A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 7.6 Heapsort 257and fast. The algorithm should take advantage of the fact that sorting is a specialpurposeapplication in that all of the values to be stored are available at the start.This means that we do not necessarily need to insert one value at a time into thetree structure.Heapsort is based on the heap data structure presented in Section 5.5. Heapsorthas all of the advantages just listed. The complete binary tree is balanced, its arrayrepresentation is space efficient, and we can load all values into the tree at once,taking advantage of the efficient buildheap function. The asymptotic performanceof Heapsort is Θ(n log n) in the best, average, and worst cases. It is not asfast as Quicksort in the average case (by a constant factor), but Heapsort has specialproperties that will make it particularly useful when sorting data sets too large to fitin main memory, as discussed in Chapter 8.A sorting algorithm based on max-heaps is quite straightforward. First we usethe heap building algorithm of Section 5.5 to convert the array into max-heap order.Then we repeatedly remove the maximum value from the heap, restoring the heapproperty each time that we do so, until the heap is empty. Note that each time weremove the maximum element from the heap, it is placed at the end of the array.Assume the n elements are stored in array positions 0 through n−1. After removingthe maximum value from the heap and readjusting, the maximum value will nowbe placed in position n − 1 of the array. The heap is now considered to be of sizen − 1. Removing the new maximum (root) value places the second largest valuein position n − 2 of the array. At the end of the process, the array will be properlysorted from least to greatest. This is why Heapsort uses a max-heap rather thana min-heap as might have been expected. Figure 7.10 illustrates Heapsort. Thecomplete Java implementation is as follows:static