Question

Collaborative filtering 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 efficient brute force algorithm to find 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 find the total number of inversions in the worst-case. What is the asymptotic runtime of your algorithm?

Answer 1

(a) Brute Force Method Inversion (int are []) 1 int contatos for (int i = 0; i < arr-length-l; i++) { for (int j zitl ; jcare length; j++) if Carr [i]> arr [j] count & count tli return count; array case, elements of are in descending order o In worst ~ At ezo 6 n-1 In-z ... 13/2/21 ~ At izl, In (D/n-2/.../3/21 1 (n-1) inversions : (n=2) inversions 130 1 1 inversion - At izn-l, InIn-11h-2) - - +3+2+1 inversions = (2-1) + (n-2) + Total ho. of = n(n-1) inversions 2

(b) Optimal algorithm merge Sort and Inie (int [] are, int I, int r) int count = 0; if (lar) int me (l+2)/2 count count & merge Sort and Ino (arr, lm): count Coundt merge sort and Inc (art milli count = count & merge and Into (are, lym, a); return Cound; merge and Inu (ant o are ind es ist ms int a). int [] left ind [] right a copy elements from (l to m) from arts copy elements from (mel toe) from alls int i=0; j zo skal, swaps = 0; While ( is left length ff je right length) if (left (i] Le right [j]) ar [k++] = left [i++]; else { arr [kar] = right [j++]; z swaps = swaps + (mil)-(lte);

subarray // Fill from the rest of the left while (ic left length) are [k++] = left [it+] Il Fill from the rest of the right subarray while (je right length) are [k++] = right [j++]; return swaps; Recurrence relation - T(h) z2T (2/2) + Oln). by using master's theorems T(m) = O(n logan) L Asymptotic notation