A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 7.12 Projects 277classmates? What does this say about the difficulty of doing empirical timingstudies?7.5 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?7.6 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.7.7 Radix Sort is typically implemented to support only a radix that is a powerof two. This allows for a direct conversion from the radix to some numberof bits in an integer key value. For example, if the radix is 16, then a 32-bitkey will be processed in 8 steps of 4 bits each. This can lead to a more efficientimplementation because bit shifting can replace the division operationsshown in the implementation of Section 7.7. Re-implement the Radix Sortcode given in Section 7.7 to use bit shifting in place of division. Comparethe running time of the old and new Radix Sort implementations.7.8 It has been proposed that Heapsort can be optimized by altering the heap’ssiftdown function. Call the value being sifted down X. Siftdown does twocomparisons per level: First the children of X are compared, then the winneris compared to X. If X is too small, it is swapped with its larger child and theprocess repeated. The proposed optimization dispenses with the test againstX. Instead, the larger child automatically replaces X, until X reaches thebottom level of the heap. At this point, X might be too large to remain inthat position. This is corrected by repeatedly comparing X with its parentand swapping as necessary to “bubble” it up to its proper level. The claimis that this process will save a number of comparisons because most nodeswhen sifted down end up near the bottom of the tree anyway. Implement bothversions of siftdown, and do an empirical study to compare their runningtimes.