Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 5.1, Problem 3E
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Introduction to Algorithms
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- write a computer program that produces the desired output from the given input. Input: A random variable and a probability distribution for a finite sample space.Output: The expected value of the random variable.arrow_forward2. A celebrity among a group of n people is a person who knows nobody but is known by everybody else. Design a decrease-and-conquer algorithm that identifies a celebrity among n people, or outputs "no celebrity identified" if there is no such person, by only asking the question "Do you know him/her?" to people. Analyze the efficiency of the algorithm.arrow_forwardYou are given a collection of n bolts of different widths and n correspondingnuts. You are allowed to try a nut and bolt together, from which you can determine whether the nut is larger than the bolt, smaller than the bolt, or matches the bolt exactly. However, there is no way to compare two nuts together or two bolts together. The problem is to match each bolt to its nut. Design an algorithm for this problem with an average-case efficiency ofΘ(n log n).arrow_forward
- Consider the functionf :: Int -> Intf n = if n==0then 0else 1 + (f(n-1))Use induction to show that the function f returns the value of n for all possible inputs n ≥0.Here are the steps:1. Verify that f 0 returns 0 to show the base case.2. Show that if f(n-1) returns n −1 then f n returns n.3. Since you have shown the base case and the induction step, you can confidentlystate that the function works for all possible nonnegative input values.Hint: To show that ”if f(n-1) returns n −1 then f n returns n” is valid you need toassume that f(n-1) returns n −1 and then argue that it must follow that f n returnsn. Use the definition of the function and just a little bit of algebra.Criteria for Success: You have clearly written down all three steps of the inductive proof. Your proof contains complete sentences which explain all the steps andthe algebra. I don’t want to see just a bunch of symbols on a page!arrow_forwardApply gaussian random walk(walking in a random direction with each step of length 1. i.e. each time at random theta it will walk cos(theta) at x, sin(theta) at y). What is the probability that the walk starting at (0,0) will end at (5,5)? How to simulate it in python?arrow_forwarddesigned an iterative algorithm for separating n VLSIchips into those that are good and those that are bad by testing two chips at a time andlearning either that they are the same or that they are different. To help, at least half ofthe chips are promised to be good. Now design (much easier) a randomized algorithmfor this problem. Here are some hints. Randomly select one of the chips. What is the probability that the chip is good? How can you learn whether or not the selected chip is good? If it is good, how can you easily partition the chips into good and bad chips? If the chip is not good, what should your algorithm do? When should the algorithm stop? What is the expected running time of this algorithm?arrow_forward
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