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Counting sort is not a comparison sort algorithm. The sorting techniques that work on comparison are:

The best time complexity you can achieve in any of these sorting techniques is O(n * \log(n)). However, is some very specific cases, you can achieve an even faster computation time of O(n). This is made possible by an assumption. We assume that the input data set is within a finite limit.

For example, if you want to sort some characters of the English alphabet. You know that there can be a maximum of 26 characters. Similarly, if you are trying to sort marks of some students, and you know that they can range only between 0-100. These are the cases where Counting Sort can come in very handy.

#### Basic Idea:

The basic idea of counting sort is to determine, for each input element `X`

, the number of elements smaller than `X`

. This information can be used to place the element directly into its correct position in the resultant sorted array. For example, if there are `10`

elements less than `X`

, then `X`

would belong at position `11`

in the output array.

In the below example, **arr[0..n – 1]** is the input array with length n. In counting sort we need two more arrays: let us assume array **count[0..n – 1]** as another array and **out[0..n – 1]** as the final sorted array.

- Determine the range of given input elements by doing a single scan.

- Count the occurrences of each of the element and store it in an auxiliary array.

- Perform a cumulative sum on the elements of this new array.

- Use this auxiliary array to place the original elements in place. Example:
`temp[2]`

would indicate the position of`2`

in the final output array. With a`0`

based indexing,`2`

would go at position number`8`

. Repeat this process for every element of the input array.

#### Implementation and Code:

Here is an implementation of the algorithm:-

```
static void countingSort(int[] arr)
{
int max = Arrays.stream(arr).max().getAsInt();
int min = Arrays.stream(arr).min().getAsInt();
int range = max - min + 1;
// Auxiliary array to store counts of each element
int[] count = new int[range];
// Auxiliary array to store the resultant sorted array
int[] output = new int[arr.length];
// Store count of each element
for (int j : arr) {
count[j - min]++;
}
// Change count[i] so that count[i] now contains actual
// position of this character in output array
for (int i = 1; i < count.length; i++) {
count[i] += count[i - 1];
}
// Build the output array
// To make it stable we are operating in reverse order.
for (int i = arr.length - 1; i >= 0; i--) {
output[count[arr[i] - min] - 1] = arr[i];
count[arr[i] - min]--;
}
System.arraycopy(output, 0, arr, 0, arr.length);
}
```

Code language: Java (java)

*Time Complexity:* O(n)*Space Complexity:* O(n + k) where k is the range of the elements.

You can also find the sample code along with test cases on GitHub.