showed that one way of comparing different algorithms for accomplishing the same
ID: 3605102 • Letter: S
Question
showed that one way of comparing different algorithms for accomplishing the same task is complexity analysis. You will recall that in complexity analysis we express the time an algorithm takes to run as a function of the size of the input, and we used the big-Oh notation. For example, if an algorithm has a complexity of O(1), then it always runs in the same amount of time, no matter what the size of the input is; if it O(n), then the time it takes for the algorithm to run is proportional to the size of the input.
However, complexity analysis has a number of limitations. For example, big-Oh analysis concerns the worst case scenario. For example, some sorting algorithms with a complexity of O(n^2) often run considerably faster if the list that it receives as input is (almost) sorted; other sorting algorithms with a complexity of O(n^2) always take the same amount of time, no matter what state the list is in. Also, in big-Oh, we look at the dominant term in our calculation of the complexity of the algorithm. Thus, when we analyze an algorithm and discover that it runs in 75,312 + n time units, we still say that it has a complexity of O(n). It is therefore deemed to be better than an algorithm that run in .007 + n^2 time units, as this algorithm has a complexity of O(n^2). We also saw the rationale behind this: If n becomes sufficiently large, the other factors become insignificant.
Fortunately, there is another way to determine how long it takes for an algorithm to run, namely timing experiments. In a timing experiment, you actually implement the algorithm in a programming language, such as Java or C++, and simply measure how long it takes for the algorithm to run.
In the term project for this course, you are going to conduct a timing experiment and compare the results with the results you would get from a complexity analysis. We will compare Bubble Sort with Selection Sort.
In its least sophisticated form, bubble sort (http://en.wikipedia.org/wiki/Bubble_sort) works as follows:
Assuming that the list contains n elements.
Compare the first and the second element in the list, and swap them if the last element is smaller than the preceding one; otherwise, do nothing to this pair.
Now, compare second and third element t and swap them if the first of them is larger than the second; otherwise, do nothing to this pair.
Move on the next pair and continue the process until you reach the end of the list.
A little reflection will show that at the end of this iteration, the last element in the list is now the largest element in the list. The last element has bubbled to the top.
Now repeat the process but rather than going to the end of the list, stop when you reach n-1.
Now repeat the process again, but rather than going to the end of the list, stop when you reach n-2.
Keep repeating this until you reach 1.
The Wikipedia entry has a little simulation that shows how bubble sort works. The code looks something like:
bubbleSort(array A){
n = length(A);
for(j = n; j > 0, j--)
for(i = 1; i < j; i++) {
if A[i-1] > A[i]
swap(A,i-1, i);
}
}
}
swap obviously swaps the elements and can be defined as:
swap(A, pos1, pos2) {
temp = A[pos1];
A[pos1] = A[pos2];
A[pos2] = temp;
}
Another sort is selection sort (http://en.wikipedia.org/wiki/Selection_sort). We saw selection sort in the question in the sub-module on how to determine the complexity of an algorithm. Array A contains n elements, the elements to be sorted. The algorithm starts at the first position in the array and looks through the array for the smallest element. Once it reaches the end of the array, it puts that element in the first cell of the array. It then restarts the whole process from the second position in the array, and continues until the entire array has been sorted.
selection_sort(array A) {
int i,j
int iMin;
for(j = 0; j < n; j++){
iMin = j;
The purpose of the project is to perform a timing experiment. You are required to complete the following activities:
1. Write a computer program that prompts the user for a number, creates an array for that number of random integers, and then uses the bubble sort to order the array. The program should print out the array prior to the call to the sorting algorithm and afterwards. You can write the program in either Java, C++, C#, or whatever language you are most comfortable in.
2. Repeat 1 but use selection sort this time.
1 and 2 are primarily intended to make sure that your algorithms work.
Once you are convinced your programs work, do the following
3. Write a computer program that prompts the user for one number, n for the number of items in the array to sort, and create and sort 1000 arrays of this size timing the run to get an average time to sort an array of this size. Then do the following:
Initiate a variable running_time to 0
Create a for loop that iterates 1000 times.
In the body of the loop,
Create an array of n random integers
Get the time and set this to start-time. You will have to figure out what the appropriate command is in the programming language you are using to find the time
Use bubble sort to sort the array
Get the time and set this to end-time
Subtract start-time from end-time and add the result to total_time
Once the program has run, note
The number of items sorted
The average running time for each array (total_time/1000)
Repeat the process six times, using 50, 250 and 500 as the size of the array for each of the two algorithms.
4. Repeat 3 using selection sort.
Explanation / Answer
Part 1:
#include <iostream>
#include <malloc.h>
#include <time.h>
using namespace std;
/*This function will swap the values between two variable a and b*/
void swap_integers(int *a, int *b){
int temp = *a;
*a = *b;
*b = temp;
}
int main() {
int n,i,j,temp;
cout<<"Please enter the size of the array :";
cin>>n;
int *int_array = (int *)malloc(sizeof(int));
/* initialize random seed: */
srand (time(NULL));
/* generate and insert random numbers between 1 and 1000: */
for(i=0;i<n;i++){
int_array[i] = rand() % 1000;
}
/* Printing array before sorting */
cout<<" Array Before Sorting:"<<" ";
for(i=0;i<n;i++){
cout<<int_array[i]<<" ";
}
/*Bubble sorting*/
for(i=0; i<n; i++){
/* taking n-i-1 to avoid segmentation fault
otherwise with n-i code will access index n when j = n-1
with the condition int_array[j+1]*/
for(j=0;j<n-i-1;++j){
/*If first number is gretaer in the pair
then swap the numbers*/
if(int_array[j]>int_array[j+1]){
swap_integers(&int_array[j],&int_array[j+1]);
}
}
}
/* Printing array After sorting */
cout<<" Array After Sorting:"<<" ";
for(i=0;i<n;i++){
cout<<int_array[i]<<" ";
}
return 0;
}
Part 2:
#include <iostream>
#include <malloc.h>
#include <time.h>
using namespace std;
/*This function will swap the values between two variable a and b*/
void swap_integers(int *a, int *b){
int temp = *a;
*a = *b;
*b = temp;
}
int main() {
int n,i,j,temp;
cout<<"Please enter the size of the array :";
cin>>n;
int *int_array = (int *)malloc(sizeof(int));
/* initialize random seed: */
srand (time(NULL));
/* generate and insert random numbers between 1 and 1000: */
for(i=0;i<n;i++){
int_array[i] = rand() % 1000;
}
/* Printing array before sorting */
cout<<" Array Before Sorting:"<<" ";
for(i=0;i<n;i++){
cout<<int_array[i]<<" ";
}
/*Selection sorting*/
for(i=0; i<n; i++){
/*taking the ith number as the smallest
to compare it with the rest of the followed list*/
int min = int_array[i];
int min_num_idx = i;
for(j=i+1;j<n;++j){
if(min > int_array[j] ){
min = int_array[j];
min_num_idx = j; /* store the idx of the minimum number*/
}
}
swap_integers(&int_array[i], &int_array[min_num_idx]);
}
/* Printing array After sorting */
cout<<" Array After Sorting:"<<" ";
for(i=0;i<n;i++){
cout<<int_array[i]<<" ";
}
return 0;
}
Part 3:
#include <iostream>
#include <malloc.h>
#include <time.h>
using namespace std;
/*This function will swap the values between two variable a and b*/
void swap_integers(int *a, int *b){
int temp = *a;
*a = *b;
*b = temp;
}
int main() {
int n,i,j,temp,k;
int running_time = 0;
time_t start_time = 0;
time_t end_time = 0;
cout<<"Please enter the size of the array :";
cin>>n;
int *int_array = (int *)malloc(sizeof(int));
int total_arrays = 1000; // Total 1000 arrays as asked in the question
int
/* initialize random seed: */
srand (time(NULL));
/*Run the sorting for 1000 arrays*/
for(k=0;k<total_arrays;++k){
/* generate and insert random numbers between 1 and 1000: */
for(i=0;i<n;i++){
int_array[i] = rand() % 1000;
}
start_time = time(NULL); /*note start time og the sort* /
/*Bubble sorting*/
for(i=0; i<n; i++){
/* taking n-i-1 to avoid segmentation fault
otherwise with n-i code will access index n when j = n-1
with the condition int_array[j+1]*/
for(j=0;j<n-i-1;++j){
/*If first number is gretaer in the pair
then swap the numbers*/
if(int_array[j]>int_array[j+1]){
swap_integers(&int_array[j],&int_array[j+1]);
}
}
}
end_time = time(NULL); /*Not end time of the sort*/
running_time += (end_time - start_time);
}
cout<<" Numbers of item sorted = "<<n*1000;
cout<<" Average riunning time = "<<((running_time*1.0)/1000)<<" ";
return 0;
}
Part 4:
#include <iostream>
#include <malloc.h>
#include <time.h>
using namespace std;
/*This function will swap the values between two variable a and b*/
void swap_integers(int *a, int *b){
int temp = *a;
*a = *b;
*b = temp;
}
int main() {
int n,i,j,temp,k;
int running_time = 0;
time_t start_time = 0;
time_t end_time = 0;
cout<<"Please enter the size of the array :";
cin>>n;
int *int_array = (int *)malloc(sizeof(int));
int total_arrays = 1000; // Total 1000 arrays as asked in the question
int
/* initialize random seed: */
srand (time(NULL));
/*Run the sorting for 1000 arrays*/
for(k=0;k<total_arrays;++k){
/* generate and insert random numbers between 1 and 1000: */
for(i=0;i<n;i++){
int_array[i] = rand() % 1000;
}
start_time = time(NULL); /*note start time og the sort* /
/*Selection sorting*/
for(i=0; i<n; i++){
/*taking the ith number as the smallest
to compare it with the rest of the followed list*/
int min = int_array[i];
int min_num_idx = i;
for(j=i+1;j<n;++j){
if(min > int_array[j] ){
min = int_array[j];
min_num_idx = j; /* store the idx of the minimum number*/
}
}
swap_integers(&int_array[i], &int_array[min_num_idx]);
}
end_time = time(NULL); /*Not end time of the sort*/
running_time += (end_time - start_time);
}
cout<<" Numbers of item sorted = "<<n*1000;
cout<<" Average riunning time = "<<((running_time*1.0)/1000)<<" ";
return 0;
}
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