Again, we skip the proof of correctness of this algorithm.(d) What is the worst-case running time of Find-Index-2(A[1 n])? What about its worst-caseexpected running time? Remember to prove your answer formally. Problem 2. Suppose we have an array A[1 : n] which consists of numbers {1,...,n} written in somearbitrary order (this means that A is a permutation of the set {1,...,n}). Our goal in this problem is todesign 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 nitself 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 isa 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 theprobability that none of these indices is a correct answer?Now, consider the following simple algorithm for this problem:Find-Index-1(A[1: n]):• Let i = 1. While A[i] mod 8 ¢ {1,2}, sample i e {1,...,n} uniformly at random. Output i.The proof of correctness of this algorithm is straightforward and we skip it in this question.(c) What is the worse-case expected running time of Find-Index-1(A[1 : n])? Remember to prove youranswer formally.The problem with Find-Index-1 is that in the worst-case (and not in expectation), it may actually neverterminate! For this reason, let us consider a simple modification to this algorithm as follows.Find-Index-2(A[1 : n]):• For j = 1 to n:- Sample i e {1,...,n} uniformly at random and if A[i] mod 8 € {1,2}, output i and terminate;otherwise, continue.• If the for-loop never terminated, go over the array A one element at a time to find an index i with A[i]mod 8 € {1,2} and output it as the answer.Again, we skip the proof of correctness of this algorithm.

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Again, we skip the proof of correctness of this algorithm. (d) What is the worst-case running time of Find-Index-2(A[1 : n)? What about its worst-case expected running time? Remember to prove your answer formally.

Again, we skip the proof of correctness of this algorithm. (d) What is the worst-case running time of Find-Index-2(A[1 : n)? What about its worst-case expected running time? Remember to prove your answer formally.

Computer Networking: A Top-Down Approach (7th Edition)

7th Edition

ISBN:9780133594140

Author:James Kurose, Keith Ross

Publisher:James Kurose, Keith Ross

Chapter1: Computer Networks And The Internet

Section: Chapter Questions

Problem R1RQ: What is the difference between a host and an end system? List several different types of end...

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