong order (i.e., if the next element is smaller than the current one). This process ensures that with each pass, the smallest unsorted element moves to its correct position at the beginning of the array, gradually sorting the entire array from the beginning to the end.

The reverse bubble sort maintains the same time complexity as the standard bubble sort, O(n2) in the worst case, and O(n) in the best case when the array is already sorted in the desired order. The main difference lies in the direction of the sorting process.