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Chapter 7: Problem 30

Show that there is a case for Heapsort in which we get the worst-case time complexity of \(W(n) \approx 2 n \lg n \in \Theta(n \lg n)\)

Short Answer

Expert verified
The worst-case scenario of heapsort happens when the input is in decreasing order. Each extraction operation takes \(O(\lg n)\) time and is run for each of the \(n\) elements, which leads to a time complexity of \(2n \lg n\). However, in big-theta terms, \(2n \lg n\) is considered as \(n \lg n\).

Step by step solution

01

Understanding Heapsort

Heapsort is a comparison-based sorting algorithm that uses a binary heap data structure. It involves building a max heap from the input data, then repeatedly extracting the maximum element and rearranging the heap until it's empty. Each extraction takes logarithmic time as we need to restore the heap property, making heapsort have an overall time complexity of \(O(n \lg n)\).
02

Worst Case Scenario

For heapsort, the worst-case scenario is when the input is in decreasing order. This forces a situation where the maximum element (the root of the heap) is swapped with an element at a leaf node, requiring a sift-down operation to ensure the heap property. This operation takes \(O(\lg n)\) time and needs to be performed for each of the \(n\) elements. Thus, the time complexity becomes around \(2n \lg n\).
03

Understanding Big-Theta Notation

Big Theta notation, \( \Theta \), defines an upper and lower bound on the computational complexity of an algorithm. When we say \(W(n) \approx 2n \lg n \in \Theta(n \lg n)\), we are saying that the time complexity of heapsort in the worst-case scenario is bounded both above and below by \(n \lg n\) within a constant factor.

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Most popular questions from this chapter

Among Selection Sort, Insertion Sort, Mergesort, Quicksort, and Heapsort, which algorithm would you choose in each list-sorting situation below? Justify your answers. a. The list has several hundred records. The records are quite long, but the keys are very short. b. The list has about 45,000 records. It is necessary that the sort be completed reasonably quickly in all cases. There is barely enough memory to hold the 45,000 records. c. The list has about 45,000 records, but it starts off only slightly out of order. d. The list has about 25,000 records. It is desirable to complete the sort as quickly as possible on the average, but it is not critical that the sort be completed quickly in every single case.

Show that there are \(n(n-1) / 2\) inversions in a permutation of \(n\) distinct ordered elements with respect to its transpose.

Show that a heap with \(n\) nodes has \([n / 2]\) leaves.

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.

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.
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