Collaborative ltering is a technique for generating recommendations, such as suggestions for products, songs, movies, news stories, and so on. The idea is to identify other users who have similar preferences, and recommend to you things that have been popular with them. It requires a formal notion of similarity between users, and some of the essence of this is captured by the problem of counting the number of inversions in an array.

An inversion in an array A[1,...,n] is a pair of indices (i,j) such that i < j and A[i] > A[j]. For example, in the array A = [1,3,5,2,4,6], there are three inversions, 5 and 2 (corresponding to i = 3 and j = 4), 3 and the 2, and the 5 and the 4. For this problem we will count the number of inversions in an array of size n.

(a) Write pseudo-code for an ecient brute force algorithm to nd the total number of inversions in the worst-case. Find the exact number of inversions. Show your work. (b) Write pseudo-code for an optimal algorithm to nd the total number of inversions in the worst-case. What is the asymptotic runtime of your algorithm?