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Chapter 1: Problem 25

Presently we can solve problem instances of size 100 in 1 minute using algorithm \(A,\) which is a \(\Theta\left(2^{n}\right)\) algorithm. On the other hand, we will soon have to solve problem instances twice this large in 1 minute. Do you think it would help to buy a faster (and more expensive) computer?

Short Answer

Expert verified
No, buying a faster computer would not significantly help in this case due to the exponential time complexity of the algorithm. The problem size increase outweighs any reasonable speed increase a new computer might offer.

Step by step solution

01

Understand the problem and algorithm time complexity

Firstly, we need to understand that the time complexity of algorithm \(A\) is \(\Theta\left(2^{n}\right)\). This means that as the size of the problem \(n\) increases, the time taken to solve the problem doubles. Right now, the algorithm can solve a problem of size 100 in 1 minute.
02

Apply the size increase to the time complexity equation

Next, let's apply the problem size increase to the time complexity. As per the problem, we will soon need to solve problems twice as large in 1 minute. So, the problem size \(n\) will increase to 200. Given the time complexity of the algorithm, this would mean that the time taken to solve the problem doubles with each increase in the problem size, i.e., the algorithm will take \(2^{200-100} = 2^{100}\) times as long as to solve a problem of size 200 as it does to solve a problem of size 100.
03

Determine the usefulness of a faster computer

For the final step, we consider if a faster (more expensive) computer would help. A faster computer could possibly execute the algorithm more quickly, but considering the time complexity of the algorithm, it wouldn't help enough to offset the exponential growth. The increase is so large (\(2^{100}\) times larger) that merely doubling the speed of processing (which is a reasonable expectation for a new computer) would still leave us with a task that takes much longer than 1 minute.

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Most popular questions from this chapter

There are two algorithms called Algl and Alg2 for a problem of size n. Algl runs in \(n^{2}\) microseconds and Alg2 runs in 100 n log \(n\) microseconds. Algl can be implemented using 4 hours of programmer time and needs 2 minutes of CPU time. On the other hand, Alg2 require 15 hours of programmer time and 6 minutes of CPU time. If programmers are paid 20 dollars per hour and CPU time costs 50 dollars per minute, how many times must a problem instance of size 500 be solved using Alg2 in order to justify its development cost?

Suppose you have a computer that requires 1 minute to solve problem instances of size \(n=1,000 .\) Suppose you buy a new computer that runs 1,000 times faster than the old one. What instance sizes can be run in 1 minute, assuming the following time complexities \(T(n)\) for our algorithm? a. \(T(n)=n\) b. \(T(n)=n^{3}\) c. \(T(n)=10^{n}\)

Write an algorithm that finds the \(m\) smallest numbers in a list of \(n\) numbers.

Give a \(\Theta(n \text { lg } n)\) algorithm that computes the reminder when \(x^{n}\) is divided by \(p .\) For simplicity, you may assume that \(n\) is a power of 2 That is, \(n=2^{k}\) for some positive integer \(k\)

Discuss the reflexive, symmetric, and transitive properties for asymptotic comparisons \((O, \Omega, \Theta, o)\)
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