Bubble sort

Bubble sort is a simple sorting algorithm. It works by repeatedly stepping through the list to be sorted, comparing two items at a time 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.

Analysis

Example of bubble sort sorting a list of random numbers.

Best-case performance

Bubble sort has best-case complexity Ω(n). When a list is already sorted, bubblesort will pass through the list once, and find that it does not need to swap any elements. Thus bubblesort will make only n comparisons and determine that list is completely sorted. It will also use considerably less time than О(n²) if the elements in the unsorted list are not too far from their sorted places.

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 does pretty well, but it still retains O(n²) worst-case complexity. Comb sort compares elements large gaps apart 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

example:
5 1 4 2 8 - unsorted array
1 4 2 5 8 - after one pass
1 2 4 5 8 - sorted array

Pseudocode implementation

A simple way to express bubble sort in pseudocode is as follows:

procedure bubbleSort( A : list of sortable items ) defined as:
  do
    swapped := false
    for each i in 0 to length( A ) - 2 do:
      if A[ i ] > A[ i + 1 ] then
        swap( A[ i ], A[ i + 1 ] )
        swapped := true
      end if
    end for
  while swapped
end procedure

The algorithm can also be expressed as:

procedure bubbleSort( A : list of sortable items ) defined as:
  for each i in 1 to length(A) do:
     for each j in length(A) downto i + 1 do:
       if A[ j -1  ] > A[ j ] then
         swap( A[ j - 1],  A[ j ] )
       end if
     end for
  end for
end procedure

The difference between this and the first pseudocode implementation is discussed later in the article.

Alternative implementations

One way to optimize bubblesort is to note that, after each pass, the largest element will always move down to the bottom. During each comparison, it is clear that the largest element will move downwards. Given a list of size n, the n^th element will be guaranteed to be in its proper place. Thus it suffices to sort the remaining n - 1 elements. Again, after this pass, the n - 1^th element will be in its final place. In pseudocode, this will cause the following change:

procedure bubbleSort( A : list of sortable items ) defined as:
  n := length( A )
  do
    swapped := false
    n := n - 1
    for each i in 0 to n do:
      if A[ i ] > A[ i + 1 ] then
        swap( A[ i ], A[ i + 1 ] )
        swapped := true
      end if
    end for
  while swapped
end procedure

We can then do bubbling passes over increasingly smaller parts of the list. More precisely, instead of doing n² comparisons (and swaps), we can use only n + (n-1) + (n-2) + ... + 1 comparisons. This sums up to n(n + 1) / 2, which is still O(n²), but which can be considerably faster in practice.

In practice

Although bubble sort is one of the simplest sorting algorithms to understand and implement, its O(n²) complexity means it is far too inefficient for use on lists having more than a few elements. Even among simple O(n²) sorting algorithms, algorithms like insertion sort are usually considerably more efficient, unless the data is already in nearly sorted order. 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 perversely awful algorithm", also calls bubble sort "the generic bad algorithm".[2] Donald Knuth, in his famous 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 discusses therein. 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.

Variations

• Odd-even sort is a parallel version of bubble sort, for message passing systems.

References

• Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89685-0. Pages 106–110 of section 5.2.2: Sorting by Exchanging.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Problem 2-2, pg.38.
• Sorting in the Presence of Branch Prediction and Caches

External links

Wikibooks Algorithm implementation has a page on the topic of

Bubble sort

• Bubble Sort in 20 languages
• Nice graphical visualization of BubbleSort with source code
• Bubble Sort Java Applet
• Bubble Sort Demo
• Bubble Sort Demonstration
• Lafore's Bubble Sort
• Sorting Applets in C++
• C++ Program - Bubble Sort
• C++ Program - Bubble Sort (with an explanation)
• Bubble Sort video and C++ Code
• Practical demonstration
• Analyze Bubble Sort in an online Javascript IDE

•  • e Sorting algorithms
Theory	Computational complexity theory | Big O notation | Total order | Lists | Stability | Comparison sort
Exchange sorts	Bubble sort | Cocktail sort | Comb sort | Gnome sort | Quicksort
Selection sorts	Selection sort | Heapsort | Smoothsort | Strand sort
Insertion sorts	Insertion sort | Shell sort | Tree sort | Library sort | Patience sorting
Merge sorts	Merge sort
Non-comparison sorts	Radix sort | Bucket sort | Counting sort | Pigeonhole sort | Tally sort
Others	Topological sorting | Sorting network