# Counting Sort

In computer science, **counting sort** is an algorithm for sorting 
a collection of objects according to keys that are small integers; 
that is, it is an integer sorting algorithm. It operates by 
counting the number of objects that have each distinct key value, 
and using arithmetic on those counts to determine the positions 
of each key value in the output sequence. Its running time is 
linear in the number of items and the difference between the 
maximum and minimum key values, so it is only suitable for direct 
use in situations where the variation in keys is not significantly 
greater than the number of items. However, it is often used as a 
subroutine in another sorting algorithm, radix sort, that can 
handle larger keys more efficiently.

Because counting sort uses key values as indexes into an array, 
it is not a comparison sort, and the `Ω(n log n)` lower bound for 
comparison sorting does not apply to it. Bucket sort may be used 
for many of the same tasks as counting sort, with a similar time 
analysis; however, compared to counting sort, bucket sort requires 
linked lists, dynamic arrays or a large amount of preallocated 
memory to hold the sets of items within each bucket, whereas 
counting sort instead stores a single number (the count of items) 
per bucket.

Counting sorting works best when the range of numbers for each array
element is very small.

## Algorithm

**Step I**

In first step we calculate the count of all the elements of the 
input array `A`. Then Store the result in the count array `C`.
The way we count is depicted below.

![Counting Sort](https://3.bp.blogspot.com/-jJchly1BkTc/WLGqCFDdvCI/AAAAAAAAAHA/luljAlz2ptMndIZNH0KLTTuQMNsfzDeFQCLcB/s1600/CSortUpdatedStepI.gif)

**Step II**

In second step we calculate how many elements exist in the input 
array `A` which are less than or equals for the given index. 
`Ci` = numbers of elements less than or equals to `i` in input array.

![Counting Sort](https://1.bp.blogspot.com/-1vFu-VIRa9Y/WLHGuZkdF3I/AAAAAAAAAHs/8jKu2dbQee4ap9xlVcNsILrclqw0UxAVACLcB/s1600/Step-II.png)

**Step III**

In this step we place the input array `A` element at sorted 
position by taking help of constructed count array `C` ,i.e what 
we constructed in step two. We used the result array `B` to store 
the sorted elements. Here we handled the index of `B` start from
zero.
 
![Counting Sort](https://1.bp.blogspot.com/-xPqylngqASY/WLGq3p9n9vI/AAAAAAAAAHM/JHdtXAkJY8wYzDMBXxqarjmhpPhM0u8MACLcB/s1600/ResultArrayCS.gif)

## Complexity

| Name                  | Best            | Average             | Worst               | Memory    | Stable    | Comments  |
| --------------------- | :-------------: | :-----------------: | :-----------------: | :-------: | :-------: | :-------- |
| **Counting sort**     | n + r           | n + r               | n + r               | n + r     | Yes       | r - biggest number in array |

## References

- [Wikipedia](https://en.wikipedia.org/wiki/Counting_sort)
- [YouTube](https://www.youtube.com/watch?v=OKd534EWcdk&index=61&t=0s&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8)
- [EfficientAlgorithms](https://efficientalgorithms.blogspot.com/2016/09/lenear-sorting-counting-sort.html)