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Chapter 2: Problem 34

Implement both Exchange Sort and Quicksort algorithms on your computer to sort a list of \(n\) elements. Find the lower bound for \(n\) that justifies application of the Quicksort algorithm with its overhead.

Short Answer

Expert verified
The lower bound for \(n\) that justifies the use of Quicksort over Exchange Sort depends on your implementation and your machine specs. You need to execute the comparison yourself to find out the exact value of \(n\).

Step by step solution

01

Understanding the Algorithms

Research and study the Quicksort and Exchange Sort algorithms. Understand their working, their time complexities (in terms of Big O notation), and how they should be implemented.
02

Implementing the Algorithms

Write the code for both Quicksort and Exchange Sort algorithms. These implementations need to be able to handle lists of variable sizes.
03

Testing the Algorithms

Test the correctness of both algorithms with lists of different sizes. Make sure the implementations are correctly sorting the lists.
04

Comparing the Algorithms

Time both implementations with various list sizes. Note down the execution times. Increase the list size gradually until Quicksort starts to consistently perform better than Exchange Sort.
05

Determining the Lower Bound

From the data collected, determine the smallest list size, \(n\), at which Quicksort started to consistently outperform Exchange Sort. This would be the lower bound for justifying the usage of Quicksort over Exchange Sort.

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Most popular questions from this chapter

Consider procedure solve \((P, I, O)\) given below. This algorithm solves problem \(P\) by finding the output (solution) \(O\) corresponding to any input \(l\) Assume \(g(n)\) basic operations for partitioning and combining and no basic operations for an instance of size 1 (a) Write a recurrence equation \(T(n)\) for the number of basic operations needed to solve \(P\) when the input size is \(n\) (b) What is the solution to this recurrence equation if \(g(n) \in \Theta(n)\) (proof not required \() ?\) (c) Assuming that \(g(n)=n^{2},\) solve the recurrence equation exactly for \(n=\) 27 (d) Find the general solution for \(n\) a power of 3

Suppose that there are \(n=2^{k}\) teams in an elimination tournament, where there are \(n / 2\) games in the first round, with the \(n / 2=2^{2-1}\) winners playing in the second round, and so on. (a) Develop a recurrence equation for the number of rounds in the tournament. (b) How many rounds are there in the tournament when there are 64 teams? (c) Solve the recurrence cquation of part (a).

Write an algorithm that sorts a list of \(n\) items by dividing it into three sublists or almost \(n / 3\) items, sorting each sublist recursively and merging the three sorted sublists. Analyze your algorithm, and give the results using order notation.

When a divide-and-conquer algorithm divides an instance of size \(n\) of a problem into subinstances each of size \(n / c\), the recurrence relation is typically given by \\[ \begin{array}{l} T(n)=a T\left(\frac{n}{c}\right)+g(n) \quad \text { for } n>1 \\ T(1)=d \end{array} \\] where \(g(n)\) is the cost of the dividing and combining processes, and \(d\) is a constant. Let \(n=c^{k}\) (a) Show that \\[ T\left(c^{k}\right)=d \times d^{k}+\sum_{j=1}^{k}\left[a^{k-j} \times g\left(c^{j}\right)\right] \\] (b) Solve the recurrence relation given that \(g(n) \in \Theta(n)\)

Write an efficient algorithm that searches for a value in an \(n \times m\) table (twodimensional array). This table is sorted along the rows and columns - that is \\[ \begin{array}{l} \text { Table }(i][j] \leq T a b l c[i][j+1] \\ \text { Table }[i][j] \leq T a b l e[i+1][j] \end{array} \\]
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