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\markright{\footnotesize CS 451 Programming Paradigms, Spring 2002}


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\section*{Homework 1 (corrected) --- Scheme review --- due Monday 28 January}

\subsection*{Palindromes}

Given a list of one-letter strings, determine whether it is a palindrome.

\subsection*{Powerset}
Represent sets using lists, and write a function that takes a set $S$ and returns
its powerset $2^S$. Let $f (0) = \emptyset$, $f(k+1) = 2^{f(k)}$.
Write a Scheme function for $f$ 
and compute the sets $f(k)$ for $k = 0,1,2,3,4$.


\subsection*{Binary trees}
For a given $n$, 
enumerate all structurally distinct binary trees with a total number
of nodes equal to $n$.


\subsection*{Safe transmission codes and molecules}

An alphabet is a set of symbols; for instance $\Sigma_1 = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$
is the alphabet of decimal digits.
A string is a finite sequence of symbols drawn from a specific alphabet;
for instance $x_1 = 2718281828$ is a string over the alphabet $\Sigma_1$.
Given two strings of equal length, the Hamming distance between them is
the number of positions in which their symbols differ.
For instance, $H(2718281828,
                 1828182827) = 6$. 
For strings of length $N$, the Hamming
distance is at least $0$ and at most $N$.

With an alphabet of $K$ symbols, there are $K^N$ strings of length $N$.
We'll say that two strings $x$ and $y$ are distinguishable if
$H(x,y) \ge D$, where $D$ is a prescribed minimum Hamming distance.
Out of the total $K^N$ strings, how many (at most) can we choose such that
each two of them are distinguishable?

Try to work out (mathematicaly) 
how large the biggest possible subset is, with $K$, $N$,
and $D$ as parameters. (This is a very difficult problem: no good
solutions are known.)

Write a Scheme function that will produce an exemplar of such a subset
of distinguishable strings.
The function should be called like this: 
\texttt{(maxSafeSubset alphabet wordLength minHammingDistance)};
for instance we might invoke it as:
\texttt{(maxSafeSubset '(0 1) 3 2)}, and it should then evaluate to
\texttt{'((0 0 0) (0 1 1) (1 0 1) (1 1 0))}.

Why is this problem really important?
If the alphabet is $\{0,1\}$, we are talking about coding theory used for
error detection and correction over telecommunication lines.
If the alphabet is $\{A,G,C,T\}$, we are talking about DNA (properly,
oligonucleotide) sequences suitable for molecular computation.

\subsection*{How to turn in}

Turn in your code by running 

 \emph{\char126dmykola/handin your-file}

on a regular UNM CS machine. 

You should use whatever filename is appropriate in place of your-file. 
You can put multiple files on the command line, or even directories. Directories will have their entire contents handed in, so please be sure to clean out any cruft.

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