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Chapter 7: Problem 43

Write an algorithm that sorts a list of \(n\) elements in nonincreasing order by finding the largest and smallest elements and exchanges those elements with the elements in the first and last positions. Then the size of the list is reduced by \(2,\) excluding the two elements that are already in the proper positions, and the process is repeated on the remaining part of the list until the entire list is sorted. Analyze your algorithm and show the results using order notation.

Short Answer

Expert verified
The algorithm sorts the list in non-increasing order by finding and swapping the smallest and largest elements in the unsorted part of the list. It continues to do so by reducing the unsorted list size by two. The time complexity of this algorithm is \(O(n^2)\).

Step by step solution

01

Write the Algorithm

Begin with a list of \(n\) elements. The algorithm is as follows:\n\n1. For each unsorted part from \(i=0\) till \((n-i-1)\), find both the largest and smallest elements.\n2. Swap the smallest element with the first element in the unsorted part, from the left side. Swap the largest element with the first element in the unsorted part, from the right side.\n3. Move the boundaries of the unsorted part from both sides, and repeat steps 1 and 2 until the entire list is sorted.
02

Calculate Time Complexity

To analyze the time complexity:\n\n1. The outer loop will run \(\frac{n}{2}\) times for each unsorted sub-list.\n2. The inner loop will run in \(O(n)\) for finding both smallest and largest elements.\n\nHence, the total time complexity of the algorithm will be \(O(\frac{n}{2} * n)\) = \(O(n^2)\).
03

Represent in Order Notation

Time complexity in order notation will be \(O(n^2)\).

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Most popular questions from this chapter

Write a nonrecursive Quicksort algorithm. Analyze your algorithm, and show the results using order notation. Note that it will be necessary to explicitly maintain a stack in your algorithm.

Is Exchange Sort (Algorithm 1.3 ) or Insertion Sort (Algorithm 7.1 ) more appropriate when we need to find in nonincreasing order the \(k\) largest (or in nondecreasing order the \(k\) smallest) keys in a list of \(n\) keys? Justify your answer.

An algorithm called Shell Sort is inspired by Insertion Sort's ability to take advantage of the order of the elements in the list. In Shell Sort, the entire list is divided into noncontiguous sublists whose elements are a distance \(h\) apart for some number \(h\). Each sublist is then sorted using Insertion Sort. During the next pass, the value of \(h\) is reduced, increasing the size of each sublist. Usually the value of each \(h\) is chosen to be relatively prime to: its previous value. The final pass uses the value 1 for \(h\) to sort the list. Write an algorithm for Shell Sort, study its performance, and compare the result with the performance of Insertion Sort.

Give two instances for which Quicksort algorithm is the most appropriate choice.

Another way to sort a list by exchanging out-of-order keys is called Bubble Sort. Bubble Sort scans adjacent pairs of records and exchanges those found to have out-of-order keys: After the first time through the list, the record with the largest key (or the smallest key) is moved to its proper position. This process is done repeatedly on the remaining, unsorted part of the list until the list is completely sorted. Write the Bubble Sort algorithm. Analyze your algorithm, and show the results using order notation. Compare the performance of the Bubble Sort algorithm to those of Insertion Sort, Exchange Sort, and Selection Sort.
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