Bubble Sort

** Bubble sort,** sometimes incorrectly referred to as **sinking 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 swaps are needed, which indicates that the

list is sorted. The algorithm gets its name from

the way smaller elements “bubble” to the top of

the list. Because it only uses comparisons to

operate on elements, it is a comparison sort. Although the algorithm is simple, most of the

other sorting algorithms are more efficient for

large lists.

Analysis

An example of bubble sort. Starting from the beginning of the list, compare every adjacent pair, swap their position if they are not in the right order (the latter one is smaller than the former one). After each iteration, one less element (the last one) is needed to be compared until there are no more elements left to be compared.

Performance

Bubble sort has worst-case and average complexity both О(n2), where n is the number of items being sorted. There exist many sorting algorithms with substantially better worst-case or average complexity of O(n log n). Even other О(n2) sorting algorithms, such as insertion sort, tend to have better performance than bubble sort. Therefore, bubble sort is not a practical sorting algorithm when n is large.

The only significant advantage that bubble sort has over most other implementations, even quicksort, but not insertion sort, is that the ability to detect that the list is sorted is efficiently built into the algorithm. Performance of bubble sort over an already-sorted list (best-case) is O(n). 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 have this mechanism too, but it also performs better on a list that is substantially sorted (having a small number of inversions).

Bubble sort should be avoided in case of large collections. It will not be efficient in case of reverse ordered collection.

Rabbits and turtles

The positions of the elements in bubble sort will play a large part in determining its performance. Large elements at the beginning of the list do not pose a problem, as they are quickly swapped. Small elements towards the end, however, move to the beginning extremely slowly. This has led to these types of elements being named rabbits and turtles, respectively.

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 O(n2) 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

Let us take the 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.

**First Pass: **

( **5 1** 4 2 8 ) ( **1 5** 4 2 8 ), Here, algorithm compares the first two elements, and swaps since 5 > 1.

( 1 **5 4** 2 8 ) ( 1 **4 5** 2 8 ), Swap since 5 > 4

( 1 4 **5 2 **8 ) ( 1 4 **2 5** 8 ), Swap since 5 > 2

( 1 4 2** 5 8** ) ( 1 4 2 **5 8** ), Now, since these elements are already in order (8 > 5), algorithm does not swap them.

**Second Pass:**

( **1 4** 2 5 8 ) ( **1 4 **2 5 8 )

( 1** 4 2** 5 8 ) ( 1 **2 4** 5 8 ), Swap since 4 > 2

( 1 2 **4 5 **8 ) ( 1 2 **4 5** 8 )

( 1 2 4 **5 8 **) ( 1 2 4 **5 8** ) Now, the array is already sorted, but our algorithm does not know if it is completed. The algorithm needs one whole pass without any swap to know it is sorted.

**Third Pass:**

(** 1 2** 4 5 8 ) ( **1 2 **4 5 8 )

( 1 **2 4** 5 8 ) ( 1 **2 4** 5 8 )

( 1 2 **4 5** 8 ) ( 1 2 **4 5** 8 )

( 1 2 4 **5 8** ) ( 1 2 4 **5 8** )

**In practice**

A bubble sort, a sorting algorithm that continuously steps through a list, swapping items until they appear in the correct order. The list was plotted in a Cartesian coordinate system, with each point (x,y) indicating that

the value y is stored at index x. Then the list would be sorted by Bubble sort according to every pixel’s value. Note that the largest

end gets sorted first, with smaller elements taking longer to move to their correct positions.

Although bubble sort is one of the simplest

sorting algorithms to understand and implement, its O(n2) complexity means that its efficiency decreases dramatically on lists of more than a small number of elements. Even among simple O(n2) 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.[1]

The Jargon file, which famously calls bogosort “the archetypical [sic] perversely awful

algorithm”, also calls bubble sort “the generic bad algorithm”.[2] Donald Knuth, in his famous book 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.[3]

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 requires 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 5 times slower than insertion sort and 40% slower than selection sort.[1]

In computer graphics it 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 (2n). 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 x axis) and with incrementing y their order changes (two elements are swapped) only at intersections of two lines.