Data Structures and Algorithms

Lab 3: QuickSort and Big-O

Due on Sunday, September 25th at 11:59 PM. This is a team lab. You and your assigned lab partner(s) will complete this lab together. Make sure that you are familiar with the Partner Etiquette guidelines. You may discuss the concepts of this lab with other classmates, but you may not share your code with anyone other than course staff and your lab partner(s). Do not look at solutions written by students other than your team. If your team needs help, please post on the Piazza forum or contact the instructor or ninjas. If you have any doubts about what is okay and what is not, it’s much safer to ask than to risk violating the Academic Integrity Policy.

Overview

This lab consists of three primary parts:

As with the previous lab, you and your partner will share a repository. The URL for your repository will be git@github.swarthmore.edu:cs35-f16/lab03-<your-teamname>.

Please remember that both lab partners must learn this material; it will feature prominently on your tests and final exam. Do not divide up this lab to give one team member the theory material and another member the sorting algorithm.

Part I: QuickSort

Piles of paper
Source: OpenClipArt.org

Your starting repository includes the following files:

Files that you will need to modify are bold.

To complete this lab, you should implement the quickSort function in quickSort.cpp just as we discussed the algorithm in class. You’ll need to be careful to follow the algorithm closely and to manipulate array indices accurately. You’ll probably have bugs in your first attempt, so remember the following tools we have discussed for finding and correcting mistakes:

Unit Tests

For this lab, you will be graded not only on whether your quickSort implementation is correct but also on whether you have properly tested it. You have been provided a couple tests to start, but you are required to write at least four more tests that investigate different aspects of sorting. Some ideas for additional tests include:

Although you have a minimum number of tests to write, you should ideally write as many as it takes for you to be confident in your code. Remember: each time you change your tests, run make tests before re-running your test program.

Part II: Written Assignment

In this part of your lab, you will apply the Big-O complexity material that we have discussed in class by writing some Big-O proofs about mathematical functions.

Electronic Submissions Only

In order to complete this assignment, you will need to produce a document containing some mathematical expressions and figures. You will commit and push this document to your GitHub repository; the document may take the following forms:

Book
Source: OpenClipArt.org

Your team must submit the document electronically in one of these forms. In particular, you are not permitted to turn in the following (to name a few examples):

A Few Words About LaTeX

LaTeX (pronounced lah-tek or lay-tek) is a document preparation system frequently used to produce conference papers, dissertations, and other academic works (in addition to other material). With LaTeX, you code your document much as you would an HTML document or a Python program. For instance, the LaTeX code

I'm \textit{not} making this up: \(e^{i\pi} = -1\)

produces

You are not required to learn LaTeX for this course. However, it may be easier to use basic LaTeX to complete your homework than to use something like Microsoft Word’s equation editor. In case you’re interested, the following files are already part of your repository:

Big-O Proofs

Use the formal definition of Big-O notation to prove the following statements:

  1. \(n^{3} + 2n^{2} + 4\) is \(O(n^{3})\)
  2. \(2n^{4} - n\) is \(O(n^{4})\)

Part III: Mystery Functions

In this part of the lab, you will first analyze six simple loop structures and determine their runtime complexity in terms of Big-\(O\) notation. Then, using the provided program function_timer, you will graph the empirical runtimes of these functions.

Functions

Penguin Detective
Source: OpenClipArt.org

Begin by identifying the Big-\(O\) runtime for each of the following functions. Provide a brief justification of your answer; a sentence or two should do. Give the strictest Big-\(O\) you can find (don’t give \(O(2^n)\) if the function is also \(O(n)\)) and leave out any unnecessary terms (give \(O(n^2)\) rather than \(O(n^2+3n)\)).

Using function_timer to Inspect the Functions

Each of the above functions has been implemented and packaged into an executable function_timer that was provided to you in your repository for this lab. The function_timer program will provide empirical runtime data for each function in a form that can be graphed by another tool called gnuplot.

To use this program, you must pick the following:

For instance, you can graph functions 1 and 3 within \(10 \leq n \leq 100\) by running this command:

./function_timer -1 -3 -n 10 -m 100 | gnuplot

After a moment, a window will pop up with an image like this:

Plot for functions 1 and 3 between n=10 and n=100

Note the “pipe” character (|) in the command above; this feeds the output of function_timer to the input of gnuplot so it can graph the results for you.

If you want to save the image that gnuplot makes for you, you can add a -s parameter to function_timer like so:

./function_timer -1 -3 -n 10 -m 100 -s "f1,f3 from 10 to 100.png" | gnuplot

Written Response

For this part of the assignment, include in WrittenLab.pdf the following parts:

  1. A description of which mystery function (-1, -2, etc.) matches which algorithm (fn1, fn2, etc.).
  2. For each mystery function, a short paragraph describing why you think it matches the algorithm you named.
  3. For two of the mystery functions, a graph (saved from function_timer) that supports your claims.

Peer Review

As this is a team assignment, you are required to complete a peer review of your lab partner or partners. You must do this even if everything is fine. You should complete this peer review after you have turned in your assignment. The peer review comes in the form of a simple Google Form and can be accessed here; in most cases, it will likely take less than a minute to complete. If you have trouble accessing this peer review, please make sure that you are logged into Google using your Swarthmore credentials.

Your peer review will be kept private; only course staff will have access to this information. You can also update your peer review after you have submitted it. However, if you do not submit a peer review, you will lose participation grade points. Please don’t forget!

Summary of Requirements

You must