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[Algorithms II] Week 5-2 Data Compression
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<time datetime="2016-01-04T00:00:00+01:00"><i class="fa fa-calendar"></i> Mon, 04 Jan 2016</time>
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<span class="label label-default">Series</span>
Part 10 of «Algorithms Princeton MOOC II»
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-introduction-to-data-compression" title="1. Introduction to Data Compression">1. Introduction to Data Compression</a><ul><li><a class="toc-href" href="#tools" title="Tools">Tools</a></li></ul></li><li><a class="toc-href" href="#2-run-length-coding_1" title="2. Run-Length Coding">2. Run-Length Coding</a></li><li><a class="toc-href" href="#3-huffman-compression" title="3. Huffman Compression">3. Huffman Compression</a><ul><li><a class="toc-href" href="#huffman-trie-node" title="Huffman Trie Node">Huffman Trie Node</a></li><li><a class="toc-href" href="#implementation" title="implementation">implementation</a></li><li><a class="toc-href" href="#huffman-algorithm" title="Huffman algorithm">Huffman algorithm</a></li></ul></li><li><a class="toc-href" href="#4-lzw-compression_1" title="4. LZW Compression">4. LZW Compression</a><ul><li><a class="toc-href" href="#lzw-compression" title="LZW compression">LZW compression</a></li><li><a class="toc-href" href="#lzw-expansion" title="LZW expansion">LZW expansion</a></li></ul></li><li><a class="toc-href" href="#summary_1" title="Summary">Summary</a></li></ul></div>
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</div>
<h1 id="1-introduction-to-data-compression">1. Introduction to Data Compression</h1>
<p>pb: reduce the size of a file, to save space/time for storing/transmitting. <br/>
applications: generic file compression(gzip), multimedia (mp3), communication(skype). </p>
<p>From binary data <code>B</code>, ⇒ generate a compressed representation <code>C(B)</code>. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_2/pasted_image.png"/><br/>
<strong>lossless compression</strong>: get exactly <code>B</code> from <code>C(B)</code><br/>
<strong>compression ratio</strong>: <code>|C(B)|/|B|</code>(||means number of <em>bits</em>) </p>
<h3 id="tools">Tools</h3>
<p>extension to stdio libraries: read/write <em>bits</em> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span> <span class="k">class</span> <span class="n">BinaryStdIn</span><span class="err">{</span> </span>
<span class="code-line"> <span class="nb">boolean</span> <span class="n">readBoolean</span><span class="p">();</span> <span class="o">//</span> <span class="k">read</span> <span class="mi">1</span> <span class="nb">bit</span> <span class="k">and</span> <span class="k">return</span> <span class="k">as</span> <span class="n">a</span> <span class="nb">boolean</span> </span>
<span class="code-line"> <span class="nb">char</span> <span class="n">readChar</span><span class="p">();</span> <span class="o">//</span> <span class="k">read</span> <span class="mi">8</span> <span class="n">bits</span> <span class="k">and</span> <span class="k">return</span> <span class="n">a</span> <span class="nb">char</span> </span>
<span class="code-line"> <span class="nb">char</span> <span class="n">readChar</span><span class="p">(</span><span class="nb">int</span> <span class="n">r</span><span class="p">);</span> <span class="o">//</span> <span class="k">read</span> <span class="n">r</span> <span class="p">(</span><span class="o"><=</span><span class="mi">8</span><span class="p">)</span> <span class="n">bits</span> <span class="k">and</span> <span class="k">return</span> <span class="n">a</span> <span class="nb">char</span> </span>
<span class="code-line"> <span class="o">//</span> <span class="k">similar</span> <span class="k">method</span> <span class="k">for</span> <span class="nb">int</span><span class="o">/</span><span class="n">long</span><span class="o">/</span><span class="n">double</span> </span>
<span class="code-line"> <span class="nb">boolean</span> <span class="n">isEmpty</span><span class="p">();</span> <span class="o">//</span> <span class="k">is</span> <span class="n">bitstream</span> <span class="n">empty</span><span class="o">?</span> </span>
<span class="code-line"> <span class="n">void</span> <span class="k">close</span><span class="p">();</span> </span>
<span class="code-line"><span class="err">}</span></span>
<span class="code-line"></span>
<span class="code-line"></span>
<span class="code-line"><span class="k">public</span> <span class="k">class</span> <span class="n">BinaryStdOut</span><span class="err">{</span> </span>
<span class="code-line"> <span class="n">void</span> <span class="k">write</span><span class="p">(</span><span class="nb">boolean</span> <span class="n">b</span><span class="p">);</span> <span class="o">//</span> <span class="k">write</span> <span class="mi">1</span> <span class="nb">bit</span> </span>
<span class="code-line"> <span class="n">void</span> <span class="k">write</span><span class="p">(</span><span class="nb">char</span> <span class="k">c</span><span class="p">);</span> <span class="o">//</span> <span class="k">write</span> <span class="mi">8</span> <span class="n">bits</span> </span>
<span class="code-line"> <span class="n">void</span> <span class="k">write</span><span class="p">(</span><span class="nb">char</span> <span class="k">c</span><span class="p">,</span> <span class="nb">int</span> <span class="n">r</span><span class="p">);</span> <span class="o">//</span> <span class="k">write</span> <span class="n">r</span> <span class="p">(</span><span class="n">least</span><span class="o">-</span><span class="n">significant</span><span class="p">)</span> <span class="n">bits</span> <span class="k">of</span> <span class="k">c</span> </span>
<span class="code-line"> <span class="o">//</span> <span class="k">similar</span> <span class="k">method</span> <span class="k">for</span> <span class="nb">int</span><span class="o">/</span><span class="n">long</span><span class="o">/</span><span class="n">double</span> </span>
<span class="code-line"> <span class="n">void</span> <span class="k">close</span><span class="p">();</span> <span class="o">//</span> <span class="n">might</span> <span class="k">add</span> <span class="k">some</span> <span class="n">byte</span> <span class="n">alignment</span> <span class="n">bits</span> </span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>usage example: store a date 12/31/1999<br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_2/pasted_image001.png"/> </p>
<p>universal date compression?<br/>
<strong>prop</strong>. NO algorithm can compress every bitstring.<br/>
pf. by contradiction: repeatedly compress the bitstring ⇒ bit length goes to 0. </p>
<h1 id="2-run-length-coding_1">2. Run-Length Coding</h1>
<p>one simple type of redundancy in bitstream: long runs of repeated bits. <br/>
⇒ use <em>4-bit counts</em> to represent alternating 1s and 0s. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_2/pasted_image002.png"/> </p>
<ul>
<li>question1: how many bits to represent counts ? ⇒ pick 8 bits (just tradeoff) </li>
<li>question2: what if run length is bigger than max length(2^8)? ⇒ just add length=0 of the alternating bit... </li>
</ul>
<p>application: JPEG... </p>
<p>code: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span> <span class="k">class</span> <span class="n">RunLength</span><span class="err">{</span> </span>
<span class="code-line"> <span class="n">private</span> <span class="k">final</span> <span class="k">static</span> <span class="nb">int</span> <span class="n">R</span> <span class="o">=</span> <span class="mi">256</span><span class="p">;</span> <span class="o">//</span> <span class="k">max</span> <span class="n">run</span> <span class="k">length</span> </span>
<span class="code-line"> <span class="n">private</span> <span class="k">final</span> <span class="k">static</span> <span class="nb">int</span> <span class="n">lgR</span> <span class="o">=</span> <span class="mi">8</span><span class="p">;</span> </span>
<span class="code-line"> <span class="k">public</span> <span class="k">static</span> <span class="n">void</span> <span class="n">compress</span><span class="p">()</span><span class="err">{</span> </span>
<span class="code-line"> <span class="nb">boolean</span> <span class="nb">bit</span><span class="p">,</span> <span class="n">oldbit</span> <span class="o">=</span> <span class="k">false</span><span class="p">;</span> </span>
<span class="code-line"> <span class="nb">int</span> <span class="n">l</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span> </span>
<span class="code-line"> <span class="n">while</span><span class="p">(</span><span class="o">!</span><span class="n">BinaryStdIn</span><span class="p">.</span><span class="n">isEmpty</span><span class="p">())</span><span class="err">{</span> </span>
<span class="code-line"> <span class="nb">bit</span> <span class="o">=</span> <span class="n">BinaryStdIn</span><span class="p">.</span><span class="n">readBoolean</span><span class="p">();</span> </span>
<span class="code-line"> <span class="k">if</span><span class="p">(</span><span class="nb">bit</span><span class="o">!=</span><span class="k">old</span><span class="p">)</span><span class="err">{</span> </span>
<span class="code-line"> <span class="n">oldbit</span> <span class="o">=</span> <span class="nb">bit</span><span class="p">;</span> </span>
<span class="code-line"> <span class="n">BinaryStdOut</span><span class="p">.</span><span class="k">write</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">lgR</span><span class="p">);</span> </span>
<span class="code-line"> <span class="n">l</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span> </span>
<span class="code-line"> <span class="err">}</span> </span>
<span class="code-line"> <span class="k">else</span> <span class="k">if</span><span class="p">(</span><span class="n">l</span><span class="o">==</span><span class="n">R</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="err">{</span> </span>
<span class="code-line"> <span class="n">BinaryStdOut</span><span class="p">.</span><span class="k">write</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">lgR</span><span class="p">);</span> </span>
<span class="code-line"> <span class="n">BinaryStdOut</span><span class="p">.</span><span class="k">write</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">lgR</span><span class="p">);</span> </span>
<span class="code-line"> <span class="n">l</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span> </span>
<span class="code-line"> <span class="err">}</span> </span>
<span class="code-line"> <span class="n">l</span> <span class="o">++</span><span class="p">;</span> </span>
<span class="code-line"> <span class="err">}</span> </span>
<span class="code-line"> <span class="n">BinaryStdOut</span><span class="p">.</span><span class="k">write</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">lgR</span><span class="p">);</span> </span>
<span class="code-line"> <span class="n">BinaryStdOut</span><span class="p">.</span><span class="k">close</span><span class="p">();</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="err">}</span> </span>
<span class="code-line"> <span class="k">public</span> <span class="k">static</span> <span class="n">void</span> <span class="n">expand</span><span class="p">()</span><span class="err">{</span> </span>
<span class="code-line"> <span class="nb">boolean</span> <span class="nb">bit</span> <span class="o">=</span> <span class="k">false</span><span class="p">;</span> </span>
<span class="code-line"> <span class="n">while</span><span class="p">(</span><span class="o">!</span><span class="n">BinaryStdIn</span><span class="p">.</span><span class="n">isEmpty</span><span class="p">())</span><span class="err">{</span> </span>
<span class="code-line"> <span class="nb">int</span> <span class="n">l</span> <span class="o">=</span> <span class="n">BinaryStdIn</span><span class="p">.</span><span class="n">readInt</span><span class="p">(</span><span class="n">lgR</span><span class="p">);</span> </span>
<span class="code-line"> <span class="k">for</span><span class="p">(</span><span class="nb">int</span> <span class="n">i</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="n">i</span><span class="o"><</span><span class="n">l</span><span class="p">;</span><span class="n">i</span><span class="o">++</span><span class="p">)</span> </span>
<span class="code-line"> <span class="n">BinaryStdOut</span><span class="p">.</span><span class="k">write</span><span class="p">(</span><span class="nb">bit</span><span class="p">);</span> </span>
<span class="code-line"> <span class="nb">bit</span> <span class="o">!=</span> <span class="nb">bit</span><span class="p">;</span> </span>
<span class="code-line"> <span class="err">}</span> </span>
<span class="code-line"> <span class="n">BinaryStdOut</span><span class="p">.</span><span class="k">close</span><span class="p">();</span> </span>
<span class="code-line"> <span class="err">}</span> </span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h1 id="3-huffman-compression">3. Huffman Compression</h1>
<p>classique... </p>
<p>idea: <em>variable length code</em>. <br/>
ex. Morse code, more freq chars use less chars. <br/>
→ pb: ambiguity? one code is the prefix of another... <br/>
→ need to use <em>prefix-free code</em> </p>
<p>⇒ use a trie to represent the prefix-free code, in bitstream, use a binary trie: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_2/pasted_image003.png"/> </p>
<h3 id="huffman-trie-node">Huffman Trie Node</h3>
<p>define trie nodes having frequences: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public static class Node implements Comparable<Node>{ </span></span>
<span class="code-line"><span class="err"> private char ch; // only used for leaves, null for internal nodes </span></span>
<span class="code-line"><span class="err"> private int freq; </span></span>
<span class="code-line"><span class="err"> private final Node left, right; // left-->0, right-->1 </span></span>
<span class="code-line"><span class="err"> public Node(char ch, int freq, Node left, Node right){//...} </span></span>
<span class="code-line"><span class="err"> public boolean isLeaf(){ </span></span>
<span class="code-line"><span class="err"> return left==null && right==null;} </span></span>
<span class="code-line"><span class="err"> public int compareTo(Node that){ </span></span>
<span class="code-line"><span class="err"> return this.freq-that.freq;} </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h3 id="implementation">implementation</h3>
<p><strong>expansion</strong><br/>
<em>in the bitstream, we first put the (serialized) trie in the head.</em> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public void expand{ </span></span>
<span class="code-line"><span class="err"> Node root = getTrie(); // </span></span>
<span class="code-line"><span class="err"> int N = BinaryStdIn.readInt(); // number of chars in string </span></span>
<span class="code-line"><span class="err"> for(int i=0; i<N; i++){ </span></span>
<span class="code-line"><span class="err"> Node x = root; </span></span>
<span class="code-line"><span class="err"> while (!x.isLeaf()){ </span></span>
<span class="code-line"><span class="err"> if(BinaryStdIn.readBoolean()) x = x.right; </span></span>
<span class="code-line"><span class="err"> else x = x.left; </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> BinaryStdOut.write(x.ch); </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> BinaryStdOut.close(); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p><strong>output trie</strong><br/>
ie. to serialize a trie. <br/>
⇒ use <em>preorder traversal</em>: <br/>
0 for internal nodes, and 1 for leaf (followed by the corresponding char)<br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_2/pasted_image004.png"/><br/>
recursive method: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private static void writeTrie(Node x){ </span></span>
<span class="code-line"><span class="err"> if(x==null) return; </span></span>
<span class="code-line"><span class="err"> if(x.isLeaf()){ </span></span>
<span class="code-line"><span class="err"> BinaryStdOut.write(true); // leaf node </span></span>
<span class="code-line"><span class="err"> BinaryStdOut.write(x.ch); // followed by the char </span></span>
<span class="code-line"><span class="err"> } else { </span></span>
<span class="code-line"><span class="err"> BinaryStdOut.write(false); // internal node </span></span>
<span class="code-line"><span class="err"> writeTrie(x.left); </span></span>
<span class="code-line"><span class="err"> writeTrie(x.right); </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p><strong>read trie</strong><br/>
Reconstruct the trie from the serialized bitstring. <br/>
preorder(x) = x+preorder(x.left)+preorder(x.right) ⇒ recursive method... </p>
<p>这个递归函数还是蛮有代表性的, 值得看. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private static Node readTrie(){ </span></span>
<span class="code-line"><span class="err"> if(BinaryStdOut.readBoolean()) { </span></span>
<span class="code-line"><span class="err"> char ch = BinaryStdOut.readChar(); </span></span>
<span class="code-line"><span class="err"> return new Node(ch, 0, null, null); </span></span>
<span class="code-line"><span class="err"> } else { </span></span>
<span class="code-line"><span class="err"> Node left = readTrie(); </span></span>
<span class="code-line"><span class="err"> Node right = readTrie(); </span></span>
<span class="code-line"><span class="err"> return new Node('\0', 0, left, right); // subtle recursion </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h3 id="huffman-algorithm">Huffman algorithm</h3>
<p>pb: how to find the best prefixless code? </p>
<p>Shannon-Fano algo: </p>
<blockquote>
<ul>
<li>partition symbols S into 2 subsets: S1, S0, with roughly equal freq </li>
<li>code in S1 start with 1 and in S0 start with 0 </li>
<li>recur on S1 and S0 </li>
</ul>
</blockquote>
<p><strong>[Huffman algo]</strong> (1950) </p>
<blockquote>
<ul>
<li>count letter freq in input </li>
<li>build a node for each char </li>
<li>select the 2 tries with minimum weight(freq), merge them, and put it back </li>
<li>repeat until we get only 1 trie </li>
</ul>
</blockquote>
<p><img alt="" class="img-responsive" src="../images/algoII_week5_2/pasted_image005.png"/> </p>
<p>application: jpeg, pdf, mp3, ... </p>
<p><strong>implementation</strong> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">private</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="n">Node</span><span class="w"> </span><span class="n">buildTrie</span><span class="p">(</span><span class="nc">int</span><span class="err">[]</span><span class="w"> </span><span class="n">freq</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">MinPQ</span><span class="o"><</span><span class="n">Node</span><span class="o">></span><span class="w"> </span><span class="n">pq</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">MinPQ</span><span class="o"><</span><span class="n">Node</span><span class="o">></span><span class="p">();</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">for</span><span class="p">(</span><span class="nc">char</span><span class="w"> </span><span class="n">i</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o"><</span><span class="n">R</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">pq</span><span class="p">.</span><span class="k">insert</span><span class="p">(</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Node</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">freq</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">while</span><span class="p">(</span><span class="n">pq</span><span class="p">.</span><span class="k">size</span><span class="p">()</span><span class="o">></span><span class="mi">1</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">Node</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pq</span><span class="p">.</span><span class="n">pop</span><span class="p">(),</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pq</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">pq</span><span class="p">.</span><span class="k">insert</span><span class="p">(</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Node</span><span class="p">(</span><span class="s1">'\0'</span><span class="p">,</span><span class="n">x</span><span class="p">.</span><span class="n">freq</span><span class="o">+</span><span class="n">y</span><span class="p">.</span><span class="n">freq</span><span class="p">,</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="err">}</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">pq</span><span class="p">.</span><span class="k">top</span><span class="p">();</span><span class="w"> </span></span>
<span class="code-line"><span class="err">}</span><span class="w"></span></span>
</pre></div>
<p><strong>prop</strong>. Huffman algorithm produces the optimal prefixless code. </p>
<p>running time: <strong>N + RlgR</strong> </p>
<h1 id="4-lzw-compression_1">4. LZW Compression</h1>
<p>idea:<br/>
adaptive model: learn and <em>update</em> the model as you read the text. <br/>
decoder will do the same thing. </p>
<p>build a table of not only mapping chars to codes, but also mapping >=2-char combinations to codes(of fixed width). <br/>
when encoding strings, look for the longest prefix that is in our table. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_2/pasted_image006.png"/> </p>
<h3 id="lzw-compression">LZW compression</h3>
<p>LZW compression algo </p>
<ul>
<li>create a symbol table mapping string keys to W-bit code, initialize as mapping only chars to codes </li>
<li>find longest prefix <code>s</code> of the unscanned input </li>
<li>write out the corresponding code </li>
<li>add <code>s+c</code> into the symbol table, where <code>c</code> is the next char in input </li>
</ul>
<p>Use a trie for representing the code table → because support longest prefix match. </p>
<p><strong>implementation</strong> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span> <span class="n">void</span> <span class="n">compress</span><span class="p">(</span><span class="n">String</span> <span class="k">input</span><span class="p">)</span><span class="err">{</span> </span>
<span class="code-line"> <span class="n">TST</span><span class="o"><</span><span class="nb">Integer</span><span class="o">></span> <span class="n">st</span> <span class="o">=</span> <span class="k">new</span> <span class="n">TST</span><span class="o"><</span><span class="nb">Integer</span><span class="o">></span><span class="p">();</span> <span class="o">//</span> <span class="n">a</span> <span class="n">trie</span> <span class="n">symbol</span> <span class="k">table</span> </span>
<span class="code-line"> <span class="k">for</span><span class="p">(</span><span class="nb">char</span> <span class="n">i</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span> <span class="n">i</span><span class="o"><</span><span class="n">R</span><span class="p">;</span> <span class="n">i</span><span class="o">++</span><span class="p">)</span> </span>
<span class="code-line"> <span class="n">st</span><span class="p">.</span><span class="n">put</span><span class="p">(</span><span class="ss">""</span><span class="o">+</span><span class="p">(</span><span class="nb">char</span><span class="p">)</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">);</span> </span>
<span class="code-line"> <span class="nb">int</span> <span class="n">code</span> <span class="o">=</span> <span class="n">R</span><span class="o">+</span><span class="mi">1</span><span class="p">;</span> <span class="o">//</span> <span class="n">R</span> <span class="k">is</span> <span class="n">used</span> <span class="k">as</span> <span class="ss">"stop"</span> </span>
<span class="code-line"> <span class="n">while</span><span class="p">(</span><span class="k">input</span><span class="p">.</span><span class="k">length</span><span class="p">()</span><span class="o">></span><span class="mi">0</span><span class="p">)</span><span class="err">{</span> </span>
<span class="code-line"> <span class="n">String</span> <span class="n">s</span> <span class="o">=</span> <span class="n">st</span><span class="p">.</span><span class="n">longestPrefixOf</span><span class="p">(</span><span class="k">input</span><span class="p">);</span> </span>
<span class="code-line"> <span class="n">BinaryStdOut</span><span class="p">.</span><span class="k">write</span><span class="p">(</span><span class="n">st</span><span class="p">.</span><span class="k">get</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">W</span><span class="p">);</span> </span>
<span class="code-line"> <span class="nb">int</span> <span class="n">l</span> <span class="o">=</span> <span class="n">s</span><span class="p">.</span><span class="k">length</span><span class="p">();</span> </span>
<span class="code-line"> <span class="k">if</span><span class="p">(</span><span class="n">l</span><span class="o"><</span><span class="k">input</span><span class="p">.</span><span class="k">length</span><span class="p">())</span> </span>
<span class="code-line"> <span class="n">st</span><span class="p">.</span><span class="n">put</span><span class="p">(</span><span class="n">s</span><span class="o">+</span><span class="k">input</span><span class="p">.</span><span class="n">charAt</span><span class="p">(</span><span class="n">l</span><span class="p">),</span> <span class="n">code</span><span class="o">++</span><span class="p">);</span> </span>
<span class="code-line"> <span class="k">input</span> <span class="o">=</span> <span class="k">input</span><span class="p">.</span><span class="k">substring</span><span class="p">(</span><span class="n">l</span><span class="p">);</span> </span>
<span class="code-line"> <span class="err">}</span> </span>
<span class="code-line"> <span class="n">BinaryStdOut</span><span class="p">.</span><span class="k">write</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">W</span><span class="p">);</span> <span class="o">//</span> <span class="k">write</span> <span class="ss">"stop"</span> </span>
<span class="code-line"> <span class="n">BinaryStdOut</span><span class="p">.</span><span class="k">close</span><span class="p">();</span></span>
<span class="code-line"></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h3 id="lzw-expansion">LZW expansion</h3>
<p><em>no need to get the codeword table</em>, the input is just the compressed bitstring. </p>
<ul>
<li>maintain a (reverse)table mapping int(code) to strings. </li>
<li>as we decode the string, we add new entries to our table. </li>
</ul>
<p>to represent the table: just an array. </p>
<p><em>note</em> when add new entries: add key = <code>s+c</code>, where <code>s</code> is the last decoded string, <code>c</code> is the first char of the currently decoded string.<br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_2/pasted_image008.png"/><br/>
example: when decoded ABR, s = BR, currently decoded string = ABR, so we add BRA. </p>
<p><strong>tricky case</strong>: ABABABA<br/>
compression<br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_2/pasted_image009.png"/><br/>
problem in expansion when reading the "83": <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_2/pasted_image011.png"/> </p>
<p>→ need to be able to update the table when encountering a code not yet defined... </p>
<h1 id="summary_1">Summary</h1>
<ul>
<li>Huffman: fixed-length symbols, with variable-length codes. </li>
<li>LZW: variable-length symbols, with fixed-length codes. </li>
</ul>
<p>theoretical limit: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week5_2/pasted_image012.png"/> </p>
<p>还剩下一周的内容, 可能要等到二月底考完以后有时间再看了, 现在先复习备考...</p>
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<div id="toc"><ul><li><a class="toc-href" href="#1-introduction-to-data-compression" title="1. Introduction to Data Compression">1. Introduction to Data Compression</a><ul><li><a class="toc-href" href="#tools" title="Tools">Tools</a></li></ul></li><li><a class="toc-href" href="#2-run-length-coding_1" title="2. Run-Length Coding">2. Run-Length Coding</a></li><li><a class="toc-href" href="#3-huffman-compression" title="3. Huffman Compression">3. Huffman Compression</a><ul><li><a class="toc-href" href="#huffman-trie-node" title="Huffman Trie Node">Huffman Trie Node</a></li><li><a class="toc-href" href="#implementation" title="implementation">implementation</a></li><li><a class="toc-href" href="#huffman-algorithm" title="Huffman algorithm">Huffman algorithm</a></li></ul></li><li><a class="toc-href" href="#4-lzw-compression_1" title="4. LZW Compression">4. LZW Compression</a><ul><li><a class="toc-href" href="#lzw-compression" title="LZW compression">LZW compression</a></li><li><a class="toc-href" href="#lzw-expansion" title="LZW expansion">LZW expansion</a></li></ul></li><li><a class="toc-href" href="#summary_1" title="Summary">Summary</a></li></ul></div>
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