I know that we are (n-1) * (n times), but why the division by 2?
It's only (n - 1) * n if you use a naive bubblesort. You can get a significant savings if you notice the following:
After each compare-and-swap, the largest element you've encountered will be in the last spot you were at.
After the first pass, the largest element will be in the last position; after the kth pass, the kth largest element will be in the kth last position.
Thus you don't have to sort the whole thing every time: you only need to sort n - 2 elements the second time through, n - 3 elements the third time, and so on. That means that the total number of compare/swaps you have to do is (n - 1) + (n - 2) + .... This is an arithmetic series, and the equation for the total number of times is (n - 1)*n / 2.
Example: if the size of the list is N = 5, then you do 4 + 3 + 2 + 1 = 10 swaps -- and notice that 10 is the same as 4 * 5 / 2.
