Bubble sort
Bubble sort, sometimes referred to as sinking sort, is a simple sorting algorithm that repeatedly steps through the list, compares adjacent elements and swaps them if they are in the wrong order. The pass through the list is repeated until the list is sorted. The algorithm, which is a comparison sort, is named for the way smaller or larger elements "bubble" to the top of the list. This simple algorithm performs poorly in real world use and is used primarily as an educational tool. More efficient algorithms such as quicksort, timsort, or merge sort are used by the  sorting libraries built into popular programming languages such as Python and Java. Analysis. Performance. Bubble sort has a worst-case and average complexity of formula_1, where formula_2 is the number of items being sorted. Most practical sorting algorithms have substantially better worst-case or average complexity, often formula_3. Even other formula_1 sorting algorithms, such as insertion sort, generally run faster than bubble sort, and are no more complex. Therefore, bubble sort is not a practical sorting algorithm. The only significant advantage that bubble sort has over most other algorithms, even quicksort, but not insertion sort, is that the ability to  efficiently detect that the list is sorted is built into the algorithm. When the list is already sorted (best-case), the complexity of bubble sort is only formula_5. By contrast, most other algorithms, even those with better average-case complexity, perform their entire sorting process on the set and thus are more complex. However, not only does insertion sort share this advantage, but it also performs better on a list that is substantially sorted (having a small number of inversions). Additionally, if this behavior is desired, it can be trivially added to any other algorithm by checking the list before  the algorithm runs. Rabbits and Turtles. The distance and direction that elements must move during the sort determine bubble sort's performance because elements move in different directions at different speeds. An element that must move toward the end of the list can move quickly because it can take part in successive swaps. For example, the largest element in the list will win every swap, so it moves to its sorted position on the first pass even if it starts near the beginning. On the other hand, an element that must move toward the beginning of the list cannot  move faster than one step per pass, so elements move toward the beginning very slowly. If the smallest element is at the end of the list, it will take formula_6 passes to move it to the beginning. This has led to these types of elements being named rabbits and turtles, respectively, after the characters in Aesop's fable of The Tortoise and the Hare. Various efforts have been made to eliminate turtles to improve upon the speed of bubble sort. Cocktail sort is a bi-directional bubble sort that goes from beginning to end, and then reverses itself, going end  to beginning. It can move turtles fairly well, but it retains formula_1 worst-case complexity. Comb sort compares elements separated by large gaps, and can move turtles extremely quickly before proceeding to smaller and smaller gaps to smooth out the list. Its average speed is comparable to faster algorithms like quicksort. Step-by-step example. Take an array of numbers "5 1 4 2 8", and sort the array from lowest number to greatest number using bubble sort. In each step, elements written in bold are being compared. Three passes will be required; Now, the array is already sorted, but  the algorithm does not know if it is completed. The algorithm needs one additional whole pass without any swap to know it is sorted. Implementation. Pseudocode implementation. In pseudocode the algorithm can be expressed as (0-based array): procedure bubbleSort(A : list of sortable items) end procedure Optimizing bubble sort. The bubble sort algorithm can be optimized by observing that the "n"-th pass finds the "n"-th largest element and puts it into its final place. So, the inner loop can avoid looking at the last "n" − 1 items when running for the "n"-th time: procedure bubbleSort(A : list of sortable items ) end procedure More generally, it can happen that more than one element is placed in their final position on a single pass. In particular, after every pass, all elements after the last swap are sorted, and do not need to be checked again. This allows to skip over many elements, resulting in about a worst case 50% improvement in comparison count (though no improvement in swap counts), and adds very little complexity because the new code subsumes the "swapped" variable: To accomplish this in pseudocode, the following can be written: procedure bubbleSort(A : list of sortable items) end procedure Alternate modifications , such as the cocktail shaker sort attempt to improve on the bubble sort performance while keeping the same idea of repeatedly comparing and swapping adjacent items. Use. Although bubble sort is one of the simplest sorting algorithms to understand and implement, its "O"("n") complexity means that its efficiency decreases dramatically on lists of more than a small number of elements. Even among simple "O"("n") sorting algorithms, algorithms like insertion sort are usually considerably more efficient. Due to its simplicity, bubble sort is often used to introduce the concept of an algorithm, or a sorting algorithm, to introductory computer science students . However, some researchers such as Owen Astrachan have gone to great lengths to disparage bubble sort and its continued popularity in computer science education, recommending that it no longer even be taught. The Jargon File, which famously calls bogosort "the archetypical [sic] perversely awful algorithm", also calls bubble sort "the generic bad algorithm". Donald Knuth, in "The Art of Computer Programming", concluded that "the bubble sort seems to have nothing to recommend it, except a catchy name and the fact that it leads to some interesting theoretical problems", some of which he then discusses. Bubble sort is asymptotically equivalent in  running time to insertion sort in the worst case, but the two algorithms differ greatly in the number of swaps necessary. Experimental results such as those of Astrachan have also shown that insertion sort performs considerably better even on random lists. For these reasons many modern algorithm textbooks avoid using the bubble sort algorithm in favor of insertion sort. Bubble sort also interacts poorly with modern CPU hardware. It produces at least twice as many writes as insertion sort, twice as many cache misses, and asymptotically more branch mispredictions. Experiments by Astrachan sorting strings in Java show bubble sort to  be roughly one-fifth as fast as an insertion sort and 70% as fast as a selection sort. In computer graphics bubble sort is popular for its capability to detect a very small error (like swap of just two elements) in almost-sorted arrays and fix it with just linear complexity (2"n"). For example, it is used in a polygon filling algorithm, where bounding lines are sorted by their "x" coordinate at a specific scan line (a line parallel to the "x" axis) and with incrementing "y" their order changes (two elements are swapped) only at intersections of two lines . Bubble sort is a stable sort algorithm, like insertion sort. Debate over name. Bubble sort has been occasionally referred to as a "sinking sort". For example, Donald Knuth describes the insertion of values at or towards their desired location as letting "[the value] settle to its proper level", and that "this method of sorting has sometimes been called the "sifting" or "sinking" technique. This debate is perpetuated by the ease with which one may consider this algorithm from two different but equally valid perspectives: In popular culture. In 2007, former Google CEO Eric Schmidt asked then-presidential candidate Barack Obama  during an interview about the best way to sort one million integers; Obama paused for a moment and replied: "I think the bubble sort would be the wrong way to go."