[Math] Formal Proof of Counting Sort and Bucket Sort Algorithms

Question

I'm new to this forum so please be patient.

I'm studying two sort algorithms: counting sort and bucket sort.

In numerous books I found examples, as a 'proof' that these algorithms work, but those test
use a specific set of values.

So I want to know how can I do a formal mathematical proof of the working of the mentioned algorithms.

Any clue will help, I don't know exactly where to start(of course if you can provide a method would be better)

Thanks in advance

Answer 1

In general, without making any reference to the two particular algorithms mentioned, there are (at least) two ways of proving the correctness of a sorting algorithm:

• Proof by induction: assume that the algorithm can correctly sort $n$ items, and show that it can then also sort $n+1$ (or $2n$ or any other number greater than $n$) items. This works particularly well for recursive sorting algorithms like quicksort or merge sort.

• Proof by termination analysis: show that the algorithm must terminate, and that it can only terminate when the data is correctly sorted. One way to show that the algorithm must terminate is to find a property (such as the number of inversions) which is bounded from below and can be shown to decrease by at least one during each iteration of the algorithm.