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Chapter 7: Problem 33

Modify Heapsort so that it stops after it finds the \(k\) largest keys in nonincreasing order. Analyze your algorithm, and show the results using order notation.

Short Answer

Expert verified
The modified Heapsort algorithm that finds the \(k\) largest keys in nonincreasing order has the time complexity of \(O(n + k \log n)\), where \(n\) is the total number of keys.

Step by step solution

01

Modifying Heapsort

The first step is to modify the Heapsort. The modification includes running the first part of Heapsort which is heapifying the array, and then instead of extracting all elements, only extract \(k\) elements. To run the first part and build a max heap, go through all non-leaf nodes, starting from the last internal node, down to the root node, and perform the heapify operation. This will ensure that the largest element is at the root of the heap, which will be the first largest key.
02

Extracting the \(k\) largest keys

After building the max heap, the next step is to extract the \(k\) largest keys. To do this, perform the second part of Heapsort which is extracting the top element from the heap and then calling heapify, but only do this \(k\) times. Each time after top extraction, call heapify on the root again to bring the next largest element to the root, which can then be extracted in the next iteration.
03

Analyzing the Algorithm

During analysis, observe that building the max heap, which is done once, has a time complexity of \(O(n)\), where \(n\) is the total number of keys. After that, extraction of \(k\) largest keys is done \(k\) times, with each having a time complexity of \(O(\log n)\) because of the heapify operation. Therefore, the overall time complexity, using big-O notation, of the modified algorithm becomes \(O(n + k \log n)\) as the time complexities for creating the heap and extracting the \(k\) largest keys are added.

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Most popular questions from this chapter

Show that the time complexity for the average number of exchanges performed by the Quicksort algorithm is approximated by \(0.69(n+1)\) lg \(n\)

Write a linear-time sorting algorithm that sorts a permutation of integers 1 through \(n\), inclusive.

Show that a heap with \(n\) nodes has \(\lceil n / 2\rceil\) leaves.

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 against those of Insertion Sort, Exchange Sort, and Selection Sort.

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