The Reason Behind Employing an Additional Array and Counting Smaller Elements in the Counting Sort Algorithm

Question

Is it feasible to streamline the counting sort algorithm by exclusively utilizing the counting array (C) after determining the frequency of each element in the input array? Instead of creating an additional output array, can we achieve the sorted array directly by traversing C in reverse order and placing the elements accordingly? For instance, if we have an input array A and a corresponding count array C:

Would it be valid to sort the array by placing one 5 at the end, followed by no fours, three 3s just before 5, adding two 2s, and concluding with no 1s and two 0s? This would result in the sorted array:

Additionally, in the existing counting sort implementation, we count the sum of each element in C with its previous element to determine how many elements are lower or equal to that particular element. Considering this, can we simplify the sorting process by relying solely on the cumulative counts in C and eliminating the conventional steps involving the creation of an extra output array and forward traversal of the input array? Are there reasons for retaining the existing process, apart from considerations such as stability and scenarios where input array elements represent attributes of objects (e.g., satellite data)?

Answer 1

Is it feasible to streamline the counting sort algorithm by exclusively utilizing the counting array (C) after determining the frequency of each element in the input array? Instead of creating an additional output array, can we achieve the sorted array directly by traversing C in reverse order and placing the elements accordingly?

That's normal if the (integer) sort keys are the items being sorted themselves, and in that case, it works just as well (and is easier to code) to traverse the array from front to back. Otherwise, no, it does not work, for the same reason that doing the same thing from front to back does not work: unless the objects are already in sorted order, you will overwrite and lose some of them.

Additionally, in the existing counting sort implementation, we count the sum of each element in C with its previous element to determine how many elements are lower or equal to that particular element. [...] can we simplify the sorting process by relying solely on the cumulative counts in C and eliminating the conventional steps involving the creation of an extra output array and forward traversal of the input array?

You can perform that simplification when the items themselves are the keys. Otherwise not, because again, you will overwrite and lose some of the items if you do not use additional storage.

Are there reasons for retaining the existing process, apart from considerations such as stability and scenarios where input array elements represent attributes of objects (e.g., satellite data)?

Again, Counting Sort can be simplified in the special case of sorting integers, where the items are themselves the sort keys. Perhaps in some other special cases, too. In general, however, you need the extra steps and storage to get the correct result.

You could also consider converting a Counting Sort into a Flash Sort, which reorders the array in-place, but that's hardly a simplification.