39 lines
1.3 KiB
TeX
39 lines
1.3 KiB
TeX
\section{Introduction}
|
|
|
|
\definition{}
|
|
An \textit{alphabet} is a set of symbols. Two examples are
|
|
$\{\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}\}$ and $\{\texttt{0}, \texttt{1}\}$.
|
|
|
|
\definition{}
|
|
A \textit{string} is a sequence of symbols from an alphabet. \par
|
|
For example, \texttt{CBCAADDD} is a string over the alphabet $\{\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}\}$.
|
|
|
|
\problem{}
|
|
Say we want to store a length-$n$ string over the alphabet $\{\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}\}$ as a binary sequence. \par
|
|
How many bits will we need? \par
|
|
\hint{
|
|
Our alphabet has four symbols, so we can encode each symbol using two bits, \par
|
|
mapping $\texttt{A} \rightarrow \texttt{00}$,
|
|
$\texttt{B} \rightarrow \texttt{01}$,
|
|
$\texttt{C} \rightarrow \texttt{10}$, and
|
|
$\texttt{D} \rightarrow \texttt{11}$.
|
|
}
|
|
|
|
\begin{solution}
|
|
$2n$ bits.
|
|
\end{solution}
|
|
|
|
\vfill
|
|
|
|
|
|
\problem{}<naivelen>
|
|
Similarly, we can encode an $n$-symbol string over an alphabet of size $k$ \par
|
|
using $n \times \lceil \log_2k \rceil$ bits. Show that this is true. \par
|
|
\note[Note]{We'll call this the \textit{na\"ive coding scheme}.}
|
|
|
|
|
|
\vfill
|
|
As you might expect, this isn't ideal: we can do much better than $n \times \lceil \log_2k \rceil$.
|
|
We will spend the rest of this handout exploring more efficient ways of encoding such sequences of symbols.
|
|
\pagebreak
|