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[Algorithms I] Week 6 Hash Tables
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-hash-functions" title="1. Hash Functions">1. Hash Functions</a><ul><li><a class="toc-href" hashing'="" href="#javas-hashing" s="" title="Java">Java's hashing</a></li><li><a class="toc-href" href="#modeular-hasing" title="Modeular hasing">Modeular hasing</a></li><li><a class="toc-href" href="#uniform-hasing-assumption" title="uniform hasing assumption">uniform hasing assumption</a></li></ul></li><li><a class="toc-href" href="#2-separate-chaining_1" title="2. Separate Chaining">2. Separate Chaining</a><ul><li><a class="toc-href" href="#implementation" title="implementation">implementation</a></li><li><a class="toc-href" href="#analysis" title="analysis">analysis</a></li></ul></li><li><a class="toc-href" href="#3-linear-probing_1" title="3. Linear Probing">3. Linear Probing</a><ul><li><a class="toc-href" href="#implementation_1" title="implementation">implementation</a></li><li><a class="toc-href" href="#cluster" title="cluster">cluster</a></li></ul></li><li><a class="toc-href" href="#4-hash-table-context_1" title="4. Hash Table Context">4. Hash Table Context</a><ul><li><a class="toc-href" href="#seperate-chaining-vs-linear-probing" title="Seperate chaining VS linear probing">Seperate chaining VS linear probing</a></li><li><a class="toc-href" href="#variations" title="variations">variations</a></li><li><a class="toc-href" href="#st-choice-hash-tables-vs-balanced-search-trees" title="ST choice: hash tables VS balanced search trees">ST choice: hash tables VS balanced search trees</a></li></ul></li></ul></div>
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<p><em>Can we do better than BST if we do not need ordered operations ?</em> </p>
<p>(No compare methods, use equals method) </p>
<p>Idea: save items in an array. <br/>
<strong>Hash function</strong>: method for calclulating the array index of a key. </p>
<p>Issues: </p>
<ul>
<li>computing hash function </li>
<li>equality tests </li>
<li>collision resolution </li>
</ul>
<p><em>Classic space-time tradeoff.</em> </p>
<h1 id="1-hash-functions">1. Hash Functions</h1>
<p>Goal: scramble the keys uniformly to produce a table index. </p>
<ul>
<li>effcient to compute </li>
<li>all indices are equally likely for any key </li>
</ul>
<p>challenge: need different approach for different key types. </p>
<h2 id="javas-hashing">Java's hashing</h2>
<p>All java object has a methode <code>int hashCode()</code> <br/>
requirement: </p>
<ul>
<li>if x.equals(y) ⇒ x.hashCode()==y.hashCode() </li>
<li>
<p>(hopefully) if x.equals(y)==false ⇒ x.hashCode!=y.hashCode() </p>
</li>
<li>
<p>Default implementation: <em>memory address</em> for x. </p>
</li>
<li>Custom implementations for standard types: Integer, Double, String, File, URL, Date... </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image.png"/> <br/>
Strings: Horner's method — compute a polynome. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image001.png"/> <br/>
recipe for user-defined types: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image002.png"/> </p>
<h2 id="modeular-hasing">Modeular hasing</h2>
<p>(hash code VS hash function) </p>
<ul>
<li><em>Hash code</em>: 32-bit integer between -2^31 and 2^31-1 <em>(can be negative!!)</em> </li>
<li><em>Hash function</em>: integer between 0 and M-1 (used directly as array index, should >=0) </li>
</ul>
<p><strong>BUG code:</strong> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private int hash(Key k){ </span></span>
<span class="code-line"><span class="err"> return k.hashCode()%M; </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>→ bug: <em>number returned can be negative !</em> </p>
<p><strong>1-in-billion bug code:</strong> <br/>
<code>return Math.abs(k.hashCode()) % M;</code> </p>
<p>→ bug: Math.abs() returns negative numbers for -2^31 <br/>
<img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image003.png"/> </p>
<p><strong>correct code:</strong> <br/>
just take the first 31 bits: x&0x7fffffff <br/>
<code>return k.hashCode()&0x7fffffff % M;</code> </p>
<p>另一个hash的思路是用random(用hashcode作为随机种子) </p>
<h2 id="uniform-hasing-assumption">uniform hasing assumption</h2>
<p>each key is equally likely to hash to an integer between 0 and M-1 </p>
<p>一些数学结论: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image004.png"/> </p>
<h1 id="2-separate-chaining_1">2. Separate Chaining</h1>
<p><em>one strategy for collision resolution.</em> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image005.png"/> </p>
<p><strong>idea</strong>: Using a table of size M < N, build a list for each of the table positions. </p>
<p><img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image006.png"/> </p>
<h2 id="implementation">implementation</h2>
<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">SeparateChainingHashST</span><span class="o"><</span><span class="k">Key</span><span class="p">,</span><span class="w"> </span><span class="k">Value</span><span class="o">></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="nc">int</span><span class="w"> </span><span class="n">M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">97</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="n">st</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">97</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">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="k">Object</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="o">//</span><span class="w"> </span><span class="k">no</span><span class="w"> </span><span class="n">generic</span><span class="w"> </span><span class="k">array</span><span class="w"> </span><span class="n">creation</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="k">next</span><span class="p">;</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">Node</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="w"> </span><span class="k">Value</span><span class="w"> </span><span class="n">v</span><span class="p">,</span><span class="w"> </span><span class="n">Node</span><span class="w"> </span><span class="n">nxt</span><span class="p">)</span><span class="err">{</span><span class="p">...</span><span class="err">}</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="nc">int</span><span class="w"> </span><span class="n">hash</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="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="p">.</span><span class="n">hashCode</span><span class="p">()</span><span class="o">&</span><span class="mh">0x7fffffff</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">M</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">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">h</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash</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">for</span><span class="p">(</span><span class="n">Node</span><span class="w"> </span><span class="n">x</span><span class="o">=</span><span class="n">st</span><span class="o">[</span><span class="n">h</span><span class="o">]</span><span class="p">;</span><span class="w"> </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="o">=</span><span class="n">x</span><span class="p">.</span><span class="k">next</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">k</span><span class="p">.</span><span class="k">equals</span><span class="p">(</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="k">return</span><span class="w"> </span><span class="p">(</span><span class="k">Value</span><span class="p">)</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="nf">cast</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="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="k">Key</span><span class="w"> </span><span class="n">k</span><span class="p">,</span><span class="w"> </span><span class="k">Value</span><span class="w"> </span><span class="n">v</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">h</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash</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">for</span><span class="p">(</span><span class="n">Node</span><span class="w"> </span><span class="n">x</span><span class="o">=</span><span class="n">st</span><span class="o">[</span><span class="n">h</span><span class="o">]</span><span class="p">;</span><span class="w"> </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="o">=</span><span class="n">x</span><span class="p">.</span><span class="k">next</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">k</span><span class="p">.</span><span class="k">equals</span><span class="p">(</span><span class="n">x</span><span class="p">.</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="err">{</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">v</span><span class="p">;</span><span class="w"> </span><span class="k">return</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">st</span><span class="o">[</span><span class="n">h</span><span class="o">]</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="n">k</span><span class="p">,</span><span class="n">v</span><span class="p">,</span><span class="n">st</span><span class="o">[</span><span class="n">h</span><span class="o">]</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>注意内部类Node里用的是Object — 因为不能声明generic array. </p>
<h2 id="analysis">analysis</h2>
<p><strong>proposition</strong> <br/>
Under the assumption of uniform hashing, the number of keys in a list is within a constant factor to N/M. <br/>
<em>proof.</em> <br/>
binominal distribution. </p>
<p>M times faster than sequential search <br/>
→ typical choice: M ~ N/5 </p>
<h1 id="3-linear-probing_1">3. Linear Probing</h1>
<p>Another approch for collision resolution. </p>
<p>idea: <strong>open addressing</strong> <br/>
Use an array of size <strong>M>N</strong>, when a key collides, <em>find next open slot</em>. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image007.png"/> </p>
<ul>
<li>insert: when occupied, move pointer until got open slot </li>
<li>search: when not found, move pointer until find or reach open slot </li>
<li><em>delete</em>: set key to null, then for all items behind this key: set to null, then insert this key...... </li>
</ul>
<p>array M <em>must</em> be larger than N ⇒ array resizing is necessary. </p>
<h2 id="implementation_1">implementation</h2>
<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">LinearProbingHashST</span><span class="o"><</span><span class="k">Key</span><span class="p">,</span><span class="w"> </span><span class="k">Value</span><span class="o">></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="nc">int</span><span class="w"> </span><span class="n">M</span><span class="p">,</span><span class="w"> </span><span class="n">N</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">Object</span><span class="err">[]</span><span class="w"> </span><span class="n">keys</span><span class="p">;</span><span class="o">//</span><span class="n">cannot</span><span class="w"> </span><span class="k">use</span><span class="w"> </span><span class="n">generic</span><span class="w"> </span><span class="k">array</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="err">[]</span><span class="w"> </span><span class="n">vals</span><span class="p">;</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="k">Key</span><span class="w"> </span><span class="n">k</span><span class="p">,</span><span class="w"> </span><span class="k">Value</span><span class="w"> </span><span class="n">v</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">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash</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">for</span><span class="p">(;</span><span class="n">keys</span><span class="o">[</span><span class="n">i</span><span class="o">]!=</span><span class="k">null</span><span class="p">;</span><span class="n">i</span><span class="o">=</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">%</span><span class="n">M</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">keys</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">.</span><span class="k">equals</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="err">{</span><span class="n">vals</span><span class="o">[</span><span class="n">i</span><span class="o">]=</span><span class="n">v</span><span class="p">;</span><span class="w"> </span><span class="k">return</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">keys</span><span class="o">[</span><span class="n">i</span><span class="o">]=</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="n">vals</span><span class="o">[</span><span class="n">i</span><span class="o">]=</span><span class="n">v</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">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">h</span><span class="o">=</span><span class="n">hash</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">for</span><span class="p">(;</span><span class="n">keys</span><span class="o">[</span><span class="n">h</span><span class="o">]!=</span><span class="k">null</span><span class="p">;</span><span class="n">h</span><span class="o">=</span><span class="p">(</span><span class="n">h</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">%</span><span class="n">M</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">k</span><span class="p">.</span><span class="k">equals</span><span class="p">(</span><span class="n">keys</span><span class="o">[</span><span class="n">h</span><span class="o">]</span><span class="p">))</span><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="n">vals</span><span class="o">[</span><span class="n">h</span><span class="o">]</span><span class="p">;</span><span class="o">//</span><span class="n">ugly</span><span class="w"> </span><span class="nf">cast</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="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>
<h2 id="cluster">cluster</h2>
<p>def. <strong>cluster</strong> <br/>
A contiguous block of items. </p>
<p>New keys are more likely to hash into a cluster. </p>
<p>Knuth parking pb: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image008.png"/> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image010.png"/> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image009.png"/> <br/>
(当年Knuth就是在证明了这个以后决定写那套书) </p>
<p>Typical choice: <strong>N/M ~ 1/2</strong> (3/2 for search hit and 5/2 for search miss) </p>
<p><img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image011.png"/> </p>
<h1 id="4-hash-table-context_1">4. Hash Table Context</h1>
<p>widely used in applicataions. </p>
<p>cost for computing hash VS cost for searching: <br/>
hashCode() for strings in java 1.1: exemain only 8-9 evenly spaced characters to save time. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image012.png"/> </p>
<p>uniform hashing assumption: performance not guaranteed... <br/>
<img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image013.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image014.png"/> </p>
<p><strong>one-way hash functions</strong> <br/>
Hard to find a key that hash to a desired value, or 2 keys having the same hash value. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image015.png"/> <br/>
used for fingurprints or store passwords, but too expensive for ST implementations. </p>
<h2 id="seperate-chaining-vs-linear-probing">Seperate chaining VS linear probing</h2>
<p><img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image016.png"/> </p>
<h2 id="variations">variations</h2>
<p><img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image017.png"/> </p>
<h2 id="st-choice-hash-tables-vs-balanced-search-trees">ST choice: hash tables VS balanced search trees</h2>
<p><img alt="" class="img-responsive" src="../images/algoI_week6/pasted_image018.png"/> </p>
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Part 13 of series «Algorithms Princeton MOOC I»：
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