Program 3 Sorting ‑ Improved Mergesort Solution

Purpose




Although merge sort shows the same execution complexity as quick sort, (i.e., O(n log n)), its practical performance is much slower than quick sort due to array copying operations at each recursive call. This programming assignments improves the performance of merge sort by implementing a non­recursive, semi­ in­place version of the merge­sort algorithm. You will learn how to estimate the complexity of an algorithm through computer simulation.




In‑Place Sorong




In­place sorting is to sort data items without using additional arrays. For instance, quicksort performs partitioning operations by simply repeating a swapping operation on two data items in a given array, which thus needs no extra arrays.




On the other hand, mergesort we have studied allocates a temporary array, sorts partial data items in that array, and copies them back to the orignal array at each recursive call. Due to this repetitive array­copying operations, mergesort is much slower than quicksort although their running time is upper­bounded to O(n log n).




It has been an research interest to develop in­place mergesort algorithms, almost all of which are however impractically complicated. Yet, we can improve the performance of mergesort with adding the following two restrictions:




Using a non­recursive method (use iterative method)



Using only one additional array, (i.e., a semi­in­place method). Merge data from the original into the additional array at the very bottom stage, and thereafter perform the next merge from the additional into the original array, in which way merge operations are applied to the original and additional array in turn as you go through each repetitive stage. In other words, at the first merge, data is from original to additional array. And the next merge, data is from additional to original array, etc.



Note that we still need to allow data items to be copied between the original and this additional arrays as many times as O(log n).




Statement of Work
Design and implement a non­recursive, semi­in­place version of the merge­sort algorithm. The framework of your mergesort function will be :






#include <vector




#include <math.h // may need to use pow( )




using namespace std;




template <class Comparable




void mergeImproved( vector<Comparable &a ) {




int size = a.size( );




vector<Comparable b( size ); // this is only one temporary array.




// implement a nonrecursive mergesort only using vectors a and b.




}










Needless to say, the above mergeImproved( ) function must not call itself or some other recursive functions.




Furthermore, the algorithm should still be based on the same divide­and­conquer approach.




Use the driver file (Files­­Program3/driver.cpp ) to verify and evaluate the performance of your non­ recursive, semi­in­place mergesort program. The code below assumes that your program is written in the "mergeImproved.cpp" file.



You can use the usual mergsort and quicksort from the Files­Programs/Program3/ (or Sample codes folder)



Modify the driver file to compare the performance among the usual quicksort, the usual mergesort, and your improved mergesort as increasing the array size. You will need a loop that increase the array size by 20 each until size=1,000. Initial size is 20.



Use Excel to draw a graph comparing the three algorithms, and estimate its Big O. You will need to draw O(nlog), O(n), and O(logn) to estimate its Big O.









What to Turn in




Clearly state in your code comments any other assumptions you have made. Turn in:




your nonrecursive, semi­in­place mergesort program, (i.e., template <class Comparable void mergeImproved( vector<Comparable &a ) in "mergeImproved.cpp".) (Don't use different function name or file name!)



your modified driver.cpp that include a function that compares performance of merge, quick and mergeImproved algorithms.



a separate report in .doc or .docx that must includes:



One­page output of your improved mergesort program (when #items = 30), and
A chart that compares the performance among the usual quicksort, the usual mergesort, and your improvedmergesort, as well as plots of nlogn, n, logn.



Estimated Big O for each algorithm.















Grading Guide and Answers




Check the following grading guide to see how your homework will be graded. Key answer will be made available after the due date through Solution page.













Program 3 Grade Guideline




1. Documentation (10 pts)




One page
output (a.out 30 will
fit one page.)


Correct(2pts)
1
~ 2 errors(1pt)
3+ errors or no results(0pt)
Performance comparison between
your algorithm and ordinary merge/quick sorts
All
three plots are given
and Big-O is estimated using nlogn, n, and logn plots (total 6 plot
s). Comparison results are clearly stated.(8 pts)




No chart: 0 pts




Each of the following cases be taken off 2 points :




No Big-O