How does bubble sort algorithm work, and what is its complexity?

Bubble sort works by repeatedly swapping adjacent elements if they are in the wrong order, with a complexity of O(n^2).

Bubble sort is a simple sorting algorithm that works by repeatedly stepping through the list to be sorted, comparing each pair of adjacent items and swapping them if they are in the wrong order. The pass through the list is repeated until no more swaps are needed, indicating that the list is sorted. The algorithm gets its name from the way smaller or larger elements "bubble" to the top of the list.

Each pass through the list places the next largest value in its proper place. In essence, each item "bubbles" up to the location where it belongs. If an array has n elements, then bubble sort would need n-1 passes to ensure that every element has been compared with every other element.

The complexity of bubble sort is O(n^2) in both average and worst-case scenarios. This is because each comparison involves a pair of elements, and in the worst case, we have to make n*(n-1)/2 comparisons, which simplifies to O(n^2). This makes bubble sort inefficient on large lists, and generally inferior to other sorting algorithms like quicksort, mergesort, or heapsort.

However, bubble sort has a best-case time complexity of O(n) when the input list is already sorted, as only one pass is needed to confirm that all elements are in order. It also has the advantage of being able to detect that the list is sorted efficiently if it is allowed to run through the entire list without having to swap any elements. This makes it useful in certain situations where a list is expected to be almost sorted, such as in real-time systems where the data is almost in sorted order and the overhead of other sorting algorithms would be unnecessary.

In terms of space complexity, bubble sort is very efficient with a complexity of O(1), as it only requires a single additional memory space for temp variable used for swapping. This makes it a good choice in situations where memory space is a limiting factor.