HSPC: Finding (or Building) a Team

This is a post in my High School Programming Competition (HSPC) series.  For an outline of current and upcoming posts, please see this introduction.
 
At the collegiate level, programming competition teams are typically formed by a chapter of the ACM (Association for Computing Machinery) or a similar club within the computer science department.  While many universities have a CS (or related) department, analogous organizations tend not to exist in high school.  In this post, I hope to share some tips on what to do if you're interested in participating in an HSPC but can't find a team to join.

I'll be writing this primarily from the perspective of a student who is looking for or building a team, as this is what I ended up doing.

School teams

Finding a coach

Most teams tend to represent and are sponsored by a single high school.  If your high school already has a team, there's not much left to do.  Thus, I'll assume there isn't one at your school.  You'll likely want to get a teacher to agree to coach a new team.  If your school has a computer science class, the right teacher to approach is pretty much obvious.

If you don't have a CS class at your school, though, this can be a little more tricky.  Some related classes which attract the type of teachers that could be interested in leading a class include math (especially higher-level math) and science (especially physics).  A robotics team coach (i.e. for a FIRST or VRC team) may be interested in coaching a programming team, too.  If none of these options apply for you, consider looking outside the school for a parent who works in a related field, like engineering or mathematics.

Realize, though, that when you approach this teacher requesting them to advise your team, you're really asking if they'd be willing to spend several hours of their time (possibly likely unpaid) to provide your team with a place to practice, advice for improvement, and the many logistical steps required to get your team to a tournament.  The more preparation and thought you put into your request before talking with your teacher, the more likely they are to agree to help.  Don't just say "I'd like to do programming contests, kthxbai!"  Instead, research a specific contest that is a reasonable distance from your school you'd like to attend, come up with an estimate on the time they'd have to spend per week on this project, and explain (in as precise a way as possible) how you need their help.  This way, they have more information to make a good decision.

Team Members

You will want to find around 3 other people to join your team (team size limits may vary based on the contest you will attend).  If you already know some friends who would be interested in competing, this isn't a very hard task.  However, it may be that you're the only one you know at your school interested in CS.  This is one area in which having a teacher as a coach is very helpful; they may be interested in making an announcement before class, which will get the word out quickly.  Depending on your school policies, it may be possible to create a flyer and post it on a bulletin board in the hall, or post to a class Facebook group.

On the other hand, it could be that you have way too many people who are interested.  This is a good problem!  Many contests will allow multiple team registrations per school, so it may be possible to send everyone who is interested.  However, it is also typical for contests to cap the number of teams coming from the same school; for instance, UVa caps our contest at 3 teams.  In this event, it is a good idea to hold a "qualifier" tournament or give all interested parties a test and take only the top students. 

Community teams

School teams are sometimes infeasible for various reasons; this was the circumstance in which I found myself.  Building a community team tends to be a little more difficult, particularly because there isn't an easy way to locate potential team members.

Finding a Coach

Finding a coach for a community team can be somewhat challenging; most often, it ends up being a parent of one of the team members who may have some relevant background.

If none of your team members' parents have any interest in CS, fear not!  It is completely possible to have a student-led team, but you do need an adult "figurehead" to handle logistical concerns and act as a point-of-contact for the contest director.  Technical mentorship can be sourced through means other than your "official coach;" for instance, you could contact a local university with a CS competition team and ask if they'd be interested in sending a few students to give a presentation and/or provide some feedback on your preparation.

My advice from above on preparing before approaching your potential coach is still applicable in this situation: the more complete of a picture you can "paint" for this person, the more easily they can make a decision to volunteer their time.

Team Members

Many programming competitions are built around the concept of a team of 3 or 4 students working to solve more problems than other teams.  While policies on "one-person teams" vary between host organizations, the general sentiment is that you need team members to do well and to optimally learn.  (My personal experience testifies to this.)  So, although it can be tempting to try to compete solo instead of going to the effort of finding possible team members, it is well worth the time to find friends interested in this topic.

Unfortunately, there is no universal way to contact anyone in a geographic region to ascertain their interest in an activity.  Some options could include posting a notice in a local library, reaching out via Facebook groups, or talking with members of a local hackerspace.

As another point of note, these team members do not even need to be well-versed in programming.  In my senior year of high school, my team consisted of two people who really knew the Java language and two people who knew enough programming to write some simple things, but who were just really good at thinking logically and solving problems.  These contests really boil down to solving problems with programming as simply the medium for recording the solution, rather than a programming event that happens to involve solving problems.

If it is relevant to your situation, homeschoolers tend to have a geographically-defined association that may have a Facebook group or email distribution list.  Although, from your perspective, it may sometimes appear that you are the only person in your community interested in programming, trying an email list like that one can be quite fruitful.

Other Thoughts

This was a very non-technical entry and probably the least important post in the series.  Finding team members and coaches is a "soft-skills" problem, and there is no right answer.  I promise the future posts in the series will be much more technical and easily applicable, but I wanted to write this first because one must have a team before getting really crazy about preparing for a contest. 

Deranged Exams [A Solved ICPC Problem]

This past week, my ICPC team worked the 2013 Greater New York Regional problem packet.  One of my favorite problems in this set was Problem E: Deranged Exams.  The code required to solve this problem isn't that complicated, but the math behind it is a little unusual.  In this post, I aim to explain the math and provide a solution to this problem.

Problem Description

The full problem statement is archived online; in shortened form, we can consider the problem to be:

Given a "matching" test of $n$ questions (each question maps to exactly one answer, and no two questions have the same answer), how many possible ways are there to answer at least the first $k$ questions wrong?

It turns out that there's a really nice solution to this problem using a topic from combinatorics called "derangements."  (Note that the problem title was a not-so-subtle hint towards the solution.)

Derangements

While the idea of a permutation should be familiar to most readers, the closely related topic of a derangement is rarely discussed in most undergraduate curriculum.  So, it is reasonable to start with a definition:

A derangement is a permutation in which no element is in its original place.  The number of derangements on $n$ elements is denoted $D_n$; this is also called the subfactorial of $n$, denoted $!n$.

The sequence $\langle D_n\rangle$ is A000166 in OEIS (a website with which, by the way, every competitive programmer should familiarize themselves).

It turns out that there is both a recursive and an explicit formula for $D_n$:
\begin{align}
D_n &= (-1)^n \sum_k\binom{n}{k} (-1)^k k! \\
&= n\cdot D_{n-1} + (-1)^n;\;(D_0=1)
\end{align}
This is significant because we can use the explicit formulation for computing single values of derangements, or we can use dynamic programming to rapidly compute $D_n$ for relatively small $n$.

Problem Approach

The key observation here is that, using the derangement formula, we may compute the number of ways to answer a given set of questions incorrectly, using only the answers corresponding to those questions.  Instead of focusing on the first $k$ questions, which we must answer incorrectly, let us look to the remaining $n-k$ questions.

Consider the case when we answer $r$ questions correctly.  There are $\binom{n-k}{r}$ ways of choosing which $r$ questions we answer correctly (since the first $k$ must be wrong).

The remaining $n-r$ questions must be answered incorrectly using only the answers to the same $n-r$ questions.  Using our knowledge of derangements, there are $!(n-r)$ ways to assign those incorrect answers.

Finally, note that the number of correct answers, $r$ is bounded by $n-k$; summing over all possible values of $r$, we obtain:
\[S(n, k) = \sum_{r=0}^{n-k} \binom{n-k}{r}!(n-r)\]

Code

Equations are great, but implementation is required for ICPC.  First, we must consider input/output size.  The problem statement gives the following ranges for $n$ and $k$:
\[1\leq n \leq 17 \\  0 \leq k \leq n \]

We can expect that this will fit in a 64-bit integer, as $n! \leq 2^{63}-1$ for $n\leq 20$.  Thus, we don't even need to be careful in computing binomial coefficients due to intermediate overflow!  I'll let the code (and comments) speak for itself:

import java.util.*;
 

public class Test {
  // Basic iterative factorial; just multiply all
  // the numbers less than or equal to n.
  // returns 1 if n < 1 (which is important for n=0)
  private static long fact(int n) {
    long retval = 1; 
    while(n > 0) 
      retval *= n--;
    return retval;
  }

  // Naive binomial coefficient computation 
  // Generally, you need to watch overflow.  But,
  // we can ignore that here because fact(17) < 2^63-1
  private static long binom(int n, int k) {
    return fact(n)/(fact(k)*fact(n-k));
  }
 
  public static void main(String[] args) { 
    //While not recommended in general, we can use 
    // a scanner because we're not reading a lot of input.
    Scanner cin = new Scanner(System.in);
 
    // Precompute the derangement numbers
    long[] d = new long[18]; // we might need values of D_n up to n=17
    d[0] = 1;
    for (int i = 1, j=-1; i < d.length; i++, j*=-1)
      d[i] = i*d[i-1] + j;
    //Process the input
    int P = cin.nextInt();
    for (int caseNum = 0; caseNum < P; caseNum++) {
      cin.nextInt();
      int n = cin.nextInt();
      int k = cin.nextInt();
 
      //S(n, k) = sum(binom(n-k, r)*d[n-r], r=0..n-k)
      long ans = 0;
      for (int r = 0; r <= n-k; r++)
         ans += binom(n-k, r)*d[n-r];
 
      System.out.printf("%d %d\n", caseNum+1, ans);
    }
  }
}

Further Reference

Derangements are discussed in Concrete Mathematics by Graham, Knuth, and Patashnik on pages 193-196.  In those pages, the identities shown in this blog entry are derived.  Also discussed is a closely related problem that may be called $r$-derangements.

In the $r$-derangement problem, we seek the number of arrangements in which exactly $r$ elements are in their original place.  (The number of $0$-derangements, then, is just $D_n$.)