CS 473: Algorithms
(Fall 2025)

What homework is for

Algorithm design and analysis is a skill that can only be developed through practice and feedback, just like cooking or basketball or integration or gardening or interviewing or teaching. Yes, there are several things that are useful to know and understand, but that knowledge and understanding is not enough. That comfortable feeling of "Oh, sure, I get it" when you follow a well-presented lecture or hear a TA carefully explain the solution to a homework problem is a seductive, dangerous trap. You can only learn to do the thing by actually doing the thing.

The homework is your opportunity to practice doing the thing. The lectures and textbook and office hours hopefully provide good intuition and motivation and justification for the skills we want you to develop, but the best way to develop those skills is by trying to solve the problems yourself. The practice is far more important than the solution.

Because the homework is intended to help you develop new skills, you are likely to get stuck; for some problems, you may have no idea how to even start. And that's okay. That's why we have a textbook and lecture videos and Discord and a library and Wikipedia; helping you get unstuck is part of our job. That's why we encourage students to work together; not so that you can share solutions, but so that you can share ideas and suggestions and feedback.

Similarly, you won't necessarily develop a complete solution to every homework problem yourself, and you may not be able to tell at first which parts of your final submitted solution are correct. And that's also okay. That's why we provide homework solutions—not just to show you the answer, but to help you see your own work more clearly. That's why we grade your homework submissions—not to give you points, but to give you feedback to help you improve.

To get the most out of any particular homework problem, it's important not just to aim for a solution to that specific problem, but to pay attention to how you're solving it. Every problem is an opportunity to practice that kind of problem; every solution is an example of that kind of solution. That goes for exams, too. And technical interviews. And research.

It's also important to aim for improvement—not perfection (which is impossible), not being better than other people (which can be toxic), but doing the thing better than you did yesterday, every day.

In practice, course grades in 473 are determined almost entirely by exams, which ask you to demonstrate the skills that the homework is meant to develop; homework scores have minimal impact. (In a typical semester, the median homework average is around 90%.) So even if your goal is to optimize your course grade, practicing with the homework problems—developing both your skill and your confidence in that skill—is more important than getting the right answers.

Deadly Sins

We've identified a small number of bad writing (and thinking) habits that are strongly correlated with poor performance in algorithms courses, but are easy to avoid. Homework and exam solutions that commit any of these sins will be penalized. We're not trying to be scary or petty (Honest!), but we do want to break a few common bad habits that seriously impede mastery of the course material.

• Write general solutions, not examples. Don't describe algorithms by showing the first two or three iterations and then writing "and so on". Similarly, don't try to prove something by demonstrating it for a few small examples and then writing "do the same thing for all n". Any solution that includes phrases like "and so on", "etc.", "do this for all n", or "repeat this process" automatically gets a score of zero. Those phrases indicate precisely where you should have used iteration, recursion, or induction but didn't.
• Write for humans, not compilers. Algorithms should be described using pseudocode and/or structured English. We reserve the right to give bare code a grade of zero, even if it is perfectly correct. Whenever you introduce a new abstract object (for example, a graph, a recurrence, a data structure, or an algorithm), you must explain in English what it means.
  • In particular, whenever you describe a function/algorithm with input parameters, you must explicitly describe the meaning of each input parameter and exactly how the output of your function/algorithm depends on those input parameters.
  • We are not asking you to separately explain your algorithm line by line in English; rather, we want a complete specification of the problem that your algorithm is solving. Describe how to use your algorithm, not just how it works.
  • If the problem statement already includes a complete specification of your algorithm, you do not need to repeat it. On the other hand, if your algorithm uses additional input parameters or solves a more general problem than the one we ask for, then the problem statement does not include a complete specification of your algorithm.
  • You should already be doing this when you write code. Just keep doing it.
• Don't use weak induction. Always, always, ALWAYS use a strong induction hypothesis, even in proofs that only apply the induction hypothesis at n-1. Why would you even want to tie n-2 hands behind your back?! Weak induction proofs will still receive full credit if they are perfect, but will be more heavily penalized otherwise, especially if the induction argument has the form P(n)⇒P(n+1) instead of P(n-1)⇒P(n). Basically, weak induction should die in a dumpster fire.
• Greedy algorithms require proofs of correctness. If you use a greedy algorithm, you must prove that your algorithm is correct, or you will get zero credit, even if the algorithm is perfectly correct. Are you absolutely sure you don't want to use dynamic programming instead?
• Don't cheat. You must write everything in your own words, and properly cite every outside source you use, including other students. Using ideas from other sources or people without citation is plagiarism. Copying other sources verbatim, even with proper citation, even with their permission, is plagiarism. Don't do that.

Format: What to submit

For each numbered homework problem, you will prepare and submit a single PDF file.

• We strongly recommend typesetting your homework, especially if you have sloppy handwriting. We recommend using LaTeX, but you are welcome to use whatever program and/or markup language you like. If you are using LaTeX, you are welcome to use our solution template.
• You are welcome to submit scans of hand-written homework solutions, provided they are clear and easy to read. We strongly recommend writing with a black ink on white unlined paper, using a scanning app instead of a raw photo, and printing your scanned document to check for readability before submitting. Please do not submit your first hand-written draft.
• Every submitted homework solution should contain the following information:
  • At the top of the first page: The homework number, the problem number, and the name and NetID of every homework group member. We use this information as error correction, in case someone submits the wrong PDF file or forgets to tell Gradescope about other group members.
  • At the end of the last page: An explicit list of all sources and collaborators, even if that list is empty. Submissions without such a list will not be graded.
  Be generous. If you're not sure whether a consulted source was useful, cite it. Similarly, if you're not sure whether someone should be included in your list of collaborators, cite them. For discussions in class, in office hours, or in other large groups, where collecting individual names is impractical, it's okay to write something like "discussions in Tanvi's office hours" or "Thursday evening Grainger study group".
• Each numbered homework problem can be answered completely in at most two typeset pages, or at most four handwritten pages. If your solution is significantly longer than that, you are probably including too much unimportant details.

Logistics: How to submit

All homework will be submitted and graded on Gradescope. You can enroll yourself on the course Gradescope page with the entry code 5KKB3X.

• Submit a separate PDF file for each numbered homework problem. Gradescope will not accept other text file formats such as plain text, HTML, LaTeX source, or Microsoft Word (.doc or .docx); homework submitted as images (.png or .jpg) will not be graded.
• For all homeworks except Homework 0, groups of at most three students can submit group solutions for each problem. You are responsible for forming your own groups.
  • For each problem, exactly one member of each homework group should submit the team's solution and identify all other group members on Gradescope.
  • If you discover that you did not receive credit for your group's homework submission, please ask someone in your group to submit a regrade request. We will use the information at the top of the first page to verify that you are a member of that homework group.
• Gradescope allows you to resubmit as often as you like, but only the last submission will actually be graded. In particular, if your last submission is after the deadline, late penalties will apply, even if you previously submitted a full-credit solution before the deadline. Similarly, all group members must be identified on the last submission; new submissions do not automatically inherit group members from previous submissions.

Form: How to write

Please make it easy for the graders to figure out what you mean in the short time they have to grade your solution. If your solutions are difficult to read or understand, you will lose points.

Be Honest

• Write everything in your own words, and properly cite every outside source you use. We strongly encourage you to use any outside source at your disposal, provided you use your sources properly and give them proper credit. If you get an idea from an outside source, citing that source will not lower your grade. Failing to properly cite an outside source—thereby taking credit for work that is not your own—is plagiarism.
  The only sources that you are not required to cite are the official course materials (lectures, lecture notes, homework solutions, and exam solutions) from this semester and sources for prerequisite material (which we assume you already know by heart).
• List everyone you worked with on each homework problem. Again, we strongly encourage you to work together, but you must give everyone proper credit.
• Please see our academic integrity policy for more details.

Be Clear

• Write legibly. You will lose points if the graders find your work difficult to read. Writing legibly also helps you think more clearly.
  • The graders have complete discretion to decide whether a submission is difficult to read. In borderline cases, the graders will first give a warning, asking the student for specific improvements in future homework submissions, rather than immediately deducting points. Please take this feedback seriously.
• Write sensibly. You will lose points for poor spelling, grammar, punctuation, arithmetic, algebra, logic, and so on. This rule is especially important for students whose first language is not English.
• Write carefully. We can only grade what you actually write, not what you mean. We cannot read your mind, and we will not try. If your answer is ambiguous, the graders are explicitly instructed to choose an interpretation that makes it wrong.
• Don't submit your first draft. Revise, revise, revise. First figure out the solution, then think about the right way to present it, and only then start writing what you plan to submit. See also "write legibly".
• State your assumptions. If a problem statement is ambiguous, explicitly state any additional assumptions that your solution requires. (Please also ask for clarification in class, in office hours, or on Ed Discussion, or on Discord!) For example, if the performance of your algorithm depends on how the input is represented, tell us exactly what representation you require. Yes, even if the assumption is "obvious".
• Don't submit code. Describe your algorithms using clean, human-readable pseudocode. Your description should allow a bright student in CS 225 to easily implement your algorithm in their favorite language, using a software library containing implementations of every algorithm we've seen in class, without knowing what problem you're trying to solve. Oh, and they think your favorite programming language sucks.

Be Concise

• Keep it short. Every homework problem can be answered completely in at most two typeset pages or five handwritten pages; most problems require considerably less. Yes, I am aware of the crushing irony.
• Omit irrelevant details. Don't write "red-black tree" when you mean "balanced binary tree" or "dictionary". Don't write "depth-first search" when you mean "whatever-first search". Don't submit code; we want to see your ideas, not syntactic sugar. If your solution requires more than two typeset pages, you are providing too many irrelevant details.
• Don't regurgitate. Don't explain binary search; just write "binary search". Don't write the pseudocode for Dijkstra's algorithm; just write "Dijkstra's algorithm". If the solution appears on page 374 of Jeff's book, just write "See page 374 of the textbook." If your answer is similar to something we've seen in class, just say so and (carefully!) describe your changes. You will lose points for vomiting, especially if you get the details wrong.
• Don't bullshit. You will not get partial credit for word salad that happens to include some correct keywords.

Content: What to write

• Answer the right question. No matter how clear and polished your solution is, it's worthless if it doesn't answer the question we asked. Make sure you understand the question before you start thinking about how to answer it. If something is unclear, ask for clarification! This is especially important on exams.
• By default, if a homework or exam problem asks you to describe an algorithm, you need to do several things to get full credit:
  • Describe the problem that your algorithm is supposed to solve. This is often the hardest part of designing an algorithm. See the second Deadly Sin.
  • Give a concise pseudocode description of your algorithm. Don't regurgitate, and don't turn in code!
  • Describe a correct algorithm.
  • Justify the correctness of your algorithm. You must provide a brief justification for your solutions, as evidence that you understand why they are correct. Unless we explicitly say otherwise, we generally do not want a complete proof of correctness (because complete proofs would be too long, tedious, and unenlightening) but rather a high-level sketch of the major steps in the proof. Correctness arguments are required on exams only when we specifically ask for them.
  • Analyze your algorithm's worst-case running time. This may be as simple as saying "There are two nested loops from 1 to n, so the running time is O(n²)." Or it may require setting up and solving a recurrence, in which case you'll also have to justify your solution. We only care about worst-case running time. Unless we specifically ask otherwise, you do not have to analyze space.
  • Describe the fastest correct algorithm you can, even if the problem does not include the words "fast" or "efficient". Faster algorithms are worth more points; brute force is usually not worth much. We usually won't tell you what time bound to shoot for; that's part of what you're trying to learn. However, if your algorithm is incorrect, you won't get any points, no matter how fast it is!
    For each problem we assign, we have a target running time in mind. Correct algorithms that meet the target running time are worth full credit. Slower algorithms are worth significant partial credit. We typically deduct 2 or 3 points out of 10 for each extra factor of n in the running time; however, every correct and clearly presented algorithm, no matter how slow, is worth partial credit. On the other hand, algorithms that are faster than the target time bound are worth extra credit; again, we typically award 2 or 3 points for each factor of n. We usually do not give the target running time in the problem statement, because when we do, a lot more students submit incorrect algorithms with the correct running time. First make it work; then make it faster.
• Some problems may deviate from these default requirements. For example, we may ask you for an algorithm that uses a particular approach, even though another approach may be more efficient. (Answer the right question!)