Question

Using C++, Python, or Java, write a program that:

In this programming exercise you will perform an empirical analysis of the QuickSort algorithm to study the actual average case behavior and compare it to the mathematically predicted behavior. That is, you will write a program that counts the number of comparisons performed by QuickSort on an array of a given size. You will run the program on a large number of arrays of a certain size and determine the average number of comparisons performed. You will do this for several different sizes and compare the observed results with what is expected.

Using C++, Python, or Java, write a program that:

prompts the user to enter the number n, of arrays to be processed and the size m, of each array

creates an array, of the given size, containing the integers between 1 and m

for the given number of arrays

randomly unsorts the array

sorts the array using the version of QuickSort developed in your text and in class

counts the number of comparisons performed during the partitioning subroutine of the sort

calculates the average number of comparisons performed in processing all the arrays

You will need to use the versions of QuickSort and Partition given in your text since that was what was used to derive the predicted average complexity you will be comparing your results with.

The program should output the number of arrays processed, the size of the arrays processed and the average number of comparisons per array.

For example, if the user enters 50 and 100 the output would be similar to:

# of arrays processed: 50

# of items in each array: 100

average number of comparisons: 926.02 (Your results may be different!)

The following algorithm can be used to unsort an array A, of size m.

1) FOR i := 1 to m

2)    Generate a random number r, between 1 and m

3)    Swap A[i] and A[r]

Use your program to determine the average number of comparisons performed in sorting 1000 arrays for each of the array sizes, 10, 50, 100, 500, 1000, 5000, 10,000, 50,000 and 100,000. Compare the observed average with the average predicted by the derivation of the average-case time complexity in section 2.4 of your text. Display your results in a table comparing the predicted average with the observed average. You will want to look at the percent difference between the predicted and observed values. Write a paragraph discussing your conclusions about any differences you notice and what they tell you about the actual asymptotic behavior of the quick sort.

Incremental Development

It is usually not a good idea to try to code a complicated algorithm from beginning to end. Taking a few extra minutes to create your program incrementally can save you HOURS of debugging. In developing this program, I would suggest the following approach:

Write functions that implement QuickSort and Partition as given in your text.

Write a “main” function that creates the array of exercise 19 on p. 91 and sorts it using your QuickSort function.

Add code to count the number of comparisons performed during the sort. This will require an additional parameter to both QuickSort and Partition.

Check that the number of comparisons used to sort the array of exercise 19 on p. 91 is what it should be.

Write your “unsort” function and test it thoroughly.

Change “main” so that it prompts the user for the size m, of the array, creates an array of size m that contains the integers 1 to m, unsorts it, then uses QuickSort to sort it and counts the comparisons.

Finally, change “main” so that it prompts the user for the number of arrays n, to test and prints the average number of comparisons done.

Answer 1

Here is the code for quick sort and counting comparisons and average after taking user inout

other part of assignment you have to do by running code with different values and comparing with what is told in class (as these are not shared in question)

I used last element as pivot in partition which is common among most textbooks

#include <iostream>
#include <stdlib.h>
using namespace std;

// A utility function to swap two elements
void swap(int* a, int* b)
{
int t = *a;
*a = *b;
*b = t;
}

/* This function takes last element as pivot, places
the pivot element at its correct position in sorted
array, and places all smaller (smaller than pivot)
to left of pivot and all greater elements to right
of pivot */
int partition (int arr[], int low, int high, int *comparisons)
{
int pivot = arr[high]; // pivot
int i = (low - 1); // Index of smaller element

for (int j = low; j <= high- 1; j++)
{
// If current element is smaller than or
// equal to pivot
*comparisons = *comparisons+1;
if (arr[j] <= pivot)
{
i++; // increment index of smaller element
swap(&arr[i], &arr[j]);
}
}
swap(&arr[i + 1], &arr[high]);
return (i + 1);
}

/* The main function that implements QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
void quickSort(int arr[], int low, int high, int *comparisons)
{
if (low < high)
{
/* pi is partitioning index, arr[p] is now
at right place */
int pi = partition(arr, low, high, comparisons);

// Separately sort elements before
// partition and after partition
quickSort(arr, low, pi - 1, comparisons);
quickSort(arr, pi + 1, high, comparisons);
}
}

void unsort(int arr[], int n)
{
int r = 0;
for(int i = 0; i < n; i++)
{
r = rand()%n;
swap(&arr[i], &arr[r]);
}
}

int main()
{
int n, m;
cout << "Enter number of times to sort array: ";
cin >> n;
  
cout << "Enter size of array to be sorted: ";
cin >> m;
  
int *arr = new int [m];
  
for(int i = 0; i < m; i++)
{
arr[i] = i+1;
}
  
long totalComparisons = 0;
  
for(int i = 0; i < n; i++)
{
unsort(arr, m);
int comparisons = 0;
quickSort(arr, 0, m-1, &comparisons);
totalComparisons += comparisons;
}
  
delete[] arr;
  
double average = ((double) totalComparisons)/n;

cout << "# of arrays processed: " << n << endl;
cout << "# of items in each array: " << m << endl;
cout << "average number of comparisons: " << average << endl;
  
return 0;
}