Problem 3. Given an array A[1 : n] of n positive integers (possibly with repetitions), your goal is to find whether there is a sub-array A[l : r] such that E A[i] = n. i=l Example. Given A = up to n = 10. On the other hand, if the input array is A = since no sub-array of A adds up to n = 7. [13, 1, 2, 3, 4, 7, 2, 3, 8, 9] for n = 10, the answer is Yes since elements in A[2 : 5] add [3, 2, 6, 8, 20, 2, 4] for n = 7, the answer is No Hint: Observe that if -, A[i] = n, then A[i] = E=1 A[i] - n; this may come handy!

Problem 3. Given an array A[1 : n] of n positive integers (possibly with repetitions), your goal is to find whether there is a sub-array A[l : r] such that E A[i] = n. i=l Example. Given A = up to n = 10. On the other hand, if the input array is A = since no sub-array of A adds up to n = 7. [13, 1, 2, 3, 4, 7, 2, 3, 8, 9] for n = 10, the answer is Yes since elements in A[2 : 5] add [3, 2, 6, 8, 20, 2, 4] for n = 7, the answer is No Hint: Observe that if -, A[i] = n, then A[i] = E=1 A[i] - n; this may come handy!

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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Question

Problem 3. Given an array A[1 : n] of n positive integers (possibly with repetitions), your goal is to find
whether there is a sub-array A[l : r] such that
EAli) = n.
i=l
Example. Given A =
up to n = 10. On the other hand, if the input array is A = [3, 2, 6, 8, 20, 2, 4| for n = 7, the answer is No
since no sub-array of A adds up to n = 7.
[13, 1, 2, 3, 4, 7, 2, 3, 8,9] for n = 10, the answer is Yes since elements in A[2 : 5] add
Hint: Observe that if - A[i] = n, then A[i] = E-1 A[i] -
- n; this may come handy!

(b) Now suppose our goal is to instead find whether there is a sub-array A[l, r] such that
Ali] = n².
i=l
Design a randomized algorithm for this case with worst-case expected runtime of O(n).

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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:…