Consider the following algorithm, which takes as input a sequence of n integers a_1, a_2,… , a_n and produces as output a matrix M = {m_(ij)} where m_(ij) is the minimum term in the sequence of integers a_i , a_(i+1),… , a_j for j ≥ i and m_(ij) = 0 otherwise. initialize M so that m_(ij) = a_i if j ≥ i and m_(ij) = 0 otherwise for i := 1 to n for j := i + 1 to n for k := i + 1 to j m_(ij) := min(m_(ij), a_k) return M= {m_(ij)} {m_(ij) is the minimum term of a_i , a_(i+1),… , a_j } a) Show that this algorithm uses O(n^3) comparisons to compute the matrix M. b) Show that this algorithm uses Ω(n^3) comparisons to compute the matrix M. Using this fact and part (a), conclude that the algorithms uses Θ(n^3) comparisons. [Hint: Only consider the cases where i ≤ n∕4 and j ≥ 3n∕4 in the two outer loops in the algorithm.]

Algorithm

Problem 2. Suppose we have an array A[1 : n] which consists of numbers {1,...,n} written in some arbitrary order (this means that A is a permutation of the set {1,...,n}). Our goal in this problem is to design a very fast randomized algorithm that can find an index i in this array such that A[i] mod 8 € {1,2}, i.e., the reminder of dividing A[i] by 8 is either 1 or 2. For simplicity, in the following, we assume that n itself is a multiple of 8 and is at least 8 (so a correct answer always exist). For instance, if n = 8 and the array is A = [8,7, 2,5, 4, 6,3, 1], we want to output either of indices 3 or 8. (a) Suppose we sample an index i from {1,...,n} uniformly at random. What is the probability that i is a correct answer, i.e., A[i] mod 8 E {1,2}? (b) Suppose we sample m indices from {1,...,n} uniformly at random and with repetition. What is the probability that none of these indices is a correct answer? Now, consider the following simple algorithm for this problem: Find-Index-1(A[1:…