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Assignment 4: Sorting

> The ability to destroy a planet is insignificant next to the power of sorting.
> — Darth Vader (if he was a computer scientist)

Your task is to run performance experiments on these sorting algorithms:

– *bubble* sort
– *insertion* sort
– *selection* sort
– *shell* sort (see below)
– *merge* sort
– *quick* sort
– *iquick* sort (see below)
– *priority queue sort* where the priority queue is implemented as a heap

You’ll create a spreadsheet to report on your results (described below).

Put your implementations of all the functions in a file named
[a4_sort_implementations.h](a4_sort_implementations.h). Implement each algorithm
as described in the textbook/notes: you can use the code from the textbook, or
from other sources (make sure to cite your sources!). Implement the *exact*
function headers listed in [a4_base.h](a4_base.h). *Don’t* change
[a4_base.h](a4_base.h) in any way: put all the sorting-related code you write in
[a4_sort_implementations.h](a4_sort_implementations.h).

**Important** The *only* #include that
[a4_sort_implementations.h](a4_sort_implementations.h) can use is `#include
“a4_base.h”`. No other #includes are allowed.

## Shell Sort

[Shell sort](https://en.wikipedia.org/wiki/Shellsort) is a sorting algorithm
that works by swapping pairs of elements in ever-decreasing gaps. The sequence
of gaps has a significant impact on the algorithm’s performance, and so for
consistency you are **required** to use this gap sequence: n/2, n/4, n/8 …, 1
where n is the size of the vector being sorted.

This means that the last gap will always be 1, and so the algorithm will always
finish with a final pass of insertion sort.

**Important** There are gap sequences that make shell sort run more efficiently
than this one, but, for the sake of consistency, do *not* use them. Use only the
gap sequence given above.

## iquick Sort

*iquick* sort is regular quick sort, except when the sub-vectors being sorted
are shorter than some predetermined threshold length (that you pick), insertion
sort is used to sort them instead of quick sort. If you choose a good threshold
length, you can get better average-time performance than regular quick sort. Do
some experimentation to find the best threshold length!

## Priority Queue Sort

Implement priority queue sort (given in the lecture notes) using a heap to
implement the priority queue. For consistency, please use the priority queue
sort code given in the lectures. Since heaps do O(log n) worst-case
insertion/removal in a priority queue, the overall worst-case running time of
priority queue sort is O(n log n).

## Testing Your Implementations

Test your implementations using [test_sorts.cpp](test_sorts.cpp). You can modify
that file in any way you like (it is *not* one of the submitted files for this
assignment).

> **Note** [test_sorts.cpp](test_sorts.cpp) only checks for basic correctness of
> the sorting algorithm, and ignores the `Sort_stats` objects they return. Feel
> free to add more test cases if you want to test the `Sort_stats` objects.

Notice that most of the functions in [a4_base.h](a4_base.h) are *template*
functions, i.e. they take a `vector<T>` as input, where `T` is a type of value
that can be sorted.

All the sorts return a `Sort_stats` object containing their running
time and comparisons counts:

“`cpp
// a4_base.h

// …

typedef unsigned long ulong;

struct Sort_stats
{
string sort_name;
size_t vector_size = 0;
ulong num_comparisons = 0;
double cpu_running_time_sec = 0.0;

string to_csv() const
{
return sort_name + “, ”
+ to_string(vector_size) + “, ”
+ to_string(num_comparisons) + “, ”
+ to_string(cpu_running_time_sec);
}
}; // struct Sort_stats

// …
“`

For this assignment, `num_comparisons` is the number of times `<` or `<=` is
called on the values in the vector. It has the type `ulong` (`unsigned long`)
because the comparison counts can get very large.

You should modify the code of the sorting functions to return accurate
`Sort_stats` objects. You can write helper functions if you like.

### Example: Bubble Sort

For example, you can implement bubble sort like this:

“`cpp
template <typename T>
Sort_stats bubble_sort(vector<T> &v)
{
ulong num_comps = 0; // <— num_comps is initialized to 0 here
clock_t start = clock();

for (int i = 0; i < v.size(); i++)
{
for (int j = 0; j < v.size() – 1; j++)
{
num_comps++; // <— num_comps is incremented here
if (v[j] > v[j + 1])
{
T temp = v[j];
v[j] = v[j + 1];
v[j + 1] = temp;
}
}
}

clock_t end = clock();
double elapsed_cpu_time_sec = double(end – start) / CLOCKS_PER_SEC;

return Sort_stats{“Bubble sort”,
v.size(),
num_comps,
elapsed_cpu_time_sec};
}
“`

### Timing Code in C++

As shown in `bubble_sort`, CPU time can be measured using `clock()` like this:

“`cpp
#include <ctime>

void timed_experiment() {
clock_t start = clock();

//
// … do stuff …
//

clock_t end = clock();

double elapsed_cpu_time_sec = double(end – start) / CLOCKS_PER_SEC;

cout << “It took ” << elapsed_cpu_time_sec << ” seconds to do stuff.\n”;
}
“`

[cpu_timer_example.cpp](cpu_timer_example.cpp) contains an example of timing
code.

## Helper Functions

In addition to the sorting routines, there are two other helper functions in
[a4_base.h](a4_base.h) that you need to implement:

– `is_sorted(v)` returns `true` if the vector is sorted in ascending order, and
`false` otherwise. Its worst-case running time should be linear in the size of
`v`.
– `rand_vec(size, min, max)` returns a vector of `size` random integers, where
the integers range from `min` to `max` (both `min` and `max` are included).
Its worst-case running time should be linear in `size`.

Both of these functions are helpful when testing sorting routines, so you should
implement them first.

## Generating the Experimental Data

When your sorting algorithms are ready, use them to generate the following data:

– For each of the 8 sorting algorithms, do the following:
– For *N* = 2000, 4000, 6000, …, 50000 (25 different values of *N*) do the
following:
– create a vector of *N* random integers, where the numbers in the vectors
range from 1 to *N*
– run the sort on the vector of random integers
– print the sort name, amount of data being sorted, number of comparisons
that were done, and the CPU time (in seconds)

Since there are 8 algorithms and 25 values of *N*, you’ll end up with 8 * 25 =
200 results.

If you print the individual results as **comma separated values** (**CSV**s),
then you can more easily import them into a spreadsheet. For example, here are 4
calls to bubble sort printed as comma separated values:

“`
Bubble sort, 2000, 3998000, 0.003197
Bubble sort, 4000, 15996000, 0.014169
Bubble sort, 6000, 35994000, 0.035031
Bubble sort, 8000, 63992000, 0.066863
“`

If you save this output in a file, the file should be readable by Excel or
Google Sheets as a CSV file.

## Visualizing the Data in a Spreadsheet

Using Excel or Google Sheets, create a table of all the data you generated. Here
us an example of what the first four rows of the table ought to look like (the
exact numbers in your table might be different):

| **Name**    | **N** | **Comparisons** | **CPU Seconds** |
|————-|——-|—————–|—————–|
| bubble sort |  2000 | 3998000         | 0.003197        |
| bubble sort |  4000 | 15996000        | 0.014169        |
| bubble sort |  6000 | 35994000        | 0.035031        |
| bubble sort |  8000 | 63992000        | 0.066863        |

The table should have 4 columns, and each column is labelled with the given
name, as shown.

> **Remember** The table will have 200 rows of data.

In addition to this table, **draw four line graphs**, where the x-axis of each
graph is *N* (the size of the vector being sorted), and has the values from the
data you generated: N=2000, 4000, 6000, …, 50000. The four graphs are:

– **Graph 1**: **CPU time** for *bubble* sort, *insertion* sort, *selection*
sort, and *shell** sort. The the y-axis is the CPU time in seconds.

– **Graph 2**: **CPU time** for *merge* sort, *quick* sort, *iquick* sort, and
*priority queue sort*. The y-axis is the CPU time in seconds.

– **Graph 3**: **comparison counts** for *bubble* sort, *insertion* sort,
*selection* sort, and *shell* sort. The y-axis is the number of comparisons.

– **Graph 4**: **comparison counts** for *merge* sort, *quick* sort, *iquick*
sort, and *priority queue sort*. The y-axis is the number of comparisons.

Each graph will have 4 lines on it, one for each of the sorting algorithms it is
plotting data for.

Make these graphs *beautiful* and *easy to read*. Give them descriptive titles
(**don’t** just name them Graph 1, Graph 2, etc.!), clearly labelled x and y
axes, and use colors or textures to distinguish different lines on the same
graph.

> **Note** It’s not too hard to make nice-looking graphs in Excel or Google
> Sheets. Spend a bit of time playing with the options.

## What to Submit

When you’re done, submit a zip file containing just these two files:

– [a4_sort_implementations.h](a4_sort_implementations.h)

– An Excel or Google Sheets file named **a4.xlsx** that contains the table of
experimental data and the 4 graphs.

## Grading

The marker will compile and run your program on Ubuntu Linux using
[makefile](makefile) like this:

“`bash
> make a4_test
g++ -O3 -std=c++17 -Wall -Wextra -Werror -Wfatal-errors -Wno-sign-compare -Wnon-virtual-dtor -g a4_test.cpp -o a4_test

> ./a4_test
… testing output …

> valgrind ./a4_test
… valgrind output …
“`

The file `a4_test.cpp` is a test program written by the marker that tests the
correctness of your sorting algorithms (it is not provided here!). It will
#include [a4_base.h](a4_base.h) and
[a4_sort_implementations.h](a4_sort_implementations.h).

Note that the `-O3` flag is one of the options: this asks g++ to optimize the
code for speed.

> **Note** The compilation commands are quite strict! Make sure your program
> compiles with them before submitting it.

## Marking Scheme

– **1 mark** for a correct and efficient implementation of `is_sorted(v)`
– **1 mark** for a correct and `rand_vec(size, min, max)`.
– **14 marks** for implementing the 7 sorting algorithms (bubble sort is not
included since it’s given in the notes); 2 marks for each correct sorting
algorithm. This includes both the correctness of the sorting algorithm itself,
and the correctness of the `Sort_stats` object it returns.
– **5 marks** for a nicely-formatted spreadsheet table with 4 columns (all
labelled) and 200 rows of data, as explained above.
– **5 marks** for graphs that are beautiful, easy to read, have good titles,
have their axes labelled, and clearly label the different lines using color or
textures. If the graphs look wrong, or have missing data, or plot the wrong
thing, etc., the highest possible mark for this part will be **2 marks**.

### Overall source code readability: 5 marks

– All code is sensibly and consistently indented, and all lines are 100
characters in length, or less.
– Whitespace is used to group related pieces of a code to make it easier for
humans to read. All whitespace has a purpose.
– Variable and function names are self-descriptive.
– Appropriate features of C++ are used, as discussed in class and in the notes.
**Note** If you use a feature that we haven’t discussed in class, **you must
explain it in a comment**, even if you think it’s obvious.
– Comments are used when needed to explain code whose purpose is not obvious
from the code itself. There should be *no* commented-out code from previous
versions.

`### Deductions

– **-5 marks** for any memory leaks, or other errors, reported by `valgrind`.
– Up to **-3 marks** if you do *not* include your full name, email, and SFU ID
in the header of your file.
– Up to **-3 marks** if your submitted files don’t following the
naming/formatting conventions given above.
– **A score of 0** if one or more of the following are true:
– your code *doesn’t* compile with the given makefile
– you *don’t* include the “Statement of Originality”, or it is modified in any
way
– you use code from some other source (e.g. the web, the textbook, ChatGPT, a
friend, a teacher, a TA…) *without* citing the source
– you submit a “wrong” non-working file, and then *after the due date* submit
the “right” file. If you can provide evidence that you finished the
assignment on time, then it may be marked

There may be other deductions, depending upon the circumstances.

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