Polish
This commit is contained in:
@ -1,6 +1,6 @@
|
||||
\section{LZ Codes}
|
||||
|
||||
The LZ-family\footnotemark{} of codes (LZ77, LZ78, LZSS, LZMA, and others) take advantage of repeated sequences of symbols
|
||||
The LZ-family\footnotemark{} of codes (LZ77, LZ78, LZSS, LZMA, and others) take advantage of repeated subsequences
|
||||
in a string. They are the basis of most modern compression algorithms, including DEFLATE, which is used in the ZIP, PNG,
|
||||
and GZIP formats.
|
||||
|
||||
@ -21,10 +21,10 @@ Pointers take the form \texttt{<pos, len>}, where \texttt{pos} is the position o
|
||||
For example, we can encode the string \texttt{ABRACADABRA} as \texttt{[ABRACAD<7, 4>]}. \par
|
||||
The pointer \texttt{<7, 4>} tells us to look back 7 positions (to the first \texttt{A}), and copy the next 4 symbols. \par
|
||||
Note that pointers refer to the partially decoded output---\textit{not} to the encoded string. \par
|
||||
This allows pointers to reference other pointers, and ensures codes like \texttt{A<1,9>} are valid.
|
||||
This allows pointers to reference other pointers, and ensures that codes like \texttt{A<1,9>} are valid.
|
||||
|
||||
\problem{}
|
||||
Encode \texttt{ABCD$\cdot$ABCD$\cdot$BABABA$\cdot$ABCD$\cdot$ABCD} using LZ.
|
||||
Encode \texttt{ABCD$\cdot$ABCD$\cdot$BABABA$\cdot$ABCD$\cdot$ABCD} using this scheme. \par
|
||||
Then, decode the following:
|
||||
\begin{itemize}
|
||||
\item \texttt{[ABCD<4,4>]}
|
||||
@ -39,7 +39,7 @@ Then, decode the following:
|
||||
\linehack{}
|
||||
|
||||
In parts two and three, remember that we're reading the \textit{output string.} \par
|
||||
The nine \texttt{A}s in part two are produced one by one, \par
|
||||
The ten \texttt{A}s in part two are produced one by one, \par
|
||||
with the decoder's \say{read head} following its \say{write head.}
|
||||
|
||||
\begin{itemize}
|
||||
@ -58,98 +58,114 @@ Convince yourself that LZ is a generalization of the run-length code we discusse
|
||||
\remark{}
|
||||
Note that we left a few things out of this section: we didn't discuss the algorithm that converts a string to an LZ-encoded blob,
|
||||
nor did we discuss how we should represent strings encoded with LZ in binary. We skipped these details because they are
|
||||
problems of implementation---they're the engineer's headache, not the mathematician's. If you're interested, a brief explanation is below.
|
||||
Ask an instructor to explain.
|
||||
problems of implementation---they're the engineer's headache, not the mathematician's. \par
|
||||
|
||||
\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);
|
||||
\pagebreak
|
||||
|
||||
\node at (0, 0) {\texttt{0}};
|
||||
\node at (1, 0) {\texttt{0}};
|
||||
\node at (2, 0) {\texttt{1}};
|
||||
\node at (3, 0) {\texttt{0}};
|
||||
\node at (4, 0) {\texttt{1}};
|
||||
\node at (5, 0) {\texttt{1}};
|
||||
\node at (6, 0) {\texttt{0}};
|
||||
\node at (7, 0) {\texttt{0}};
|
||||
\node at (8, 0) {\texttt{1}};
|
||||
|
||||
\draw (-0.5, 0.25) -- (8.5, 0.25);
|
||||
\draw (-0.5, -0.25) -- (8.5, -0.25);
|
||||
\draw (-0.5, -0.75) -- (8.5, -0.75);
|
||||
|
||||
\draw (-0.5, 0.25) -- (-0.5, -0.75);
|
||||
\draw (0.5, 0.25) -- (0.5, -0.75);
|
||||
\draw (8.5, 0.25) -- (8.5, -0.75);
|
||||
|
||||
\node at (0, -0.5) {flag};
|
||||
\node at (4.5, -0.5) {if flag \texttt{<pos, len>}, else eight-bit symbol};
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
% Text tape
|
||||
\node[color=gray] at (-0.75, 0) {\texttt{...}};
|
||||
\node[color=gray] at (0.0, 0) {\texttt{D}};
|
||||
\node at (0.5, 0) {\texttt{A}};
|
||||
\node at (1.0, 0) {\texttt{B}};
|
||||
\node at (1.5, 0) {\texttt{C}};
|
||||
\node at (2.0, 0) {\texttt{D}};
|
||||
\node at (2.5, 0) {\texttt{A}};
|
||||
\node at (3.0, 0) {\texttt{B}};
|
||||
\node at (3.5, 0) {\texttt{C}};
|
||||
\node at (4.0, 0) {\texttt{D}};
|
||||
\node[color=gray] at (4.5, 0) {\texttt{B}};
|
||||
\node[color=gray] at (5.0, 0) {\texttt{D}};
|
||||
\node[color=gray] at (5.5, 0) {\texttt{A}};
|
||||
\node[color=gray] at (6.0, 0) {\texttt{C}};
|
||||
\node[color=gray] at (6.75, 0) {\texttt{...}};
|
||||
|
||||
\draw (-1.75, 0.25) -- (7.25, 0.25);
|
||||
\draw (-1.75, -0.25) -- (7.25, -0.25);
|
||||
|
||||
|
||||
\draw[line width = 0.7mm, color=oblue, dotted] (2.25, 0.5) -- (2.25, -0.5);
|
||||
\draw[line width = 0.7mm, color=oblue]
|
||||
(-1.25, 0.5)
|
||||
-- (4.25, 0.5)
|
||||
-- (4.25, -0.5)
|
||||
-- (-1.25, -0.5)
|
||||
-- cycle
|
||||
;
|
||||
|
||||
\draw
|
||||
(4.2, -0.625)
|
||||
-- (4.2, -0.75)
|
||||
to node[anchor=north, midway] {lookahead} (2.3, -0.75)
|
||||
-- (2.3, -0.625)
|
||||
;
|
||||
|
||||
\draw
|
||||
(2.2, -0.625)
|
||||
-- (2.2, -0.75)
|
||||
to node[anchor=north, midway] {search buffer} (-1.1, -0.75)
|
||||
-- (-1.1, -0.625)
|
||||
;
|
||||
|
||||
\draw[color=gray]
|
||||
(2.2, 0.625)
|
||||
-- (2.2, 0.75)
|
||||
to node[anchor=south, midway] {match!} (0.3, 0.75)
|
||||
-- (0.3, 0.625)
|
||||
;
|
||||
|
||||
%\draw[->, color=gray] (2.5, 0.3) -- (2.5, 0.8) to[out=90,in=90] (0.5, 0.8);
|
||||
\node at (7.0, -0.75) {Result: \texttt{[$\cdot\cdot\cdot$DABCD<4,4>$\cdot\cdot\cdot$]}};
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
%\begin{instructornote}
|
||||
% A simple LZ-scheme can work as follows. We encode our string into a sequence of
|
||||
% nine-bit blocks, drawn below. The first bit of each block tells us whether or not
|
||||
% this block is a pointer, and the next eight bits contain either a \texttt{pos, len} pair
|
||||
% (using, say, for bits for each number) or a plain eight-bit symbol code.
|
||||
% \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{0}};
|
||||
% \node at (4, 0) {\texttt{1}};
|
||||
% \node at (5, 0) {\texttt{1}};
|
||||
% \node at (6, 0) {\texttt{0}};
|
||||
% \node at (7, 0) {\texttt{0}};
|
||||
% \node at (8, 0) {\texttt{1}};
|
||||
%
|
||||
% \draw (-0.5, 0.25) -- (8.5, 0.25);
|
||||
% \draw (-0.5, -0.25) -- (8.5, -0.25);
|
||||
% \draw (-0.5, -0.75) -- (8.5, -0.75);
|
||||
%
|
||||
% \draw (-0.5, 0.25) -- (-0.5, -0.75);
|
||||
% \draw (0.5, 0.25) -- (0.5, -0.75);
|
||||
% \draw (8.5, 0.25) -- (8.5, -0.75);
|
||||
%
|
||||
% \node at (0, -0.5) {flag};
|
||||
% \node at (4.5, -0.5) {if flag \texttt{<pos, len>}, else eight-bit symbol};
|
||||
% \end{tikzpicture}
|
||||
% \end{center}
|
||||
%
|
||||
% To encode a string, we read it using a \say{window}, shown below. This window consists of
|
||||
% a search buffer and a lookahead buffer, both of which have a fixed (but configurable) size.
|
||||
% This window passes over the string one character at a time, inserting a pointer if it finds
|
||||
% the lookahead buffer inside its search buffer, and a plain character otherwise.
|
||||
%
|
||||
%
|
||||
% \begin{center}
|
||||
% \begin{tikzpicture}
|
||||
% % Text tape
|
||||
% \node[color=gray] at (-0.75, 0) {\texttt{...}};
|
||||
% \node[color=gray] at (0.0, 0) {\texttt{D}};
|
||||
% \node at (0.5, 0) {\texttt{A}};
|
||||
% \node at (1.0, 0) {\texttt{B}};
|
||||
% \node at (1.5, 0) {\texttt{C}};
|
||||
% \node at (2.0, 0) {\texttt{D}};
|
||||
% \node at (2.5, 0) {\texttt{A}};
|
||||
% \node at (3.0, 0) {\texttt{B}};
|
||||
% \node at (3.5, 0) {\texttt{C}};
|
||||
% \node at (4.0, 0) {\texttt{D}};
|
||||
% \node[color=gray] at (4.5, 0) {\texttt{B}};
|
||||
% \node[color=gray] at (5.0, 0) {\texttt{D}};
|
||||
% \node[color=gray] at (5.5, 0) {\texttt{A}};
|
||||
% \node[color=gray] at (6.0, 0) {\texttt{C}};
|
||||
% \node[color=gray] at (6.75, 0) {\texttt{...}};
|
||||
%
|
||||
% \draw (-1.75, 0.25) -- (7.25, 0.25);
|
||||
% \draw (-1.75, -0.25) -- (7.25, -0.25);
|
||||
%
|
||||
%
|
||||
% \draw[line width = 0.7mm, color=oblue, dotted] (2.25, 0.5) -- (2.25, -0.5);
|
||||
% \draw[line width = 0.7mm, color=oblue]
|
||||
% (-1.25, 0.5)
|
||||
% -- (4.25, 0.5)
|
||||
% -- (4.25, -0.5)
|
||||
% -- (-1.25, -0.5)
|
||||
% -- cycle
|
||||
% ;
|
||||
%
|
||||
% \draw
|
||||
% (4.2, -0.625)
|
||||
% -- (4.2, -0.75)
|
||||
% to node[anchor=north, midway] {lookahead} (2.3, -0.75)
|
||||
% -- (2.3, -0.625)
|
||||
% ;
|
||||
%
|
||||
% \draw
|
||||
% (2.2, -0.625)
|
||||
% -- (2.2, -0.75)
|
||||
% to node[anchor=north, midway] {search buffer} (-1.1, -0.75)
|
||||
% -- (-1.1, -0.625)
|
||||
% ;
|
||||
%
|
||||
% \draw[color=gray]
|
||||
% (2.2, 0.625)
|
||||
% -- (2.2, 0.75)
|
||||
% to node[anchor=south, midway] {match!} (0.3, 0.75)
|
||||
% -- (0.3, 0.625)
|
||||
% ;
|
||||
%
|
||||
% %\draw[->, color=gray] (2.5, 0.3) -- (2.5, 0.8) to[out=90,in=90] (0.5, 0.8);
|
||||
% \node at (7.0, -0.75) {Result: \texttt{[$\cdot\cdot\cdot$DABCD<4,4>$\cdot\cdot\cdot$]}};
|
||||
% \end{tikzpicture}
|
||||
% \end{center}
|
||||
%
|
||||
% This is not the exact process used in practice---but it's close enough. \par
|
||||
% This process may be tweaked in any number of ways.
|
||||
%\end{instructornote}
|
||||
%
|
||||
%\makeatletter\if@solutions
|
||||
% \vfill
|
||||
% \pagebreak
|
||||
%\fi\makeatother
|
||||
Reference in New Issue
Block a user