The bubble sort algorithm is a straightforward sorting technique that operates by repeatedly comparing and swapping adjacent elements that are out of order.

In essence, bubble sort systematically traverses the list to be sorted, comparing each pair of adjacent elements and swapping them if they are not in the desired order. This process continues until the entire list is sorted. The name “bubble sort” is derived from the way smaller or larger elements “bubble” up to their correct positions in the list.

An optimized version of bubble sort introduces a flag that tracks whether any swaps occurred during the current iteration. If no swaps are made, it indicates that the list is already sorted, eliminating the need for further iterations. This optimization reduces unnecessary passes over the list, thereby enhancing the algorithm’s efficiency.

To illustrate, let’s consider an example with a list of five elements: $[5, 3, 8, 4, 2]$. The bubble sort algorithm begins at the start of the list, comparing the first two elements, $5$ and $3$. Since $5$ is greater than $3$ and we are sorting in ascending order, these two elements are swapped. Next, the algorithm compares the subsequent pair, $5$ and $8$. As these elements are already in the correct order, no swap occurs. This process is repeated for each adjacent pair until the algorithm has traversed the entire list.

After the first pass, the largest element will have “bubbled up” to the end of the list. The algorithm then restarts from the beginning but only processes up to the second-to-last element, as the last element is now correctly positioned. This procedure continues, with each subsequent pass examining one fewer element than the previous pass, until no swaps are necessary.

From a coding perspective, while bubble sort is not the most efficient algorithm for large datasets due to its relatively high computational complexity, it is praised for its simplicity and ease of implementation. It performs particularly well with nearly sorted or smaller datasets. Moreover, the optimized version that employs a swap-checking flag can significantly enhance performance in best-case scenarios, such as when the list is already sorted.