Minor edits

Advanced/Compression/parts/1 runlength.tex

@@ -88,6 +88,15 @@ We'll encode our string into a sequence of 6-bit blocks, interpreted as follows:
 So, the sequence \texttt{BBB} will be encoded as \texttt{[0011-01]}. \par
 \note[Notation]{Just like spaces, dashes in a binary blob are added for readability.}
 
+
+\remark{Notation}
+In this handout, encoded binary blobs will always be written in square brackets. \par
+Ignore spaces and dashes, they are provided for convenience. \par
+For example, the binary sequences \texttt{[000 011 100 001 010 100]} and \texttt{[000011100001010100]} \par
+are identical. The first, however, is easier to read.
+
+\pagebreak
+
 \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}?
@@ -98,7 +107,7 @@ Is this more or less efficient than \ref{runlenone}?
 \end{solution}
 
 \vfill
-\pagebreak
+
 
 \problem{}
 Is run-length coding always efficient? When does it work well, and when does it fail?

Advanced/Compression/parts/3 huffman.tex

@@ -1,8 +1,8 @@
 \section{Huffman Codes}
 
 
-\remark{}
-As a first example, consider the alphabet $\{\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}, \texttt{E}\}$. \par
+\example{}
+Now consider the alphabet $\{\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}, \texttt{E}\}$. \par
 With a na\"ive coding scheme, we can encode a length-$n$ string with $3n$ bits, by mapping...
 \begin{itemize}
 	\item $\texttt{A}$ to $\texttt{000}$
@@ -11,17 +11,119 @@ With a na\"ive coding scheme, we can encode a length-$n$ string with $3n$ bits,
 	\item $\texttt{D}$ to $\texttt{011}$
 	\item $\texttt{E}$ to $\texttt{100}$
 \end{itemize}
-With this scheme, the string \texttt{ADEBCE} becomes \texttt{[000 011 100 001 010 100]}. \par
-This matches what we computed in \ref{naivelen}: ~ $6 \times \lceil \log_2(5) \rceil = 6 \times 3 = 18$. \par
-\note[Notation]{
-	The spaces in \texttt{[000 011 100 001 010 100]} are provided for convenience. \par
-	This is equivalent to \texttt{[000011100001010100]}, but is easier to read. \par
-	In this handout, encoded binary blobs will always be written in square brackets.
-}
+For example, this encodes \texttt{ADEBCE} as \texttt{[000 011 100 001 010 100]}. \par
+To encoding strings over $\{\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}, \texttt{E}\}$ with this scheme, we
+need an average of three bits per symbol.
 
 \vspace{2mm}
 
-You could argue that this coding scheme is wasteful: we're not using three of the eight possible three-bit sequences!
+One could argue that this coding scheme is wasteful: \par
+we're not using three of the eight possible three-bit sequences!
+
+\example{}
+There is, of course, a better way. \par
+Consider the following mapping:
+
+\begin{itemize}
+	\item $\texttt{A}$ to $\texttt{00}$
+	\item $\texttt{B}$ to $\texttt{01}$
+	\item $\texttt{C}$ to $\texttt{10}$
+	\item $\texttt{D}$ to $\texttt{110}$
+	\item $\texttt{E}$ to $\texttt{111}$
+\end{itemize}
+
+\problem{}
+\begin{itemize}
+	\item Using the above code, encode \texttt{ADEBCE}.
+	\item Then, decode \texttt{[110011001111]}.
+\end{itemize}
+
+\begin{solution}
+	\texttt{ADEBCE} becomes \texttt{[00 110 111 01 10 111]}, \par
+	and \texttt{[110 01 10 01 111]} is \texttt{DBCBE}.
+\end{solution}
+
+\vfill
+
+\problem{}
+How many bits does this code need per symbol, on average?
+
+\begin{solution}
+	\begin{equation*}
+		\frac{2 + 2 + 2 + 3 + 3}{5} = \frac{12}{5} = 2.4
+	\end{equation*}
+\end{solution}
+
+\vfill
+
+\problem{}
+Consider the code below. How is it different from the one above? \par
+Is this a good way to encode five-letter strings?
+\begin{itemize}
+	\item $\texttt{A}$ to $\texttt{00}$
+	\item $\texttt{B}$ to $\texttt{01}$
+	\item $\texttt{C}$ to $\texttt{10}$
+	\item $\texttt{D}$ to $\texttt{110}$
+	\item $\texttt{E}$ to $\texttt{11}$
+\end{itemize}
+
+\begin{solution}
+	No. The code for \texttt{E} occurs inside the code for \texttt{D},
+	and we thus can't decode sequences uniquely. For example, we could
+	decode the fragment \texttt{[11001$\cdot\cdot\cdot$]} as \texttt{EA}
+	or as \texttt{DB}.
+\end{solution}
+
+\vfill
+\pagebreak
+
+
+\remark{}
+Huffman codes can be visualized as a tree which we traverse while decoding our sequence. \par
+We start at the topmost node, taking the left edge if we see a \texttt{0} and the right edge if we see a \texttt{1}. \par
+As an example, consider the code for $\{\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}, \texttt{E}\}$
+on the previous page:
+
+
+\begin{itemize}
+	\item $\texttt{A}$ encodes as $\texttt{00}$
+	\item $\texttt{B}$ encodes as $\texttt{01}$
+	\item $\texttt{C}$ encodes as $\texttt{10}$
+	\item $\texttt{D}$ encodes as $\texttt{110}$
+	\item $\texttt{E}$ encodes as $\texttt{111}$
+\end{itemize}
+
+Drawing this scheme as a tree, we get the following:
+
+
+\begin{center}
+\begin{tikzpicture}[scale=1.0]
+	\begin{scope}[layer = nodes]
+		\node[int] (x) at (0, 0) {};
+		\node[int] (0) at (-0.75, -1) {};
+		\node[int] (1) at (0.75, -1) {};
+		\node[end] (00) at (-1.25, -2) {\texttt{A}};
+		\node[end] (01) at (-0.25, -2) {\texttt{B}};
+		\node[end] (10) at (0.25, -2) {\texttt{C}};
+		\node[int] (11) at (1.25, -2) {};
+		\node[end] (110) at (0.75, -3) {\texttt{D}};
+		\node[end] (111) at (1.75, -3) {\texttt{E}};
+	\end{scope}
+
+	\draw[-]
+		(x) to node[midway, fill=white, text=gray] {\texttt{0}} (0)
+		(x) to node[midway, fill=white, text=gray] {\texttt{1}} (1)
+		(0) to node[midway, fill=white, text=gray] {\texttt{0}} (00)
+		(0) to node[midway, fill=white, text=gray] {\texttt{1}} (01)
+		(1) to node[midway, fill=white, text=gray] {\texttt{0}} (10)
+		(1) to node[midway, fill=white, text=gray] {\texttt{1}} (11)
+		(11) to node[midway, fill=white, text=gray] {\texttt{0}} (110)
+		(11) to node[midway, fill=white, text=gray] {\texttt{1}} (111)
+	;
+\end{tikzpicture}
+\end{center}
+
+
 
 \vfill
 \pagebreak

Advanced/Compression/tikzset.tex

@@ -30,11 +30,9 @@
 	},
 	%
 	% Nodes
-	main/.style = {
-		draw,
-		circle,
-		fill = white,
-		line width = 0.35mm
+	int/.style = {},
+	end/.style = {
+		anchor=north
 	},
 	%
 	% Loop tweaks