CS 202 Homework 1 – Algorithm Efficiency and Sorting

Question 1 – 15 points

    (a) [5 points] Show that   (  ) = 8  4 + 5  3 + 7 is   (  5) by specifying appropriate c and n0 values in Big-O definition

    (b) [10 points] Trace the following sorting algorithms to sort the array [ 22, 8, 49, 25, 18, 30, 20, 15, 35, 27 ] in ascending order. Use the array implementation of the algorithms as described in the textbook and show all major steps.

        ◦ Selection sort

        ◦ Bubble sort


Question 2 – 70 points

Implement the following functions in the sorting.cpp file:

    (a) [40 points] Implement the insertion sort, bubble sort, merge sort, and quick sort algorithms. Your functions should take an array of integers and the size of that array and then sort it in ascending order. Add two counters to count and return the number of key comparisons and the number of data moves for all sorting algorithms. For the quick sort algorithm, you are supposed to take the first element of the array as pivot. Your functions should have the following prototypes (all prototypes should be in sorting.h):


void insertionSort(int *arr, const int size, int &compCount, int &moveCount); void bubbleSort(int *arr, const int size, int &compCount, int &moveCount); void mergeSort(int *arr, const int size, int &compCount, int &moveCount); void quickSort(int *arr, const int size, int &compCount, int &moveCount);


For key comparisons, you should count each comparison like k1 < k2 as one comparison, where k1 and k2 correspond to the value of an array entry (that is, they are either an array entry like arr[i] or a local variable that temporarily keeps the value of an array entry).

For data moves, you should count each assignment as one move, where either the right-hand side of this assignment or its left-hand side or both of its sides correspond to the value of an array entry (e.g., a swap function has three data moves).

    (b) [15 points] To test your implementation and conduct the experiments described below, you must write additional auxiliary global functions as follows. The prototypes of these functions are given below. The first function displays the array items on the screen. The other ones are to create four identical arrays that will be used for testing the sorting algorithms with random numbers (generated using the random number generator function rand), numbers in almost sorted order, and numbers in almost unsorted order, respectively.

        ◦ The "almost sorted" array is created by starting with an int array of size n containing the sequence 0, 1, 2, ..., n-1 and perturbing it slightly. To perturb the array, choose n/20 random pairs of indices (in the range 0 through n-1) and for each pair of indices (i, j), swap the elements at indexes i and j. Note that ~10% of the array will be perturbed.

        ◦ The "almost unsorted" array is generated in the same way that the "almost sorted" array is created, except that you start with the sequence n-1, n-2, ..., 3, 2, 1, 0 and perturb it slightly in the same way described above.

void displayArray(const int *arr, const int size);

void createRandomArrays(int *&arr1, int *&arr2, int *&arr3, int *&arr4, const int size);

void createAlmostSortedArrays(int *&arr1, int *&arr2, int *&arr3, int *&arr4, const int size); void createAlmostUnsortedArrays(int *&arr1, int *&arr2, int *&arr3, int *&arr4, const int size);


    (c) [5 points, mandatory] Create a main.cpp file which does the following in order:

        ◦ creates an array from the following: {9, 6, 7, 16, 18, 5, 2, 12, 20, 1, 16, 17, 4, 11, 13, 8}.

        ◦ calls the insertionSort function, displays the number of key comparisons and the number of data moves to sort this array, and calls the displayArray function to show the contents of the array after insertion sorting

        ◦ calls the bubbleSort function, displays the number of key comparisons and the number of data moves to sort this array, and calls the displayArray function to show the contents of the array after bubble sorting

        ◦ calls the mergeSort function, displays the number of key comparisons and the number of data moves to sort this array, and calls the displayArray function to show the contents of the array after merge sorting

        ◦ calls the quickSort function, displays the number of key comparisons and the number of data moves to sort this array, and calls the displayArray function to show the contents of the array after quick sorting

At the end, write a basic Makefile which compiles all of your code and creates an executable file named hw1. Check out this tutorial for writing a simple makefile:

http://www.cs.colby.edu/maxwell/courses/tutorials/maketutor/

Please make sure that your Makefile works properly, otherwise you will not get any points from Question 2.


Important: Then run your executable and add the screenshot of the console to the solution of Question 2 in the pdf submission.


    (d) [10 points] In this part, you will analyze the performance of the sorting algorithms that you will have implemented. Write a performanceAnalysis function to systematically call these algorithms. This function should call the auxiliary global functions to create the arrays with different contents. For each scenario (random, almost sorted, almost unsorted), use the following sizes for the arrays: 5000, 10000, 15000, 20000, 25000, 30000, 35000, 40000. Run each of the sorting algorithms with each scenario and output the elapsed time in milliseconds, the number of key comparisons and the number of data moves. Do not include the time required for creating these arrays.

The performanceAnalysis function needs to produce an output similar to the one given on the next page. Include this output to the answer of Question 2 in the pdf submission.

-----------------------------------------------------

Analysis of
Insertion Sort


Array SizeElapsed timecompCount
moveCount
5000
x
ms
x
x
10000
x
ms
x
x
...





-----------------------------------------------------

Analysis of
Bubble Sort


Array SizeElapsed timecompCount
moveCount
5000
x
ms
x
x
10000
x
ms
x
x
...





-----------------------------------------------------

Analysis of
Merge
Sort

Array SizeElapsed
timecompCount
moveCount
5000
x
ms
x
x
10000
x
ms
x
x
...





-----------------------------------------------------

Analysis of
Quick
Sort

Array SizeElapsed
timecompCount
moveCount
5000
x
ms
x
x
10000
x
ms
x
x
...







Question 3 – 15 points

After running your programs, prepare a single page report about the experimental results that you will have obtained in Question 2. With the help of a spreadsheet program (Microsoft Excel, Matlab or other tools), plot elapsed time versus the array size for each sorting algorithm implemented in Question 2. Combine the outputs of each sorting algorithm in a single graph.

In your report, interpret and compare your empirical results with the theoretical ones. Explain any differences between the empirical and theoretical results, if any.

Do not forget to discuss how the time complexity of your program changed when you applied the sorting algorithms to almost sorted arrays and almost unsorted arrays instead of an array containing randomly generated numbers. Also briefly explain the rationality behind this change.