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Chapter 1: Problem 13

Algorithm A performs \(10 n^{2}\) basic operations, and algorithm B performs \(300 \ln n\) basic operations. For what value of \(n\) does algorithm \(\mathrm{B}\) start to show its better performance?

Short Answer

Expert verified
Algorithm B starts to show its better performance for \(n\) values greater than about 7.395

Step by step solution

01

Set up the equation

We start by setting up the equality between the two time complexity equations to find the 'crossover point' where Algorithm B becomes faster than Algorithm A. The equation is therefore: \(10n^2 = 300\ln{n}\)
02

Simplify the equation

Next, we simplify the equation by dividing all terms by 10 to reduce complexity: \(n^2 = 30\ln{n}\)
03

Solving the equation

This is a transcendental equation and it's not simple to solve analytically. We can use numerical methods or graphical methods to solve it. A widely accepted method is using a computational tool (like a graphing calculator or a software like MATLAB or Python) to solve this equation.
04

Approximate the solution

Using a numerical approximator, you'll find that the equation \(n^2 = 30\ln{n}\) becomes true for \(n \approx 7.395\). That is, when \(n\) is greater than this number, the value of \(30\ln{n}\) will exceed \(n^2\), meaning algorithm B performs better.

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Most popular questions from this chapter

Algorithm 1.7 (nth Fibonacci Term, Iterative) is clearly linear in \(n,\) but is it a linear-time algorithm? In Section 1.3 .1 we defined the input size as the size of the input. In the case of the \(nth\) Fibonacci term, \(n\) is the input, and the number of bits it takes to encode \(n\) could be used as the input size. Using this measure the size of 64 is \(\lg 64=6,\) and the size of 1024 is \(\lg 1024=\) 10\. Show that Algorithm 1.7 is exponential-time in terms of its input size. Show further that any algorithm for computing the \(nth\) Fibonacci term must be an exponential-time algorithm because the size of the output is exponential in the input size. See Section 9.2 for a related discussion of the input size.

Group the following functions by complexity category. $$n \ln n \quad(\lg n)^{2} \quad 5 n^{2}+7 n \quad n^\frac{5}{2} $$ $$n ! \quad 2^{n !} \quad 4^{n} \quad n^{n} \quad n^{n}+\ln n$$ $$5^{\text {Ig n}} \quad \lg (n !) \quad(\lg n) ! \quad \sqrt{n} \quad e^{n} \quad 8 n+12 \quad 10^{n}+n^{20}$$

Give an algorithm for the following problem. Given a list of \(n\) distinct positive integers, partition the list into two sublists, each of size \(n / 2,\) such that the difference between the sums of the integers in the two sublists is minimized. Determine the time complexity of your algorithm. You may assume that \(n\) is a multiple of 2

Give an algorithm for the following problem, and determine its time complexity. Given a list of \(n\) distinct positive integers, partition the list into two sublists, each of size \(n / 2\), such that the difference between the sums of the integers in the two sublists is maximized. You may assume that \(n\) is a multiple of 2

Explain in English what functions are in the following sets. (a) \(n^{C(1)}\) (b) \(O\left(n^{O(1)}\right)\) (c) \(O\left(O\left(n^{O(1)}\right)\right)\)
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