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2024-04-24 15:33:33 -07:00
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Advanced/Compression/parts/1 runlength.tex
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@ -49,8 +49,13 @@ Using a na\"ive coding scheme, encode \texttt{AAAA$\cdot$AAAA$\cdot$BCD$\cdot$AA
\vfill
In \ref{runlenone}---and often, in the real world---the strings we want to encode have fairly low \textit{entropy}. \par
They have predictable patterns, sequences of symbols that don't contain a lot of information. \par
We can exploit this fact to develop efficient encoding schemes.
That is, they have predictable patterns, sequences of symbols that don't contain a lot of information. \par
\note{
For example, consider the text in this document. \par
The symbols \texttt{e}, \texttt{t}, and \texttt{<space>} are much more common than any others. \par
Also, certain subsequences are repeated: \texttt{th}, \texttt{and}, \texttt{encode}, and so on.
}
We can exploit this fact to develop encoding schemes that need relatively few bits per letter.
\example{}
A simple example of such a coding scheme is \textit{run-length encoding}. Instead of simply listing letters of a string
@ -88,10 +93,18 @@ We'll encode our string into a sequence of 6-bit blocks, interpreted as follows:
\end{center}
So, the sequence \texttt{BBB} will be encoded as \texttt{[0011-01]}. \par
\note[Notation]{
Just like dots, dashes and spaces are added for readability. \par
Just like dots, dashes and spaces are added for readability. Pretend they don't exist. \par
Encoded binary sequences will always be written in square brackets. \texttt{[]}.
}
\problem{}
Decode \texttt{[010000001111]} using this scheme.
\begin{solution}
\texttt{AAAADDD}
\end{solution}
\vfill
\problem{}
Encode \texttt{AAAA$\cdot$AAAA$\cdot$BCD$\cdot$AAAA$\cdot$AAAA} using this scheme. \par
Is this more or less efficient than \ref{runlenone}?
@ -109,12 +122,26 @@ Is this more or less efficient than \ref{runlenone}?
\problem{}
Give an example of a message on $\{\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}\}$
that uses $n$ bits when encoded with a na\"ive scheme, and \textit{fewer} than $\nicefrac{n}{2}$ bits
when encoded using the scheme described on the previous page.
\vfill
\problem{}
Give an example of a message on $\{\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}\}$
that uses $n$ bits when encoded with a na\"ive scheme, and \textit{more} than $2n$ bits
when encoded using the scheme described on the previous page.
\vfill
\problem{}
Is run-length coding always efficient? When does it work well, and when does it fail?
Is run-length coding always more efficient than na\"ive coding? \par
When does it work well, and when does it fail?
\vfill