Data Structures and Algorithm Analysis - Computer Science at ...

262 Chap. 7 Internal SortingModify the Bubble Sort implementation to add this flag and test. Comparethe modified implementation on a range of inputs to determine if it does ordoes not improve performance in practice.7.2 Double Insertion Sort is a variation on Insertion Sort that works from themiddle of the array out. At each iteration, some middle portion of the arrayis sorted. On the next iteration, take the two adjacent elements to the sortedportion of the array. If they are out of order with respect to each other, thanswap them. Now, push the left element toward the right in the array so longas it is greater than the element to its right. And push the right elementtoward the left in the array so long as it is less than the element to its left.The algorithm begins by processing the middle two elements of the array ifthe array is even. If the array is odd, then skip processing the middle itemand begin with processing the elements to its immediate left and right.First, explain what the cost of Double Insertion Sort will be in comparison tostandard Insertion sort, and why. (Note that the two elements being processedin the current iteration, once initially swapped to be sorted with with respectto each other, cannot cross as they are pushed into sorted position.) Then, implementDouble Insertion Sort, being careful to properly handle both whenthe array is odd and when it is even. Compare its running time in practiceagainst standard Insertion Sort. Finally, explain how this speedup might affectthe threshold level and running time for a Quicksort implementation.7.3 Perform a study of Shellsort, using different increments. Compare the versionshown in Section 7.3, where each increment is half the previous one,with others. In particular, try implementing “division by 3” where the incrementson a list of length n will be n/3, n/9, etc. Do other increment schemeswork as well?7.4 The implementation for Mergesort given in Section 7.4 takes an array as inputand sorts that array. At the beginning of Section 7.4 there is a simplepseudocode implementation for sorting a linked list using Mergesort. Implementboth a linked list-based version of Mergesort and the array-basedversion of Mergesort, and compare their running times.7.5 Starting with the Java code for Quicksort given in this chapter, write a seriesof Quicksort implementations to test the following optimizations on a widerange of input data sizes. Try these optimizations in various combinations totry and develop the fastest possible Quicksort implementation that you can.(a) Look at more values when selecting a pivot.(b) Do not make a recursive call to qsort when the list size falls below agiven threshold, and use Insertion Sort to complete the sorting process.Test various values for the threshold size.(c) Eliminate recursion by using a stack and inline functions.