Question 4 Suppose we are given an array A[0...n – 1] of size n, where each array entry has a value that is chosen uniformly at random in {1,...,95}. Consider an algorithm that takes as input a value r € {1,..., 95}, and searches for a in the array by querying array entries A[i] for ie {0,...,n- 1} in increasing order of i. Whenever the algorithm queries an array entry A[i] that has value r, it immediately outputs the index i. (a) What is the expected number of indices that the algorithm outputs? (b) For any k E {0,..., n – 1}, what is the probability that the smallest index the algorithm outputs is k? (c) How does the answer to (b) change, if we know that at least one array entry has value æ? Explain and show all your calculations.

Question 4 Suppose we are given an array A[0...n – 1] of size n, where each array entry has a value that is chosen uniformly at random in {1,...,95}. Consider an algorithm that takes as input a value r € {1,..., 95}, and searches for a in the array by querying array entries A[i] for ie {0,...,n- 1} in increasing order of i. Whenever the algorithm queries an array entry A[i] that has value r, it immediately outputs the index i. (a) What is the expected number of indices that the algorithm outputs? (b) For any k E {0,..., n – 1}, what is the probability that the smallest index the algorithm outputs is k? (c) How does the answer to (b) change, if we know that at least one array entry has value æ? Explain and show all your calculations.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.3: Divisibility

Problem 49E: 49. a. The binomial coefficients are defined in Exercise of Section. Use induction on to prove...

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