Cocktail Sort

Cocktail Sort is a variation of Bubble sort. The Bubble sort algorithm always traverses elements from left and moves the largest element to its correct position in the first iteration and second-largest in the second iteration and so on. Cocktail Sort traverses through a given array in both directions alternatively. Cocktail sort does not go through the unnecessary iteration making it efficient for large arrays.

Cocktail sorts break down barriers that limit bubble sorts from being efficient enough on large arrays by not allowing them to go through unnecessary iterations on one specific region (or cluster) before moving onto another section of an array.
Algorithm:
Each iteration of the algorithm is broken up into 2 stages: 

1. The first stage loops through the array from left to right, just like the Bubble Sort. During the loop, adjacent items are compared and if the value on the left is greater than the value on the right, then values are swapped. At the end of the first iteration, the largest number will reside at the end of the array.
2. The second stage loops through the array in opposite direction- starting from the item just before the most recently sorted item, and moving back to the start of the array. Here also, adjacent items are compared and are swapped if required.

Example : 

Let us consider an example array (5 1 4 2 8 0 2)

First Forward Pass:
(5 1 4 2 8 0 2) ? (1 5 4 2 8 0 2), Swap since 5 > 1 
(1 5 4 2 8 0 2) ? (1 4 5 2 8 0 2), Swap since 5 > 4 
(1 4 5 2 8 0 2) ? (1 4 2 5 8 0 2), Swap since 5 > 2 
(1 4 2 5 8 0 2) ? (1 4 2 5 8 0 2) 
(1 4 2 5 8 0 2) ? (1 4 2 5 0 8 2), Swap since 8 > 0 
(1 4 2 5 0 8 2) ? (1 4 2 5 0 2 8), Swap since 8 > 2
After the first forward pass, the greatest element of the array will be present at the last index of the array.
First Backward Pass:
(1 4 2 5 0 2 8) ? (1 4 2 5 0 2 8) 
(1 4 2 5 0 2 8) ? (1 4 2 0 5 2 8), Swap since 5 > 0 
(1 4 2 0 5 2 8) ? (1 4 0 2 5 2 8), Swap since 2 > 0 
(1 4 0 2 5 2 8) ? (1 0 4 2 5 2 8), Swap since 4 > 0 
(1 0 4 2 5 2 8) ? (0 1 4 2 5 2 8), Swap since 1 > 0
After the first backward pass, the smallest element of the array will be present at the first index of the array.
Second Forward Pass:
(0 1 4 2 5 2 8) ? (0 1 4 2 5 2 8) 
(0 1 4 2 5 2 8) ? (0 1 2 4 5 2 8), Swap since 4 > 2 
(0 1 2 4 5 2 8) ? (0 1 2 4 5 2 8) 
(0 1 2 4 5 2 8) ? (0 1 2 4 2 5 8), Swap since 5 > 2
Second Backward Pass:
(0 1 2 4 2 5 8) ? (0 1 2 2 4 5 8), Swap since 4 > 2
Now, the array is already sorted, but our algorithm doesn’t know if it is completed. The algorithm needs to complete this whole pass without any swap to know it is sorted. 
(0 1 2 2 4 5 8) ? (0 1 2 2 4 5 8) 
(0 1 2 2 4 5 8) ? (0 1 2 2 4 5 8)
Below is the implementation of the above algorithm :

Sorted array :
0 1 2 2 4 5 8 

People have given many different names to cocktail sort:

• Shaker Sort. 
• Bi-Directional Sort.
• Cocktail Shaker Sort.
• Shuttle Sort.
• Happy Hour Sort.
• Ripple Sort.

Sorting In Place: Yes
Stable: Yes
Comparison with Bubble Sort: 
Time complexities are the same, but Cocktail performs better than Bubble Sort. Typically cocktail sort is less than two times faster than bubble sort. Consider the example (2, 3, 4, 5, 1). Bubble sort requires four traversals of an array for this example, while Cocktail sort requires only two traversals. (Source Wiki)

Cocktail sort, also known as cocktail shaker sort or bidirectional bubble sort, is a variation of the bubble sort algorithm. Like the bubble sort algorithm, cocktail sort sorts an array of elements by repeatedly swapping adjacent elements if they are in the wrong order. However, cocktail sort also moves in the opposite direction after each pass through the array, making it more efficient in certain cases.

The basic idea of cocktail sort is as follows:

1. Start from the beginning of the array and compare each adjacent pair of elements. If they are in the wrong order, swap them.
2. Continue iterating through the array in this manner until you reach the end of the array.
3. Then, move in the opposite direction from the end of the array to the beginning, comparing each adjacent pair of elements and swapping them if necessary.
4. Continue iterating through the array in this manner until you reach the beginning of the array.
5. Repeat steps 1-4 until the array is fully sorted.

Here's an example of cocktail sort in action, sorting an array of integers in ascending order:

Sure, here are some potential advantages and disadvantages of using the cocktail sort algorithm:

Advantages:

1. Cocktail sort can be more efficient than bubble sort in certain cases, especially when the array being sorted has a small number of unsorted elements near the end.
2. Cocktail sort is a simple algorithm to understand and implement, making it a good choice for educational purposes or for sorting small datasets.

Disadvantages:

1. Cocktail sort has a worst-case time complexity of O(n^2), which means that it can be slow for large datasets or datasets that are already partially sorted.
2. Cocktail sort requires additional bookkeeping to keep track of the starting and ending indices of the subarrays being sorted in each pass, which can make the
3. algorithm less efficient in terms of memory usage than other sorting algorithms.
4. There are more efficient sorting algorithms available, such as merge sort and quicksort, that have better average-case and worst-case time complexity than cocktail sort.

 One example :

[2, 3, 5, 6, 7, 9]

Complexity Analysis :

The time complexity of Cocktail Sort is O(n^2) in the worst and average cases, where n is the number of elements in the array. This is because the algorithm iterates through the array multiple times, and in the worst case, each iteration involves comparing every element to its neighbor and possibly swapping them.
The best case time complexity is O(n) when the array is already sorted, but this is a rare case.
The space complexity of Cocktail Sort is O(1), which is constant. This is because the algorithm sorts the array in place, without using any extra memory.

References:

• https://en.wikipedia.org/wiki/Cocktail_shaker_sort
• https://thimbleby.net/algorithms/doku.php?id=cocktail_sort
• http://www.programming-algorithms.net/article/40270/Shaker-sort