Question

How to compute/prove average case complexity for sorting algorithms?

I have to compute the best case, worst case and average case time complexity for various sorting algorithms. I am ok with identifying the basic operations and proving the number of basic operations for best and worst case. However, I am at a loss of how to compute/prove the average case for these sorting algorithms. For example I can find in sources that the average case of say Shaker Sort and Insertion Sort is O(n^2) but I am not sure how to prove this.

Is there a typical approach to proving/computing the average case? Can someone provide an example?

Answer 1

Let's understand with the help of an example. Suppose you have to find the average time complexity of finding the element in the array.

In average case, we have to consider all the cases and then take the average of all the time complexities.

In this case , it may happen that element is founded at index 0 or index 1 or (2,3,4) or not present at all.

So

answer would be (1+2+3....+n)/n

(1+2+3+4....n)/n=(n+1)/2

So, on an average there woud be (n+1)/2 comparisons for searching the element in an array.

Hope it helps :)