% TODO:
% Basic run-length
% LZ77

\section{Run-length Coding}


%\definition{}
%\textit{Entropy} is a measure of information in a certain sequence. \par
%A sequence with high entropy contains a lot of information, and a sequence with low entropy contains relatively little.
%For example, consider the following two ten-symbol ASCII\footnotemark{} strings:
%\begin{itemize}
%	\item \texttt{AAAAAAAAAA}
%	\item \texttt{pDa3:7?j;F}
%\end{itemize}
%The first string clearly contains less information than the second.
%It's much harder to describe \texttt{pDa3:7?j;F} than it is \texttt{AAAAAAAAAA}.
%Thus, we say that the first has low entropy, and the second has fairly high entropy.
%
%\vspace{2mm}
%
%The definition above is intentionally hand-wavy. \par
%Formal definitions of entropy exist, but we won't need them today---we just need
%an intuitive understanding of the \say{density} of information in a given string.

%
%\footnotetext{
%	American Standard Code for Information Exchange, an early character encoding for computers. \par
%	It contains 128 symbols, including numbers, letters, and
%	\texttt{!"\#\$\%\&`()*+,-./:;<=>?@[\textbackslash]\^\_\{|\}\textasciitilde}
%}


%\vspace{5mm}


\problem{}<runlenone>
Using a na\"ive coding scheme, encode \texttt{AAAA$\cdot$AAAA$\cdot$BCD$\cdot$AAAA$\cdot$AAAA} in binary. \par
\note[Note]{
	We're still using the four-symbol alphabet $\{\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}\}$. \par
	Dots ($\cdot$) in the string are drawn for readability. Ignore them.
}

\begin{solution}
	There are eight \texttt{A}s on each end of that string. Mapping symbols as before, \par
	we get \texttt{[00 00 00 00 00 00 00 00 01 10 11 00 00 00 00 00 00 00 00]}
\end{solution}


\vfill
In \ref{runlenone}---and often, in the real world---the strings we want to encode have fairly low \textit{entropy}. \par
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
in their binary form, we'll add a \textit{count} to each letter, shortening repeated instances of the same symbol.

\vspace{2mm}

We'll encode our string into a sequence of 6-bit blocks, interpreted as follows:

\begin{center}
	\begin{tikzpicture}
		\node[anchor=west,color=gray] at (-2.3, 0) {Bits};
		\node[anchor=west,color=gray] at (-2.3, -0.5) {Meaning};
		\draw[color=gray] (-2.3, -0.25) -- (5.5, -0.25);
		\draw[color=gray] (-2.3, 0.15) -- (-2.3, -0.65);

		\node at (0, 0) {\texttt{0}};
		\node at (1, 0) {\texttt{0}};
		\node at (2, 0) {\texttt{1}};
		\node at (3, 0) {\texttt{1}};
		\node at (4, 0) {\texttt{0}};
		\node at (5, 0) {\texttt{1}};

		\draw (-0.5, 0.25) -- (5.5, 0.25);
		\draw (-0.5, -0.25) -- (5.5, -0.25);
		\draw (-0.5, -0.75) -- (5.5, -0.75);

		\draw (-0.5, 0.25) -- (-0.5, -0.75);
		\draw (3.5, 0.25) -- (3.5, -0.75);
		\draw (5.5, 0.25) -- (5.5, -0.75);

		\node at (1.5, -0.5) {number of copies};
		\node at (4.5, -0.5) {symbol};
	\end{tikzpicture}
\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. 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}?

\begin{solution}
	\texttt{[1000-00 0001-01 0001-10 0001-11 1000-00]} \par
	This requires 30 bits, as compared to 38 in \ref{runlenone}.
\end{solution}

\vfill
\pagebreak






\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 more efficient than na\"ive coding? \par
When does it work well, and when does it fail?

\vfill


\problem{}
Our coding scheme wastes a lot of space when our string has few runs of the same symbol. \par
Fix this problem: modify the scheme so that single occurrences of symbols do not waste space. \par
\hint{We don't need a run length for every symbol. We only need one for \textit{repeated} symbols.}

\begin{solution}
	One idea is as follows: \par
	\begin{itemize}
		\item Encode single symbols na\"ively: \texttt{ABCD} becomes \texttt{[00 01 10 11]}
		\item Signal runs using two copies of the same symbol: \texttt{AAAAAA} becomes \texttt{[00 00 0110]}. \par
		When our decoder sees two copies of the same symbol, it will interpret the next four bits as
		a run length.
	\end{itemize}
	\texttt{BDC$\cdot$DDDDD$\cdot$AADBDC} will be encoded as \texttt{[01 11 10 11-11-0101 01-01-0010 11 01 11 10]}.
\end{solution}

\vfill

\problem{}<firstlz>
Consider the following string: \texttt{ABCD$\cdot$ABCD$\cdot$BABABA$\cdot$ABCD$\cdot$ABCD}. \par
\begin{itemize}
	\item How many bits do we need to encode this na\"ively? \par
	\item How about with the (unmodified) run-length scheme described on the previous page?
\end{itemize}
\hint{You don't need to encode this string---just find the length of its encoded form.}

\begin{solution}
	Na\"ively: \tab 22 bits \par
	Run-length: \tab $6 \times 21 = 126$ bits. Watch out for the two repeated \texttt{A}s!
\end{solution}


\vfill

Neither solution to \ref{firstlz} is ideal. Run-length is very wasteful due to the lack of runs, and na\"ive coding
does not take advantage of repetition in the string. We'll need a better coding scheme.
\pagebreak
