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[Algorithms II] Week 4-1 Tries
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-r-way-tries" title="1. R-way Tries">1. R-way Tries</a><ul><li><a class="toc-href" href="#string-symbol-table-api" title="String Symbol Table API">String Symbol Table API</a></li><li><a class="toc-href" href="#r-way-tries" title="R-way tries">R-way tries</a></li><li><a class="toc-href" href="#java-implementation" title="Java implementation">Java implementation</a></li><li><a class="toc-href" href="#analysis" title="analysis">analysis</a></li></ul></li><li><a class="toc-href" href="#2-ternary-search-tries_1" title="2. Ternary Search Tries">2. Ternary Search Tries</a><ul><li><a class="toc-href" href="#implementation" title="Implementation">Implementation</a></li><li><a class="toc-href" href="#hybrid-of-tst-and-rway-trie" title="Hybrid of TST and Rway trie">Hybrid of TST and Rway trie</a></li><li><a class="toc-href" href="#analysis_1" title="analysis">analysis</a></li><li><a class="toc-href" href="#hashing-vs-tst" title="Hashing vs. TST">Hashing vs. TST</a></li></ul></li><li><a class="toc-href" href="#3-character-based-operations_1" title="3. Character-Based Operations">3. Character-Based Operations</a><ul><li><a class="toc-href" href="#api" title="API">API</a></li><li><a class="toc-href" href="#keys-ordered-iteration" title="keys(): ordered iteration">keys(): ordered iteration</a></li><li><a class="toc-href" href="#prefix-match" title="prefix match">prefix match</a></li><li><a class="toc-href" href="#longest-prefix" title="longest prefix">longest prefix</a><ul><li><a class="toc-href" href="#application-t9-texting" title="application: T9 texting">application: T9 texting</a></li></ul></li><li><a class="toc-href" href="#other-variants_1" title="Other Variants">Other Variants</a><ul><li><a class="toc-href" href="#patricia-trie" title="Patricia trie">Patricia trie</a></li><li><a class="toc-href" href="#suffix-tree" title="suffix tree">suffix tree</a></li></ul></li><li><a class="toc-href" href="#summery_1" title="Summery">Summery</a></li></ul></li></ul></div>
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<p>More efficient version of symbol-table where the keys are strings. </p>
<h1 id="1-r-way-tries">1. R-way Tries</h1>
<p>Two implementations of symbol tables that we've seen: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week4_1/pasted_image.png"/><br/>
when keys are strings:<br/>
(<code>L</code>=string length, <code>N</code>=number of strings, <code>R</code>=radix)<br/>
<img alt="" class="img-responsive" src="../images/algoII_week4_1/pasted_image001.png"/> </p>
<p>for string keys ⇒ do better by avoiding examing the entire key. </p>
<p>goal: faster than hashtable, more flexible than BST ! </p>
<h2 id="string-symbol-table-api">String Symbol Table API</h2>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class StringST<V>{// implements ST<String, V> </span></span>
<span class="code-line"><span class="err"> public void put(String key, V val); </span></span>
<span class="code-line"><span class="err"> public V get(String key); </span></span>
<span class="code-line"><span class="err"> public void delete(String key); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h2 id="r-way-tries">R-way tries</h2>
<p>A trie is a tree where: </p>
<ul>
<li>except the root, each node store <em>characters</em> (instead of string keys) — in fact the chars are stored in <em>links</em> </li>
<li>each node has R children </li>
<li>store <em>value</em> in node if the node corresponds to the<em> last char</em> in key. </li>
</ul>
<p>example: (<em>a trie<integer>)</integer></em><br/>
<img alt="" class="img-responsive" src="../images/algoII_week4_1/pasted_image002.png"/> </p>
<ul>
<li><strong>search in a trie </strong><code>get()</code> </li>
</ul>
<p><em>hit</em> if when search ends the node has a non-null value<br/>
<em>miss</em> if reach a null link or have null value when search ends. </p>
<ul>
<li><strong>insertion in a trie</strong> <code>put()</code> </li>
</ul>
<p>follow the links as the chars indicate in the key<br/>
→ if meet null links: create new node<br/>
→ when reach last char in key: set the value of the node </p>
<h2 id="java-implementation">Java implementation</h2>
<p>trie node class:<br/>
in each node use an array of size R to store links... </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="k">class</span><span class="w"> </span><span class="n">Node</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="k">Object</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="o">//</span><span class="w"> </span><span class="n">because</span><span class="w"> </span><span class="n">cannot</span><span class="w"> </span><span class="k">create</span><span class="w"> </span><span class="n">arries</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">generic</span><span class="w"> </span><span class="n">type</span><span class="w"> </span><span class="p">(</span><span class="k">array</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">Node</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="n">Node</span><span class="err">[]</span><span class="w"> </span><span class="k">next</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">Node</span><span class="o">[</span><span class="n">R</span><span class="o">]</span><span class="p">;</span><span class="o">//</span><span class="w"> </span><span class="n">chars</span><span class="w"> </span><span class="k">are</span><span class="w"> </span><span class="n">implicitly</span><span class="w"> </span><span class="n">defined</span><span class="w"> </span><span class="k">by</span><span class="w"> </span><span class="n">link</span><span class="w"> </span><span class="k">index</span><span class="w"> </span></span>
<span class="code-line"><span class="err">}</span><span class="w"></span></span>
</pre></div>
<p>另外关于为什么内部类Node声明为static, 参考这里: <br/>
<a href="http://www.geeksforgeeks.org/static-class-in-java/">http://www.geeksforgeeks.org/static-class-in-java/</a> </p>
<p>implementation: use private recursive methods, 和BST的实现类似, 定义一个递归的private函数, 返回插入后的Node, 很有用. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span><span class="w"> </span><span class="k">class</span><span class="w"> </span><span class="n">TrieST</span><span class="o"><</span><span class="k">Value</span><span class="o">></span><span class="w"> </span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">pirvate</span><span class="w"> </span><span class="n">final</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">R</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">256</span><span class="p">;</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="nf">ASCII</span><span class="w"> </span><span class="n">chars</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="n">Node</span><span class="w"> </span><span class="n">root</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">Node</span><span class="p">();</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="k">class</span><span class="w"> </span><span class="n">Node</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="k">Object</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="o">//</span><span class="w"> </span><span class="n">because</span><span class="w"> </span><span class="n">cannot</span><span class="w"> </span><span class="k">create</span><span class="w"> </span><span class="n">arries</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">generic</span><span class="w"> </span><span class="n">type</span><span class="w"> </span><span class="p">(</span><span class="k">array</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">Node</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="n">Node</span><span class="err">[]</span><span class="w"> </span><span class="k">next</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">Node</span><span class="o">[</span><span class="n">R</span><span class="o">]</span><span class="p">;</span><span class="o">//</span><span class="w"> </span><span class="n">chars</span><span class="w"> </span><span class="k">are</span><span class="w"> </span><span class="n">implicitly</span><span class="w"> </span><span class="n">defined</span><span class="w"> </span><span class="k">by</span><span class="w"> </span><span class="n">link</span><span class="w"> </span><span class="k">index</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">public</span><span class="w"> </span><span class="n">void</span><span class="w"> </span><span class="n">put</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="k">key</span><span class="p">,</span><span class="w"> </span><span class="k">Value</span><span class="w"> </span><span class="n">val</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">this</span><span class="p">.</span><span class="n">put</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="k">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span><span class="o">//</span><span class="w"> </span><span class="k">use</span><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="k">recursive</span><span class="w"> </span><span class="n">helper</span><span class="w"> </span><span class="k">method</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="n">private</span><span class="w"> </span><span class="n">Node</span><span class="w"> </span><span class="n">put</span><span class="p">(</span><span class="n">Node</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="k">key</span><span class="p">,</span><span class="w"> </span><span class="k">Value</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">d</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="err">`</span><span class="n">d</span><span class="err">`</span><span class="w"> </span><span class="k">is</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="k">index</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nc">char</span><span class="w"> </span><span class="k">to</span><span class="w"> </span><span class="n">put</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="k">returns</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="n">inserted</span><span class="w"> </span><span class="n">node</span><span class="w"></span></span>
<span class="code-line"></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="o">==</span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="n">x</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">Node</span><span class="p">();</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">key</span><span class="p">.</span><span class="n">length</span><span class="p">()</span><span class="o">==</span><span class="n">d</span><span class="p">)</span><span class="w"> </span><span class="n">x</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">else</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="nc">char</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">key</span><span class="p">.</span><span class="n">charAt</span><span class="p">(</span><span class="n">d</span><span class="p">);</span><span class="w"></span></span>
<span class="code-line"></span>
<span class="code-line"><span class="w"> </span><span class="n">x</span><span class="p">.</span><span class="k">next</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">put</span><span class="p">(</span><span class="n">x</span><span class="p">.</span><span class="k">next</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="k">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">d</span><span class="o">+</span><span class="mi">1</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">x</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">public</span><span class="w"> </span><span class="k">boolean</span><span class="w"> </span><span class="k">contains</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="k">key</span><span class="p">)</span><span class="err">{</span><span class="k">return</span><span class="w"> </span><span class="k">get</span><span class="p">(</span><span class="k">key</span><span class="p">)</span><span class="o">!=</span><span class="k">null</span><span class="p">;</span><span class="err">}</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">public</span><span class="w"> </span><span class="k">Value</span><span class="w"> </span><span class="k">get</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="k">key</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">nd</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">this</span><span class="p">.</span><span class="k">get</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="k">key</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">nd</span><span class="o">==</span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">null</span><span class="p">;</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="p">(</span><span class="k">Value</span><span class="p">)</span><span class="w"> </span><span class="n">x</span><span class="p">.</span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="nf">cast</span><span class="w"> </span><span class="n">back</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="n">private</span><span class="w"> </span><span class="n">Node</span><span class="w"> </span><span class="k">get</span><span class="p">(</span><span class="n">Node</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="k">key</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">d</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="k">returns</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="n">that</span><span class="w"> </span><span class="k">contains</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="k">key</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">x</span><span class="o">==</span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">null</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">d</span><span class="o">==</span><span class="k">key</span><span class="p">.</span><span class="n">length</span><span class="p">())</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="nc">char</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">key</span><span class="p">.</span><span class="n">charAt</span><span class="p">(</span><span class="n">d</span><span class="p">);</span><span class="w"></span></span>
<span class="code-line"></span>
<span class="code-line"><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">get</span><span class="p">(</span><span class="n">x</span><span class="p">.</span><span class="k">next</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="k">key</span><span class="p">,</span><span class="w"> </span><span class="n">d</span><span class="o">+</span><span class="mi">1</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="err">}</span><span class="w"></span></span>
</pre></div>
<p>delete node in trie: </p>
<ul>
<li>find the node and set the val to null </li>
<li>if a node has 0 links(leaf) and val==null: delete it and recursivly go up. </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week4_1/pasted_image003.png"/> </p>
<h2 id="analysis">analysis</h2>
<p><strong>time:</strong><br/>
search hit — L nodes examined. <br/>
serach miss — sublinear in L<br/>
<strong>space:</strong> <br/>
each node has R links (possibly null) — <em>too much memory for large R</em> !<br/>
<img alt="" class="img-responsive" src="../images/algoII_week4_1/pasted_image004.png"/><br/>
<strong>Application</strong><br/>
interview question: data structure for spell checking. </p>
<h1 id="2-ternary-search-tries_1">2. Ternary Search Tries</h1>
<p>Solution to the memory issue of R-way tries — much fewer null links. ternery=tuple of 3...<br/>
TST: </p>
<ul>
<li>store chars (and vals) <em>explicitly</em> in nodes </li>
<li>each node has <strong>3</strong> children: <ul>
<li><em>smaller, larger: </em>TST that starts with char smaller/bigger than its char c. </li>
<li><em>equal: </em>the TST with keys that <strong>starts with</strong> this char c </li>
<li>(所以除了用smaller/larger连接的节点在Rway trie里属于同一层 </li>
</ul>
</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week4_1/pasted_image005.png"/><br/>
类似于3-way radix sort(同一篇paper里提出的), 有有点RBTree的意思... </p>
<p><strong>searching</strong> in TST: quite similar to BST search, will go down ("equal" or "middle") when the current node matchs current char... </p>
<h2 id="implementation">Implementation</h2>
<p>Node class: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private class Node{ </span></span>
<span class="code-line"><span class="err"> private Value val; </span></span>
<span class="code-line"><span class="err"> private char c;// store char explicitly </span></span>
<span class="code-line"><span class="err"> private Node left, mid, right; </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>TST class (again use recursive helper functions)<br/>
有一种写BST代码的感觉, 另外由于这次char是显式存在node里, 代码反而更好理解了... </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class TST<Value>{ </span></span>
<span class="code-line"><span class="err"> private class Node{...}; </span></span>
<span class="code-line"><span class="err"> private Node root; </span></span>
<span class="code-line"><span class="err"> public void put(String key, Value val){ </span></span>
<span class="code-line"><span class="err"> this.put(root, key, val, 0); </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> private Node put(Node x, String key, Value val, int d){ </span></span>
<span class="code-line"><span class="err"> char c = key.charAt(d);// char to process </span></span>
<span class="code-line"><span class="err"> if(x==null) x = new Node(c); </span></span>
<span class="code-line"><span class="err"> if(c==x.c) { </span></span>
<span class="code-line"><span class="err"> if(d==key.length()-1) x.val = val; </span></span>
<span class="code-line"><span class="err"> else x.mid = put(x.mid, key, val, d+1);// d+1 means we go down one level </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> else if (c<x.c) x.left = put(x.left, key, val, d);// do not increment if the current char is not matchd </span></span>
<span class="code-line"><span class="err"> else x.right = put(x.right, key, val, d); </span></span>
<span class="code-line"><span class="err"> return x; </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> public Value get(String key){ </span></span>
<span class="code-line"><span class="err"> Node nd = this.get(root, key, 0); </span></span>
<span class="code-line"><span class="err"> if(nd==null) return null; </span></span>
<span class="code-line"><span class="err"> return nd.val; </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> private Node get(Node x, String key, int d){ </span></span>
<span class="code-line"><span class="err"> if(x==null) return null; </span></span>
<span class="code-line"><span class="err"> char c = key.charAt(d); </span></span>
<span class="code-line"><span class="err"> if(x.c==c) { </span></span>
<span class="code-line"><span class="err"> if(d==key.length()-1) return x; </span></span>
<span class="code-line"><span class="err"> else return get(x.mid, key, d+1); </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> else if(c<x.c) return get(x.left, key, d); </span></span>
<span class="code-line"><span class="err"> else return get(x.right, d); </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h2 id="hybrid-of-tst-and-rway-trie">Hybrid of TST and Rway trie</h2>
<ul>
<li>at root: do <em>R^2 branching</em> </li>
<li>other nodes are TSTs </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week4_1/pasted_image007.png"/> </p>
<h2 id="analysis_1">analysis</h2>
<p>space cost: linear in N.<br/>
time: if keys arrive in rand order... (can use rotation to get worst-case guarantee..)<br/>
<img alt="" class="img-responsive" src="../images/algoII_week4_1/pasted_image008.png"/> </p>
<h2 id="hashing-vs-tst">Hashing vs. TST</h2>
<p>Hashing: </p>
<ul>
<li>need to check entire key no matter hit or miss </li>
<li>performance relies on hash functions </li>
<li>donot support ordered operations </li>
</ul>
<p>TST: </p>
<ul>
<li>examines just enough chars </li>
<li>support ordered operations </li>
</ul>
<h1 id="3-character-based-operations_1">3. Character-Based Operations</h1>
<p>some very useful char-based operations: </p>
<ul>
<li><strong>prefix match</strong> </li>
<li><strong>wildcard match</strong>: use a dot to represent any char </li>
<li><strong>longest prefix</strong>: find the key that is the longest prefix of a string </li>
</ul>
<h2 id="api">API</h2>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class TrieST<Value>{ </span></span>
<span class="code-line"><span class="err"> // functions decalred before.. </span></span>
<span class="code-line"><span class="err"> Iterable<String> keys; </span></span>
<span class="code-line"><span class="err"> Iterable<String> keysWithPrefix(String s); </span></span>
<span class="code-line"><span class="err"> Iterable<String> keysThatMatch(String s); </span></span>
<span class="code-line"><span class="err"> Iterable<String> longestPrefixOf(String s);</span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h2 id="keys-ordered-iteration">keys(): ordered iteration</h2>
<p><code>keys()</code>: just an inorder-traversal of the <em>Rway trie</em> →dfs, + maintain the chars in the path (root to current node). <br/>
<img alt="" class="img-responsive" src="../images/algoII_week4_1/pasted_image009.png"/> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span><span class="w"> </span><span class="n">Iterable</span><span class="o"><</span><span class="n">String</span><span class="o">></span><span class="w"> </span><span class="n">keys</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">Queue</span><span class="o"><</span><span class="n">String</span><span class="o">></span><span class="w"> </span><span class="n">q</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">Queue</span><span class="o"><</span><span class="n">String</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">collect</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="ss">""</span><span class="p">,</span><span class="w"> </span><span class="n">q</span><span class="p">);</span><span class="o">//</span><span class="w"> </span><span class="n">helper</span><span class="w"> </span><span class="n">fcn</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">q</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="err">}</span><span class="w"> </span></span>
<span class="code-line"><span class="n">private</span><span class="w"> </span><span class="n">void</span><span class="w"> </span><span class="k">collect</span><span class="p">(</span><span class="n">Node</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">pathstr</span><span class="p">,</span><span class="w"> </span><span class="n">Queue</span><span class="w"> </span><span class="n">q</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">x</span><span class="o">==</span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">x</span><span class="p">.</span><span class="n">val</span><span class="o">!=</span><span class="k">null</span><span class="p">)</span><span class="o">//</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">this</span><span class="w"> </span><span class="k">is</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">key</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">q</span><span class="p">.</span><span class="n">enqueue</span><span class="p">(</span><span class="n">pathstr</span><span class="p">);</span><span class="o">//</span><span class="w"> </span><span class="n">pathstr</span><span class="w"> </span><span class="k">is</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">chars</span><span class="w"> </span><span class="k">from</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="k">to</span><span class="w"> </span><span class="n">x</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">c</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="n">c</span><span class="o"><</span><span class="n">R</span><span class="p">;</span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="n">dfs</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">collect</span><span class="p">(</span><span class="n">x</span><span class="p">.</span><span class="k">next</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">pathstr</span><span class="o">+</span><span class="n">c</span><span class="p">,</span><span class="w"> </span><span class="n">q</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>
<h2 id="prefix-match">prefix match</h2>
<p>very useful: ex. autocomplete, search bar, ...<br/>
implementation⇒ just find the end of that prefix, then call <code>keys()</code> on the <em>subtrie</em>.<br/>
之前写的俩helper function这里发挥作用了: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public Iterable<String> keysWithPrefix(String prefix){ </span></span>
<span class="code-line"><span class="err"> Node subtrieRoot = get(root, prefix, 0); </span></span>
<span class="code-line"><span class="err"> Queue<String> q = new Queue<String>(); </span></span>
<span class="code-line"><span class="err"> collect(subtrieRoot, prefix, q); </span></span>
<span class="code-line"><span class="err"> return q; </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h2 id="longest-prefix">longest prefix</h2>
<p>⇒ just do a search and keep track of the longest key that we encounterd. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">longestPrefixOf</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="n">s</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="nf">len</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">search</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</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">s</span><span class="p">.</span><span class="nf">substring</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nf">len</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="err">}</span><span class="w"> </span></span>
<span class="code-line"><span class="n">private</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="k">search</span><span class="p">(</span><span class="n">Node</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">d</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="nf">len</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">x</span><span class="o">==</span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nf">len</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">x</span><span class="p">.</span><span class="n">val</span><span class="o">!=</span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="nf">len</span><span class="o">=</span><span class="n">d</span><span class="p">;</span><span class="o">//</span><span class="w"> </span><span class="n">this</span><span class="w"> </span><span class="k">is</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="k">current</span><span class="w"> </span><span class="n">longest</span><span class="w"> </span><span class="k">prefix</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">d</span><span class="o">==</span><span class="n">s</span><span class="p">.</span><span class="n">length</span><span class="p">())</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">length</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="nc">char</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">charAt</span><span class="p">(</span><span class="n">d</span><span class="p">);</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="k">search</span><span class="p">(</span><span class="n">x</span><span class="p">.</span><span class="k">next</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">d</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nf">len</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>
<h3 id="application-t9-texting">application: T9 texting</h3>
<p>classique... G面试以及X的TP里都有涉及过... </p>
<ul>
<li>first: generate all string combinations </li>
<li>second: call get() on the trie of dictionaries. </li>
</ul>
<h2 id="other-variants_1">Other Variants</h2>
<h3 id="patricia-trie">Patricia trie</h3>
<p>improvement: remove the one-eay branching (put >1 chars in a node).<br/>
<img alt="" class="img-responsive" src="../images/algoII_week4_1/pasted_image010.png"/> </p>
<h3 id="suffix-tree">suffix tree</h3>
<ul>
<li>patricia tree of the suffix </li>
<li>linear time construction </li>
<li>ongest repeated substring, longest common substring, longest palindromic substring, substring search, tandem repeats... 好有用!!... </li>
</ul>
<h2 id="summery_1">Summery</h2>
<p><img alt="" class="img-responsive" src="../images/algoII_week4_1/pasted_image011.png"/> </p>
</div>
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目录</h4>
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<div id="toc"><ul><li><a class="toc-href" href="#1-r-way-tries" title="1. R-way Tries">1. R-way Tries</a><ul><li><a class="toc-href" href="#string-symbol-table-api" title="String Symbol Table API">String Symbol Table API</a></li><li><a class="toc-href" href="#r-way-tries" title="R-way tries">R-way tries</a></li><li><a class="toc-href" href="#java-implementation" title="Java implementation">Java implementation</a></li><li><a class="toc-href" href="#analysis" title="analysis">analysis</a></li></ul></li><li><a class="toc-href" href="#2-ternary-search-tries_1" title="2. Ternary Search Tries">2. Ternary Search Tries</a><ul><li><a class="toc-href" href="#implementation" title="Implementation">Implementation</a></li><li><a class="toc-href" href="#hybrid-of-tst-and-rway-trie" title="Hybrid of TST and Rway trie">Hybrid of TST and Rway trie</a></li><li><a class="toc-href" href="#analysis_1" title="analysis">analysis</a></li><li><a class="toc-href" href="#hashing-vs-tst" title="Hashing vs. TST">Hashing vs. TST</a></li></ul></li><li><a class="toc-href" href="#3-character-based-operations_1" title="3. Character-Based Operations">3. Character-Based Operations</a><ul><li><a class="toc-href" href="#api" title="API">API</a></li><li><a class="toc-href" href="#keys-ordered-iteration" title="keys(): ordered iteration">keys(): ordered iteration</a></li><li><a class="toc-href" href="#prefix-match" title="prefix match">prefix match</a></li><li><a class="toc-href" href="#longest-prefix" title="longest prefix">longest prefix</a><ul><li><a class="toc-href" href="#application-t9-texting" title="application: T9 texting">application: T9 texting</a></li></ul></li><li><a class="toc-href" href="#other-variants_1" title="Other Variants">Other Variants</a><ul><li><a class="toc-href" href="#patricia-trie" title="Patricia trie">Patricia trie</a></li><li><a class="toc-href" href="#suffix-tree" title="suffix tree">suffix tree</a></li></ul></li><li><a class="toc-href" href="#summery_1" title="Summery">Summery</a></li></ul></li></ul></div>
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