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Chapter 7: Problem 32

Show that the worst-case time complexity of the number of assignments of records for Heapsort is approximated by \(W(n) \approx n\) lg \(n\)

Short Answer

Expert verified
In the worst-case scenario, the number of assignments of records for Heapsort is approximately \(W(n) \approx n \log n\). This is due to each heapify operation requiring potentially log(n) assignments, and there being n such operations in total.

Step by step solution

01

Understand Heapsort

Heapsort is a comparison based sorting algorithm with a time complexity of O(n log n) for all cases. The algorithm works by visualizing the data as a binary heap, an almost complete binary tree, where each parent node is larger than or equal to its child nodes. This property maintains the so-called heap property, which is essential to the workings of heapsort.
02

Heapsort Operations and their time complexity

In heapsort, we can broadly classify operations into two categories: buildHeap and heapify. The buildHeap operation is performed to create the initial heap from the unsorted array and has a time complexity of O(n). The heapify operation maintains the heap property by percolating down a node until the heap property is restored. Its time complexity is O(log n). Next, in each of the n iterations, the maximum element in the heap (the root) is swapped with the last element in the heap and removed, then the heapify operation is invoked to restore the heap property. This gives a time complexity of O(n log n).
03

Assignments in Heapsort

In the worst case, each exchange or assignment operation in heapsort - namely, during heapify and extracting the maximum - will take place. Specifically, during heapify, each node may need to be swapped with one of its children nodes until the heap property is restored. This results in an average of log(n) assignments per operation, since the structure is a nearly complete binary tree. Over n iterations, this gives a total of n log(n) assignments.
04

Wrap Up

To sum up, in the worst-case scenario, Heapsort will perform approximately n log(n) assignment operations.

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Most popular questions from this chapter

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

Use the divide-and-conquer approach to write a nonrecursive Mergesort 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.

Give the transpose of the permutation \([2,5,1,6,3,4],\) and find the number of inversions in both permutations. What is the total number of inversions?

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 noncontinuous 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 per. formance of Insertion Sort.

Study the idea of designing a sorting algorithm based on a ternary heap. A temary heap is like an ordinary heap except that each internal node has three children.
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