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Archivers.html
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<h1 class="title">Information theory: HW #3 solution</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgheadline1">1. Task objectives, format, prerequisites</a></li>
<li><a href="#orgheadline2">2. Two-pass huffman coding</a></li>
<li><a href="#orgheadline3">3. Adaptive arithmetic coding</a></li>
<li><a href="#orgheadline4">4. Enumerative</a></li>
<li><a href="#orgheadline5">5. LZ77</a></li>
<li><a href="#orgheadline6">6. LZW</a></li>
<li><a href="#orgheadline7">7. PPM</a></li>
<li><a href="#orgheadline8">8. Burrows-Wheeler + book stack</a></li>
<li><a href="#orgheadline9">9. Standart archiving</a></li>
<li><a href="#orgheadline10">10. Comparison and summary</a></li>
</ul>
</div>
</div>
<div id="outline-container-orgheadline1" class="outline-2">
<h2 id="orgheadline1"><span class="section-number-2">1</span> Task objectives, format, prerequisites</h2>
<div class="outline-text-2" id="text-1">
<p>
So the task is to implement several algorithms of data compression
and provide reports on them. Reports will contain logs from
algorithms that will show how it operates this data. Tables with
explanation will be attached. I won't attach haskell snippets this
time as they became really big and complex. Instead, i'll put it
into another pdf and send it with this one.
</p>
<p>
Proverb correspondent to my task is the following:
</p>
<blockquote>
<p>
<i>Love the heart that hurts you, but never hurt the heart that loves you.</i>
</p>
</blockquote>
<p>
It's in english that's why I'll stick to utf-8/ascii encoding as
byte representation. It contains \(71\) letters, so it takes \(568\)
bytes if used without any coding.
</p>
<p>
Remark: sorry, but my rendering format doesn't support proper
underscore marks (_) – they turn into underlines and i see no way
to escape them. "␣" can be used instead, but it's too ugly. I'll use
it when ambiguous, and just leave empty place in other places. Yes,
and sorry for the tables. They're ugly both when they're are and are
not full-width. Rule of thumb: use the same table formatting as in
the textbook. So I choose to use full width.
</p>
</div>
</div>
<div id="outline-container-orgheadline2" class="outline-2">
<h2 id="orgheadline2"><span class="section-number-2">2</span> Two-pass huffman coding</h2>
<div class="outline-text-2" id="text-2">
<p>
I've implemented regural huffman coding, as it can be seen from the
first table: code words of the same length are sorted
lexicographically:
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="all" frame="border" style="line-height: 7px; width: 100%;">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">\(Char\)</th>
<th scope="col" class="org-right">\(Codeword\)</th>
<th scope="col" class="org-right">\(Frequency\)</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left"> </td>
<td class="org-right">00</td>
<td class="org-right">13</td>
</tr>
<tr>
<td class="org-left">e</td>
<td class="org-right">010</td>
<td class="org-right">8</td>
</tr>
<tr>
<td class="org-left">h</td>
<td class="org-right">011</td>
<td class="org-right">8</td>
</tr>
<tr>
<td class="org-left">t</td>
<td class="org-right">100</td>
<td class="org-right">11</td>
</tr>
<tr>
<td class="org-left">r</td>
<td class="org-right">1010</td>
<td class="org-right">5</td>
</tr>
<tr>
<td class="org-left">u</td>
<td class="org-right">1011</td>
<td class="org-right">5</td>
</tr>
<tr>
<td class="org-left">s</td>
<td class="org-right">11000</td>
<td class="org-right">2</td>
</tr>
<tr>
<td class="org-left">y</td>
<td class="org-right">11001</td>
<td class="org-right">2</td>
</tr>
<tr>
<td class="org-left">v</td>
<td class="org-right">11010</td>
<td class="org-right">3</td>
</tr>
<tr>
<td class="org-left">a</td>
<td class="org-right">11011</td>
<td class="org-right">4</td>
</tr>
<tr>
<td class="org-left">o</td>
<td class="org-right">11100</td>
<td class="org-right">4</td>
</tr>
<tr>
<td class="org-left">,</td>
<td class="org-right">111010</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-left">.</td>
<td class="org-right">111011</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-left">l</td>
<td class="org-right">111100</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-left">n</td>
<td class="org-right">111101</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-left">L</td>
<td class="org-right">111110</td>
<td class="org-right">1</td>
</tr>
<tr>
<td class="org-left">b</td>
<td class="org-right">111111</td>
<td class="org-right">1</td>
</tr>
</tbody>
</table>
<p>
And the second pass completes encoding:
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="all" frame="border" style="line-height: 7px; width: 100%;">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">\(C\)</th>
<th scope="col" class="org-left">\(P(C)\)</th>
<th scope="col" class="org-right">\(Codeword\)</th>
<th scope="col" class="org-right">\(L(S)\)</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">L</td>
<td class="org-left">1 / 71</td>
<td class="org-right">111110</td>
<td class="org-right">6</td>
</tr>
<tr>
<td class="org-left">o</td>
<td class="org-left">4 / 71</td>
<td class="org-right">11100</td>
<td class="org-right">11</td>
</tr>
<tr>
<td class="org-left">v</td>
<td class="org-left">3 / 71</td>
<td class="org-right">11010</td>
<td class="org-right">16</td>
</tr>
<tr>
<td class="org-left">e</td>
<td class="org-left">8 / 71</td>
<td class="org-right">010</td>
<td class="org-right">19</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">21</td>
</tr>
<tr>
<td class="org-left">t</td>
<td class="org-left">11 / 71</td>
<td class="org-right">100</td>
<td class="org-right">24</td>
</tr>
<tr>
<td class="org-left">h</td>
<td class="org-left">8 / 71</td>
<td class="org-right">011</td>
<td class="org-right">27</td>
</tr>
<tr>
<td class="org-left">e</td>
<td class="org-left">8 / 71</td>
<td class="org-right">010</td>
<td class="org-right">30</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">32</td>
</tr>
<tr>
<td class="org-left">h</td>
<td class="org-left">8 / 71</td>
<td class="org-right">011</td>
<td class="org-right">35</td>
</tr>
<tr>
<td class="org-left">e</td>
<td class="org-left">8 / 71</td>
<td class="org-right">010</td>
<td class="org-right">38</td>
</tr>
<tr>
<td class="org-left">a</td>
<td class="org-left">4 / 71</td>
<td class="org-right">11011</td>
<td class="org-right">43</td>
</tr>
<tr>
<td class="org-left">r</td>
<td class="org-left">5 / 71</td>
<td class="org-right">1010</td>
<td class="org-right">47</td>
</tr>
<tr>
<td class="org-left">t</td>
<td class="org-left">11 / 71</td>
<td class="org-right">100</td>
<td class="org-right">50</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">52</td>
</tr>
<tr>
<td class="org-left">t</td>
<td class="org-left">11 / 71</td>
<td class="org-right">100</td>
<td class="org-right">55</td>
</tr>
<tr>
<td class="org-left">h</td>
<td class="org-left">8 / 71</td>
<td class="org-right">011</td>
<td class="org-right">58</td>
</tr>
<tr>
<td class="org-left">a</td>
<td class="org-left">4 / 71</td>
<td class="org-right">11011</td>
<td class="org-right">63</td>
</tr>
<tr>
<td class="org-left">t</td>
<td class="org-left">11 / 71</td>
<td class="org-right">100</td>
<td class="org-right">66</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">68</td>
</tr>
<tr>
<td class="org-left">h</td>
<td class="org-left">8 / 71</td>
<td class="org-right">011</td>
<td class="org-right">71</td>
</tr>
<tr>
<td class="org-left">u</td>
<td class="org-left">5 / 71</td>
<td class="org-right">1011</td>
<td class="org-right">75</td>
</tr>
<tr>
<td class="org-left">r</td>
<td class="org-left">5 / 71</td>
<td class="org-right">1010</td>
<td class="org-right">79</td>
</tr>
<tr>
<td class="org-left">t</td>
<td class="org-left">11 / 71</td>
<td class="org-right">100</td>
<td class="org-right">82</td>
</tr>
<tr>
<td class="org-left">s</td>
<td class="org-left">2 / 71</td>
<td class="org-right">11000</td>
<td class="org-right">87</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">89</td>
</tr>
<tr>
<td class="org-left">y</td>
<td class="org-left">2 / 71</td>
<td class="org-right">11001</td>
<td class="org-right">94</td>
</tr>
<tr>
<td class="org-left">o</td>
<td class="org-left">4 / 71</td>
<td class="org-right">11100</td>
<td class="org-right">99</td>
</tr>
<tr>
<td class="org-left">u</td>
<td class="org-left">5 / 71</td>
<td class="org-right">1011</td>
<td class="org-right">103</td>
</tr>
<tr>
<td class="org-left">,</td>
<td class="org-left">1 / 71</td>
<td class="org-right">111010</td>
<td class="org-right">109</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">111</td>
</tr>
<tr>
<td class="org-left">b</td>
<td class="org-left">1 / 71</td>
<td class="org-right">111111</td>
<td class="org-right">117</td>
</tr>
<tr>
<td class="org-left">u</td>
<td class="org-left">5 / 71</td>
<td class="org-right">1011</td>
<td class="org-right">121</td>
</tr>
<tr>
<td class="org-left">t</td>
<td class="org-left">11 / 71</td>
<td class="org-right">100</td>
<td class="org-right">124</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">126</td>
</tr>
<tr>
<td class="org-left">n</td>
<td class="org-left">1 / 71</td>
<td class="org-right">111101</td>
<td class="org-right">132</td>
</tr>
<tr>
<td class="org-left">e</td>
<td class="org-left">8 / 71</td>
<td class="org-right">010</td>
<td class="org-right">135</td>
</tr>
<tr>
<td class="org-left">v</td>
<td class="org-left">3 / 71</td>
<td class="org-right">11010</td>
<td class="org-right">140</td>
</tr>
<tr>
<td class="org-left">e</td>
<td class="org-left">8 / 71</td>
<td class="org-right">010</td>
<td class="org-right">143</td>
</tr>
<tr>
<td class="org-left">r</td>
<td class="org-left">5 / 71</td>
<td class="org-right">1010</td>
<td class="org-right">147</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">149</td>
</tr>
<tr>
<td class="org-left">h</td>
<td class="org-left">8 / 71</td>
<td class="org-right">011</td>
<td class="org-right">152</td>
</tr>
<tr>
<td class="org-left">u</td>
<td class="org-left">5 / 71</td>
<td class="org-right">1011</td>
<td class="org-right">156</td>
</tr>
<tr>
<td class="org-left">r</td>
<td class="org-left">5 / 71</td>
<td class="org-right">1010</td>
<td class="org-right">160</td>
</tr>
<tr>
<td class="org-left">t</td>
<td class="org-left">11 / 71</td>
<td class="org-right">100</td>
<td class="org-right">163</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">165</td>
</tr>
<tr>
<td class="org-left">t</td>
<td class="org-left">11 / 71</td>
<td class="org-right">100</td>
<td class="org-right">168</td>
</tr>
<tr>
<td class="org-left">h</td>
<td class="org-left">8 / 71</td>
<td class="org-right">011</td>
<td class="org-right">171</td>
</tr>
<tr>
<td class="org-left">e</td>
<td class="org-left">8 / 71</td>
<td class="org-right">010</td>
<td class="org-right">174</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">176</td>
</tr>
<tr>
<td class="org-left">h</td>
<td class="org-left">8 / 71</td>
<td class="org-right">011</td>
<td class="org-right">179</td>
</tr>
<tr>
<td class="org-left">e</td>
<td class="org-left">8 / 71</td>
<td class="org-right">010</td>
<td class="org-right">182</td>
</tr>
<tr>
<td class="org-left">a</td>
<td class="org-left">4 / 71</td>
<td class="org-right">11011</td>
<td class="org-right">187</td>
</tr>
<tr>
<td class="org-left">r</td>
<td class="org-left">5 / 71</td>
<td class="org-right">1010</td>
<td class="org-right">191</td>
</tr>
<tr>
<td class="org-left">t</td>
<td class="org-left">11 / 71</td>
<td class="org-right">100</td>
<td class="org-right">194</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">196</td>
</tr>
<tr>
<td class="org-left">t</td>
<td class="org-left">11 / 71</td>
<td class="org-right">100</td>
<td class="org-right">199</td>
</tr>
<tr>
<td class="org-left">h</td>
<td class="org-left">8 / 71</td>
<td class="org-right">011</td>
<td class="org-right">202</td>
</tr>
<tr>
<td class="org-left">a</td>
<td class="org-left">4 / 71</td>
<td class="org-right">11011</td>
<td class="org-right">207</td>
</tr>
<tr>
<td class="org-left">t</td>
<td class="org-left">11 / 71</td>
<td class="org-right">100</td>
<td class="org-right">210</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">212</td>
</tr>
<tr>
<td class="org-left">l</td>
<td class="org-left">1 / 71</td>
<td class="org-right">111100</td>
<td class="org-right">218</td>
</tr>
<tr>
<td class="org-left">o</td>
<td class="org-left">4 / 71</td>
<td class="org-right">11100</td>
<td class="org-right">223</td>
</tr>
<tr>
<td class="org-left">v</td>
<td class="org-left">3 / 71</td>
<td class="org-right">11010</td>
<td class="org-right">228</td>
</tr>
<tr>
<td class="org-left">e</td>
<td class="org-left">8 / 71</td>
<td class="org-right">010</td>
<td class="org-right">231</td>
</tr>
<tr>
<td class="org-left">s</td>
<td class="org-left">2 / 71</td>
<td class="org-right">11000</td>
<td class="org-right">236</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">13 / 71</td>
<td class="org-right">00</td>
<td class="org-right">238</td>
</tr>
<tr>
<td class="org-left">y</td>
<td class="org-left">2 / 71</td>
<td class="org-right">11001</td>
<td class="org-right">243</td>
</tr>
<tr>
<td class="org-left">o</td>
<td class="org-left">4 / 71</td>
<td class="org-right">11100</td>
<td class="org-right">248</td>
</tr>
<tr>
<td class="org-left">u</td>
<td class="org-left">5 / 71</td>
<td class="org-right">1011</td>
<td class="org-right">252</td>
</tr>
<tr>
<td class="org-left">.</td>
<td class="org-left">1 / 71</td>
<td class="org-right">111011</td>
<td class="org-right">258</td>
</tr>
</tbody>
</table>
<p>
Encoded proverb encoded is \(l_2 = 258\) bits, while data amount needed to
transfer huffman tree should be calculated manually. First, to
transfer the tree itself, it's sufficient to pass only \(19\) bits
(calculated manually using table 3.3 from the study book).
</p>
\begin{align*}
\left\lceil\log{\dbinom{256}{1}}\right\rceil +
\left\lceil\log{\dbinom{255}{3}}\right\rceil +
\left\lceil\log{\dbinom{254}{2}}\right\rceil +
\left\lceil\log{\dbinom{253}{5}}\right\rceil +
\left\lceil\log{\dbinom{252}{6}}\right\rceil = 117
\end{align*}
<p>
So totally \(l_1 = 19 + 117 = 136\), and total length is \(l = l_1+l_2 =
394\) bits. Much better than raw \(568\) bits.
</p>
</div>
</div>
<div id="outline-container-orgheadline3" class="outline-2">
<h2 id="orgheadline3"><span class="section-number-2">3</span> Adaptive arithmetic coding</h2>
<div class="outline-text-2" id="text-3">
<p>
Implementation uses renormalization together with fixed-point
precision arithmetics (16 bits words). Algorithm "A" from textbook
is used (has \(n+1\) in denominator).
</p>
<p>
Here is the table algorithm outputs (backslash stands for escape
symbol):
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="all" frame="border" style="line-height: 7px; width: 100%;">
<colgroup>
<col class="org-left" />
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">\(C\)</th>
<th scope="col" class="org-left">\(P(C)\)</th>
<th scope="col" class="org-right">\(Codeword\)</th>
<th scope="col" class="org-right">\(L(S)\)</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">esc</td>
<td class="org-left">1 / 1</td>
<td class="org-right"> </td>
<td class="org-right">0</td>
</tr>
<tr>
<td class="org-left">L</td>
<td class="org-left">1 / 256</td>
<td class="org-right">0100110</td>
<td class="org-right">7</td>
</tr>
<tr>
<td class="org-left">esc</td>
<td class="org-left">1 / 2</td>
<td class="org-right"> </td>
<td class="org-right">7</td>
</tr>
<tr>
<td class="org-left">o</td>
<td class="org-left">1 / 255</td>
<td class="org-right">00110111</td>
<td class="org-right">15</td>
</tr>
<tr>
<td class="org-left">esc</td>
<td class="org-left">1 / 3</td>
<td class="org-right"> </td>
<td class="org-right">15</td>
</tr>
<tr>
<td class="org-left">v</td>
<td class="org-left">1 / 254</td>
<td class="org-right">0001010</td>
<td class="org-right">22</td>
</tr>
<tr>
<td class="org-left">esc</td>
<td class="org-left">1 / 4</td>
<td class="org-right">00010</td>
<td class="org-right">27</td>
</tr>
<tr>
<td class="org-left">e</td>
<td class="org-left">1 / 253</td>
<td class="org-right">0100101</td>
<td class="org-right">34</td>
</tr>
<tr>
<td class="org-left">esc</td>
<td class="org-left">1 / 5</td>
<td class="org-right">010</td>
<td class="org-right">37</td>
</tr>
<tr>
<td class="org-left"> </td>
<td class="org-left">1 / 252</td>