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[Algorithms I] Week 4-2a Elementry Symbol Tables
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-symbol-table-api" title="1. Symbol Table API">1. Symbol Table API</a><ul><li><a class="toc-href" href="#association-array-abstraction" title="Association array abstraction">Association array abstraction</a></li></ul></li><li><a class="toc-href" href="#2-elementary-implementations_1" title="2. Elementary implementations">2. Elementary implementations</a><ul><li><a class="toc-href" href="#using-unordered-linked-list" title="using unordered linked list">using unordered linked list</a></li><li><a class="toc-href" href="#using-ordered-array" title="using ordered array">using ordered array</a></li></ul></li><li><a class="toc-href" href="#3-ordered-opeartions_1" title="3. Ordered Opeartions">3. Ordered Opeartions</a></li></ul></div>
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<h1 id="1-symbol-table-api">1. Symbol Table API</h1>
<p>key-value pair abstraction </p>
<ul>
<li><strong>insert</strong> a value with a key </li>
<li>given a key, <strong>search</strong> for its value </li>
</ul>
<h2 id="association-array-abstraction">Association array abstraction</h2>
<p>Associate a value to a key — generalized array: <em>a[key]=val</em>. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class ST<Key, Value>{ </span></span>
<span class="code-line"><span class="err"> void put(Key k, Value v);//remove key if value=null </span></span>
<span class="code-line"><span class="err"> Value get(Key k);//return null if key is absent </span></span>
<span class="code-line"><span class="err"> void delete(Key k); </span></span>
<span class="code-line"><span class="err"> boolean contains(Key k); </span></span>
<span class="code-line"><span class="err"> boolean isEmpty(); </span></span>
<span class="code-line"><span class="err"> int size(); </span></span>
<span class="code-line"><span class="err"> Iterable<Key> keys();//better to return an ordered sequence of keys </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>conventions: </p>
<ul>
<li>values are not null </li>
<li>get() returns null if key not present </li>
<li>put() can <em>overwrite</em> older value </li>
</ul>
<p>→ some one-line implementations: </p>
<ul>
<li>contains: <code>return get(k)!=null;</code> </li>
<li>delete: <code>put(k, null);</code> </li>
</ul>
<p>Assume keys to be comparable: <code>class ST<Key implements Comparable<Key>, Value></code>— can thus use <code>compareTo()</code> method. <br/>
Else → we can only use the <code>equals()</code> method... <br/>
Be careful when implementing the equals method: 坑不少... <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_2a/pasted_image.png"/> </p>
<h1 id="2-elementary-implementations_1">2. Elementary implementations</h1>
<p>naive implementations </p>
<h2 id="using-unordered-linked-list">using unordered linked list</h2>
<p><em>ListNode{key, value, next}</em> </p>
<ul>
<li>search: scan through all keys ~N </li>
<li>insert: scan through, if not found, add to front ~N </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week4_2a/pasted_image001.png"/> </p>
<h2 id="using-ordered-array">using ordered array</h2>
<p>using 2 arrays: <em>keys[] (sorted), vals[]</em> <br/>
⇒ can improve performance by binary search </p>
<p><strong>search operation</strong> <br/>
write a function rank() that returns the number of keys < k searched. <br/>
找不到的时候: <em>比k小的元素个数=lo (lo>hi, 可以想想当hi=lo以后是怎么移动的)</em> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">private</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="nf">rank</span><span class="p">(</span><span class="k">Key</span><span class="w"> </span><span class="n">k</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="n">lo</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">hi</span><span class="o">=</span><span class="n">keys</span><span class="p">.</span><span class="n">length</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="k">while</span><span class="p">(</span><span class="n">hi</span><span class="o">>=</span><span class="n">lo</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="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">lo</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">hi</span><span class="o">-</span><span class="n">lo</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">;</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="n">cmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">keys</span><span class="o">[</span><span class="n">mid</span><span class="o">]</span><span class="p">.</span><span class="n">compareTo</span><span class="p">(</span><span class="n">keys</span><span class="o">[</span><span class="n">k</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">if</span><span class="p">(</span><span class="n">cmp</span><span class="o">==</span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">mid</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="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">cmp</span><span class="o">></span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="n">hi</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mid</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="k">else</span><span class="w"> </span><span class="n">lo</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mid</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">lo</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>Using rank() to implement the get() method: </p>
<div class="highlight"><pre><span class="code-line"><span></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="k">Key</span><span class="w"> </span><span class="n">k</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="n">rk</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">rank</span><span class="p">(</span><span class="n">k</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">rk</span><span class="o"><</span><span class="n">N</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="n">keys</span><span class="o">[</span><span class="n">rk</span><span class="o">]</span><span class="p">.</span><span class="n">compareTo</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">==</span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">vals</span><span class="o">[</span><span class="n">rk</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">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="err">}</span><span class="w"></span></span>
</pre></div>
<p><strong>insert operation</strong> <br/>
Like insertion sort, time complexity is ~N for each insert. </p>
<p>summery: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_2a/pasted_image002.png"/> </p>
<h1 id="3-ordered-opeartions_1">3. Ordered Opeartions</h1>
<p>When keys are comparable ⇒ provide <em>more functionalities</em> in the API. <br/>
for example: </p>
<ul>
<li><code>min()/max()</code>: min/max key </li>
<li><code>deleteMin()/deleteMax()</code> </li>
<li><code>floor(Key k)/ceiling(Key k)</code>: largest key <=k / smallest key >=k </li>
<li><code>rank(Key k)</code>: nb of keys < key </li>
<li><code>select(int i)</code>: key with rank=i </li>
<li><code>Iterator<Key> keys(lo, hi)</code>: iterates through [lo, hi] </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week4_2a/pasted_image004.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/algoI_week4_2a/pasted_image003.png"/> </p>
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