 Cocktail sort

Cocktail sort Class Sorting algorithm Data structure Array Worst case performance О(n²) Best case performance O(n) Average case performance О(n²) Cocktail sort, also known as bidirectional bubble sort, cocktail shaker sort, shaker sort (which can also refer to a variant of selection sort), ripple sort, shuffle sort,^{[1]} shuttle sort or happy hour sort, is a variation of bubble sort that is both a stable sorting algorithm and a comparison sort. The algorithm differs from bubble sort in that it sorts in both directions on each pass through the list. This sorting algorithm is only marginally more difficult to implement than bubble sort, and solves the problem with socalled turtles in bubble sort.
Contents
Pseudocode
The simplest form of cocktail sort goes through the whole list each time:
procedure cocktailSort( 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 // test whether the two elements are in the wrong order swap( A[ i ], A[ i + 1 ] ) // let the two elements change places swapped := true end if end for if swapped = false then // we can exit the outer loop here if no swaps occurred. break dowhile loop end if swapped := false for each i in length( A )  2 to 0 do: if A[ i ] > A[ i + 1 ] then swap( A[ i ], A[ i + 1 ] ) swapped := true end if end for while swapped // if no elements have been swapped, then the list is sorted end procedure
The first rightward pass will shift the largest element to its correct place at the end, and the following leftward pass will shift the smallest element to its correct place at the beginning. The second complete pass will shift the second largest and second smallest elements to their correct places, and so on. After i passes, the first i and the last i elements in the list are in their correct positions, and do not need to be checked. By shortening the part of the list that is sorted each time, the number of operations can be halved (see bubble sort).
procedure cocktailSort( A : list of sortable items ) defined as: // `begin` and `end` marks the first and last index to check begin := 1 end := length( A )  2 do swapped := false // increases `begin` because the elements before `begin` are in correct order begin := begin + 1 for each i in begin to end do: if A[ i ] > A[ i + 1 ] then swap( A[ i ], A[ i + 1 ] ) swapped := true end if end for if swapped = false then break dowhile loop end if swapped := false // decreases `end` because the elements after `end` are in correct order end := end  1 for each i in end to begin do: if A[ i ] > A[ i + 1 ] then swap( A[ i ], A[ i + 1 ] ) swapped := true end if end for while swapped end procedure
Differences from bubble sort
Cocktail sort is a slight variation of bubble sort. It differs in that instead of repeatedly passing through the list from bottom to top, it passes alternately from bottom to top and then from top to bottom. It can achieve slightly better performance than a standard bubble sort. The reason for this is that bubble sort only passes through the list in one direction and therefore can only move items backward one step each iteration.
An example of a list that proves this point is the list (2,3,4,5,1), which would only need to go through one pass of cocktail sort to become sorted, but if using an ascending bubble sort would take four passes. However one cocktail sort pass should be counted as two bubble sort passes. Typically cocktail sort is less than two times faster than bubble sort.
Another optimization can be that the algorithm remembers where the last actual swap has been done. In the next iteration, there will be no swaps beyond this limit and the algorithm has shorter passes. As the Cocktail sort goes bidirectionally, the range of possible swaps, which is the range to be tested, will reduce per pass, thus reducing the overall running time.
Complexity
The complexity of cocktail sort in big O notation is O(n^{2}) for both the worst case and the average case, but it becomes closer to O(n) if the list is mostly ordered before applying the sorting algorithm, for example, if every element is at a position that differs at most k (k ≥ 1) from the position it is going to end up in, the complexity of cocktail sort becomes O(k * n).
Cocktail sort is also briefly discussed in the book The Art of Computer Programming, along with similar refinements of bubble sort. In conclusion, Knuth states about bubble sort and its improvements (Knuth 1998, p. 110):
But none of these refinements leads to an algorithm better than straight insertion [that is, insertion sort]; and we already know that straight insertion isn't suitable for large N. [...] In short, 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.—D. E. Knuth^{[2]}Notes
 ^ Martin Duhl: Die schrittweise Entwicklung und Beschreibung einer ShuffleSortArray Schaltung in HYPERKARL aus der Algorithmischen Darstellung des BUBBLESORTALGORITHMUS, Projektarbeit, 1986, Technical University of Kaiserslautern
 ^ The Art of Computer Programming, Vol. 3: Sorting and Searching, Second Edition, AddisonWesley, 1998.
References
 Paul E. Black and Bob Bockholt, "bidirectional bubble sort", in Dictionary of Algorithms and Data Structures (online), Paul E. Black, ed., U.S. National Institute of Standards and Technology. 24 August 2009. (accessed: 5 Feb 2010)
 R. Hartenstein: THE GRAND CHALLENGE TO REINVENT COMPUTING  A new World Model of Computing; Proc. CSBC_2010, July 20–23, 2010, Belo Horizonte, Brasil, [1]
External links
 Java source code and an animated demo of cocktail sort (called bidirectional bubble sort) and several other algorithms
 .NET Implementation of cocktail sort and several other algorithms
 Interactive demo of cocktail sort
Sorting algorithms Theory Exchange sorts  Bubble sort
 Cocktail sort
 Odd–even sort
 Comb sort
 Gnome sort
 Quicksort
 Stooge sort
 Bogosort
Selection sorts  Selection sort
 Heapsort
 Smoothsort
 Cartesian tree sort
 Tournament sort
 Cycle sort
Insertion sorts Merge sorts Distribution sorts Concurrent sorts  Bitonic sorter
 Batcher odd–even mergesort
 Pairwise sorting network
Hybrid sorts  Timsort
 Introsort
 Spreadsort
 UnShuffle sort
 JSort
Quantum sorts Other Categories: Sorting algorithms
 Comparison sorts
 Stable sorts
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