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Lesson 6, Topic 8
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# Counting Sort

##### Akash
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Counting sort is a sorting algorithm that sorts the elements of an array by counting the number of occurrences of each unique element in the array. The count is stored in an auxiliary array and the sorting is done by mapping the count as an index of the auxiliary array.

## How Counting Sort Works?

1. Find out the maximum element (let it be `max`) from the given array.

2. Initialize an array of length `max+1` with all elements 0. This array is used for storing the count of the elements in the array.

3. Store the count of each element at their respective index in `count` array

For example: If the count of element “4” occurs 2 times then 2 is stored in the 4th position in the `count` array. If element “5” is not present in the array, then 0 is stored in 5th position.

Store cumulative sum of the elements of the count array.

It helps in placing the elements into the correct index.

If there are x elements less than y, its position should be at x-1.

4.For example: In the array below, the count of 4 is 6. It denotes that there are 5 elements smaller than 4. Thus, the position of 4 in the sorted array is 5th.

5. Find the index of each element of the original array in count array. This gives the cumulative count. Place the element at the index calculated.

6. After placing each element at its correct position, decrease the its count by one.

## Counting Sort Algorithm

``````countingSort(array, size)
max <- find largest element in array
initialize count array with all zeros
for j <- 0 to size
find the total count of each unique element and
store the count at jth index in count array
for i <- 1 to max
find the cumulative sum and store it in count array itself
for j <- size down to 1
restore the elements to array
decrease count of each element restored by 1``````

## Counting Sort in C

``````// Counting sort in C programming
#include <stdio.h>
void countingSort(int array[], int size)
{
int output[10];
int max = array[0];
for (int i = 1; i < size; i++)
{
if (array[i] > max)
max = array[i];
}
// The size of count must be at least the (max+1) but
// we cannot assign declare it as int count(max+1) in C as
// it does not support dynamic memory allocation.
// So, its size is provided statically.
int count[10];
for (int i = 0; i <= max; ++i)
{
count[i] = 0;
}
for (int i = 0; i < size; i++)
{
count[array[i]]++;
}
for (int i = 1; i <= max; i++)
{
count[i] += count[i - 1];
}
for (int i = size - 1; i >= 0; i--)
{
output[count[array[i]] - 1] = array[i];
count[array[i]]--;
}
for (int i = 0; i < size; i++)
{
array[i] = output[i];
}
}
void printArray(int array[], int size)
{
for (int i = 0; i < size; ++i)
{
printf("%d  ", array[i]);
}
printf("\n");
}
int main()
{
int array[] = {4, 2, 2, 8, 3, 3, 1};
int n = sizeof(array) / sizeof(array[0]);
countingSort(array, n);
printArray(array, n);
}``````

## Complexity

Time Complexities:
There are mainly four main loops. (Finding the greatest value can be done outside the function.)

Overall complexity = `O(max)+O(size)+O(max)+O(size)` = `O(max+size)`

• Worst Case Complexity: `O(n+k)`
• Best Case Complexity: `O(n+k)`
• Average Case Complexity: `O(n+k)`

In all the above cases, the complexity is same because no matter how the elements are placed in the array, the algorithm goes through `n+k` times.

There is no comparison between any elements so, it is better than comparison based sorting techniques. But, it is bad if the integers are very large because the array of that size should be made.

Space Complexity:

The space complexity of Counting Sort is `O(max)`. Larger the range of elements, larger is the space complexity.

## Counting Sort Applications

Counting sort is used when:

• there are smaller integers of multiple counts.
• linear complexity is the need.