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[Algorithms II] Week 3-2 Radix Sorts
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-strings-in-java" title="1. Strings in Java">1. Strings in Java</a><ul><li><a class="toc-href" href="#char-data-type" title="char data type">char data type</a></li><li><a class="toc-href" href="#string-data-type" title="String data type">String data type</a></li><li><a class="toc-href" href="#stringbuilder-data-type" title="StringBuilder data type">StringBuilder data type</a></li><li><a class="toc-href" href="#alphabet" title="Alphabet">Alphabet</a></li></ul></li><li><a class="toc-href" href="#2-key-indexed-counting_1" title="2. Key-Indexed Counting">2. Key-Indexed Counting</a><ul><li><a class="toc-href" href="#key-indexed-counting" title="Key-indexed counting">Key-indexed counting</a></li><li><a class="toc-href" href="#analysis" title="analysis">analysis</a></li></ul></li><li><a class="toc-href" href="#3-lsd-radix-sort_1" title="3. LSD Radix Sort">3. LSD Radix Sort</a><ul><li><a class="toc-href" href="#analysis_1" title="analysis">analysis</a></li><li><a class="toc-href" href="#implementation" title="implementation">implementation</a></li></ul></li><li><a class="toc-href" href="#4-msd-radix-sort_1" title="4. MSD Radix Sort">4. MSD Radix Sort</a><ul><li><a class="toc-href" href="#implementation_1" title="implementation">implementation</a></li><li><a class="toc-href" href="#analysis_2" title="analysis">analysis</a></li><li><a class="toc-href" href="#msd-vs-quicksort" title="MSD vs. quicksort">MSD vs. quicksort</a></li></ul></li><li><a class="toc-href" href="#5-3-way-radix-quicksort_1" title="5. 3-way Radix Quicksort">5. 3-way Radix Quicksort</a><ul><li><a class="toc-href" href="#implementation_2" title="Implementation">Implementation</a></li><li><a class="toc-href" href="#analysis_3" title="analysis">analysis</a></li></ul></li><li><a class="toc-href" href="#6-suffix-arrays_1" title="6. Suffix Arrays">6. Suffix Arrays</a><ul><li><a class="toc-href" href="#keyword-in-context-search" title="keyword-in-context search">keyword-in-context search</a></li><li><a class="toc-href" href="#longest-repeated-substring" title="longest repeated substring">longest repeated substring</a></li><li><a class="toc-href" href="#string-sorting-summery_1" title="String sorting summery">String sorting summery</a></li></ul></li></ul></div>
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</div>
<p>This week: string sort. </p>
<h1 id="1-strings-in-java">1. Strings in Java</h1>
<h3 id="char-data-type">char data type</h3>
<ul>
<li><strong>char in C</strong> </li>
</ul>
<p>8-bit integer, 256 characters, 7-bit ASCII code </p>
<ul>
<li><strong>char in Java</strong> </li>
</ul>
<p><em>16-bit Unicode</em> </p>
<h3 id="string-data-type">String data type</h3>
<p><code>String</code>: <em>immutable</em> sequence of characters<br/>
operations: lengthe, ith char, substring, concatenate </p>
<p>implementation: using a <code>char[]</code>, maintain a <code>length</code> and an <code>offset.</code> ⇒ substring methode is O(1) time.<br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image.png"/><br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image001.png"/> </p>
<h3 id="stringbuilder-data-type">StringBuilder data type</h3>
<p><code>StringBuilder</code>: <em>mutable</em> data type.<br/>
implementation: using a resizing <code>char[]</code> array (ArrayList). <br/>
⇒ contat in (amortized) constant time, <em>substring in linear time</em>!<br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image002.png"/> </p>
<p>ex. </p>
<ul>
<li>reverse a string: linear using StringBuilder, quad using String. </li>
<li>form an array of suffixes: quad (time&space) using StringBuilder, linear (time&space) using String. </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image003.png"/><br/>
<strong>Longest Common Prefix:</strong><br/>
runs in linear/subinear time ⇒ <code>compareTo()</code> for strings takes (sub)linear time! </p>
<h3 id="alphabet">Alphabet</h3>
<p>alphabet different for different type of string (ex. binary numbers, DNA, ...)<br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image004.png"/><br/>
def. <strong>Radix</strong> <code>R</code> is number of digits in alphabet. </p>
<h1 id="2-key-indexed-counting_1">2. Key-Indexed Counting</h1>
<p>review of <em>compare-based</em> sorting algorithms:<br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image005.png"/><br/>
lower-bound for compare-based algorithms: ~<em>NlgN</em> (=Lg(N!))<br/>
⇒ goal: do better by avoiding using compares. </p>
<h3 id="key-indexed-counting">Key-indexed counting</h3>
<p>is an algorithm to sort <em>by character(ex. sort array of string by their 1st character).</em> <br/>
Assumption: keys are integers between 0 and R-1 (<<N, small integer).<br/>
⇒ use keys as array index, to sort an <em>array of N integers between 0 and R-1</em>. </p>
<p><strong>[algo]</strong> </p>
<blockquote>
<ul>
<li>count freq of each key index (of size <em>R+1</em>) </li>
<li>using count array, compute a <em>cumulated freq</em> (cumsum of <code>count[]</code>) </li>
<li>the <code>cumsum[]</code> array stores the <strong>index range</strong> of each key index: </li>
</ul>
</blockquote>
<p>index range of key-i in sorted array is <code>[cumsum[i], cumsum[i+1]]</code> </p>
<ul>
<li>then get the sorted array by going through the array and using cumsum[] array </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image007.png"/>⇒ <img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image006.png"/> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span><span class="w"> </span><span class="n">void</span><span class="w"> </span><span class="n">keyIndexCounting</span><span class="p">(</span><span class="nc">int</span><span class="err">[]</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">R</span><span class="p">)</span><span class="err">{</span><span class="o">//</span><span class="w"> </span><span class="n">entries</span><span class="w"> </span><span class="ow">in</span><span class="w"> </span><span class="n">a</span><span class="err">[]</span><span class="w"> </span><span class="k">are</span><span class="w"> </span><span class="ow">in</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="o">[</span><span class="n">0,R-1</span><span class="o">]</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">N</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="p">.</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">int</span><span class="err">[]</span><span class="w"> </span><span class="nf">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nc">int</span><span class="o">[</span><span class="n">R+1</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="nf">count</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">freq</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">key</span><span class="w"> </span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nf">count</span><span class="o">[</span><span class="n">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</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">int</span><span class="w"> </span><span class="nl">ai</span><span class="p">:</span><span class="n">a</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="nf">count</span><span class="o">[</span><span class="n">ai+1</span><span class="o">]++</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="err">[]</span><span class="w"> </span><span class="n">cumsum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">count</span><span class="p">;</span><span class="w"> </span><span class="o">//</span><span class="n">cumsum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cumulated</span><span class="w"> </span><span class="n">freq</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">int</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="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 class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">cumsum</span><span class="o">[</span><span class="n">i+1</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">cumsum</span><span class="o">[</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="o">//</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="n">ranges</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">key</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="ow">in</span><span class="w"> </span><span class="n">sorted</span><span class="w"> </span><span class="k">array</span><span class="w"> </span><span class="n">should</span><span class="w"> </span><span class="n">be</span><span class="w"> </span><span class="o">[</span><span class="n">cumsum[i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">cumsum</span><span class="o">[</span><span class="n">i+1</span><span class="o">]</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="err">[]</span><span class="w"> </span><span class="n">aux</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nc">int</span><span class="o">[</span><span class="n">N</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">int</span><span class="w"> </span><span class="nl">ai</span><span class="p">:</span><span class="n">a</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">aux</span><span class="o">[</span><span class="n">cumsum[ai</span><span class="o">]++</span><span class="err">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ai</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">aux</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="analysis">analysis</h3>
<ul>
<li>running time: <strong>linear</strong> in time and in space. </li>
<li><strong>stable</strong> sorting: that's why we need the cumsum and aux array... </li>
</ul>
<h1 id="3-lsd-radix-sort_1">3. LSD Radix Sort</h1>
<p>LSD = least significant digit (for string sorting)<br/>
(assume strings all of same length)<br/>
idea: </p>
<ul>
<li>consider chars from right to left </li>
<li>sort using dth character as key (using key-indexed counting) </li>
</ul>
<p>important: the key-indexed counting should be stable.<br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image008.png"/> </p>
<h3 id="analysis_1">analysis</h3>
<p>time: W * N (W=length of string) </p>
<p>correctness:<br/>
<strong>prop</strong>. LSD sorting works<br/>
pf. by induction on i<br/>
prove that: after pass i, strings are sorted by last i characters. </p>
<h3 id="implementation">implementation</h3>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="n">LSDsort</span><span class="p">(</span><span class="n">String</span><span class="err">[]</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">W</span><span class="p">)</span><span class="err">{</span><span class="o">//</span><span class="n">W</span><span class="o">=</span><span class="n">fixed</span><span class="w"> </span><span class="n">width</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">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="o">//</span><span class="k">for</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="nc">int</span><span class="w"> </span><span class="n">N</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="p">.</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="n">String</span><span class="err">[]</span><span class="w"> </span><span class="n">aux</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">String</span><span class="o">[</span><span class="n">N</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">int</span><span class="w"> </span><span class="n">d</span><span class="o">=</span><span class="n">W</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="n">d</span><span class="o">>=</span><span class="mi">0</span><span class="p">;</span><span class="n">d</span><span class="c1">--){//W passes </span></span>
<span class="code-line"><span class="w"> </span><span class="o">//</span><span class="k">key</span><span class="o">-</span><span class="n">indexed</span><span class="w"> </span><span class="n">counting</span><span class="w"> </span><span class="k">at</span><span class="w"> </span><span class="n">digit</span><span class="w"> </span><span class="n">d</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">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nc">int</span><span class="o">[</span><span class="n">R+1</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="n">String</span><span class="w"> </span><span class="nl">s</span><span class="p">:</span><span class="n">a</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="nf">count</span><span class="o">[</span><span class="n">s.charAt[d</span><span class="o">]+</span><span class="mi">1</span><span class="err">]</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">int</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="n">i</span><span class="o"><</span><span class="n">N</span><span class="p">;</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="nf">count</span><span class="o">[</span><span class="n">i+1</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nf">count</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span><span class="o">//</span><span class="nf">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cumsum</span><span class="p">(</span><span class="nf">count</span><span class="p">),</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">each</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">for</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="nl">s</span><span class="p">:</span><span class="n">a</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">aux</span><span class="o">[</span><span class="n">s.charAt[d</span><span class="o">]++</span><span class="err">]</span><span class="o">=</span><span class="n">s</span><span class="p">;</span><span class="o">//</span><span class="n">put</span><span class="w"> </span><span class="k">each</span><span class="w"> </span><span class="k">key</span><span class="w"> </span><span class="ow">in</span><span class="w"> </span><span class="nf">right</span><span class="w"> </span><span class="n">place</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">int</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="n">i</span><span class="o"><</span><span class="n">n</span><span class="p">;</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">a</span><span class="o">[</span><span class="n">i</span><span class="o">]=</span><span class="n">aux</span><span class="o">[</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="err">}</span><span class="w"> </span></span>
<span class="code-line"><span class="err">}</span><span class="w"></span></span>
</pre></div>
<ul>
<li>if keys are binary numbers → break into bit characters then apply LSD. </li>
<li>if not fixed length → some fix.. </li>
<li>sort 1 million 32-bit integers? (Google/Obama interview) ⇒ LSD string sort~ </li>
</ul>
<h1 id="4-msd-radix-sort_1">4. MSD Radix Sort</h1>
<p>most-significant-digit first (from left to right)<br/>
Idea. </p>
<ul>
<li>partition array into R pieces according to first character (the count[] array is the partition) </li>
<li><em>recursively</em> sort each subarrays </li>
</ul>
<p>variable length: end-of-string are treated as before any char<br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image010.png"/> </p>
<h3 id="implementation_1">implementation</h3>
<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">void</span><span class="w"> </span><span class="n">sort</span><span class="p">(</span><span class="n">String</span><span class="err">[]</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="err">[]</span><span class="w"> </span><span class="n">aux</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">lo</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">hi</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="k">if</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="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="o">//</span><span class="n">sort</span><span class="w"> </span><span class="k">by</span><span class="w"> </span><span class="n">dth</span><span class="w"> </span><span class="k">character</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">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nc">int</span><span class="o">[</span><span class="n">R+2</span><span class="o">]</span><span class="p">;</span><span class="o">//</span><span class="w"> </span><span class="n">R</span><span class="o">+</span><span class="mi">2</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="n">we</span><span class="w"> </span><span class="k">are</span><span class="w"> </span><span class="n">taking</span><span class="w"> </span><span class="n">account</span><span class="w"> </span><span class="k">into</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="k">end</span><span class="o">-</span><span class="k">of</span><span class="o">-</span><span class="n">strings</span><span class="err">!!</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">int</span><span class="w"> </span><span class="n">i</span><span class="o">=</span><span class="n">lo</span><span class="p">;</span><span class="n">i</span><span class="o"><=</span><span class="n">hi</span><span class="p">;</span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="nf">count</span><span class="o">[</span><span class="n"> a[i</span><span class="o">]</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="o">+</span><span class="mi">2</span><span class="w"> </span><span class="err">]</span><span class="o">++</span><span class="p">;</span><span class="o">//</span><span class="n">charAt</span><span class="p">(</span><span class="k">end</span><span class="o">-</span><span class="k">of</span><span class="o">-</span><span class="n">string</span><span class="p">)</span><span class="o">=-</span><span class="mi">1</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">int</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="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 class="w"> </span><span class="nf">count</span><span class="o">[</span><span class="n">i+1</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nf">count</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="o">//</span><span class="n">cumsum</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">int</span><span class="w"> </span><span class="n">i</span><span class="o">=</span><span class="n">lo</span><span class="p">;</span><span class="n">i</span><span class="o"><=</span><span class="n">hi</span><span class="p">;</span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="n">aux</span><span class="o">[</span><span class="n">count[a[i</span><span class="o">]</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="o">+</span><span class="mi">1</span><span class="err">]</span><span class="o">++</span><span class="err">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="o">[</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="k">for</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">i</span><span class="o">=</span><span class="n">lo</span><span class="p">;</span><span class="n">i</span><span class="o"><=</span><span class="n">hi</span><span class="p">;</span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">aux</span><span class="o">[</span><span class="n">i-lo</span><span class="o">]</span><span class="p">;</span><span class="o">//</span><span class="w"> </span><span class="nl">attention</span><span class="p">:</span><span class="w"> </span><span class="n">aux</span><span class="w"> </span><span class="k">is</span><span class="w"> </span><span class="n">filled</span><span class="w"> </span><span class="k">from</span><span class="w"> </span><span class="k">index</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="n">instead</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">lo</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="o">//</span><span class="n">recursively</span><span class="w"> </span><span class="n">sort</span><span class="w"> </span><span class="k">each</span><span class="w"> </span><span class="n">subarray</span><span class="w"> </span><span class="p">(</span><span class="n">R</span><span class="w"> </span><span class="n">subarrays</span><span class="w"> </span><span class="ow">in</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="n">ranges</span><span class="w"> </span><span class="n">stored</span><span class="w"> </span><span class="ow">in</span><span class="w"> </span><span class="nf">count</span><span class="err">[]</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">int</span><span class="w"> </span><span class="n">r</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="n">r</span><span class="o"><</span><span class="n">R</span><span class="p">;</span><span class="n">r</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">sort</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">aux</span><span class="p">,</span><span class="n">lo</span><span class="o">+</span><span class="nf">count</span><span class="o">[</span><span class="n">r</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">lo</span><span class="o">+</span><span class="nf">count</span><span class="o">[</span><span class="n">r+1</span><span class="o">]-</span><span class="mi">1</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="err">}</span><span class="w"></span></span>
</pre></div>
<h3 id="analysis_2">analysis</h3>
<ul>
<li>can recycle <code>aux[]</code>, but <em>not</em> <code>count[]</code>. </li>
<li>too slow for small subarrays (if len(subarray)<<R) </li>
<li>huge nb of subarrays because of recursion </li>
</ul>
<p>improvement ⇒ cutoff to insertion sort... </p>
<p>characteristics of MSD sort: <br/>
examines <strong>just enough chars</strong> to sort. <br/>
→ can be sublinear in N. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image011.png"/> </p>
<h3 id="msd-vs-quicksort">MSD vs. quicksort</h3>
<p>disadvantages for MSD: </p>
<ul>
<li>random access of memory (cache inefficient) </li>
<li>too many instructions in inner loop </li>
<li>extra space for count[] </li>
<li>extra space for aux[] </li>
</ul>
<p>disadvantages for qsort: </p>
<ul>
<li>NlgN nb of string compares </li>
<li>has to rescan many chars for keys with <em>long prefix matches</em> </li>
</ul>
<h1 id="5-3-way-radix-quicksort_1">5. 3-way Radix Quicksort</h1>
<p><em>⇒ combine benefits of qsort and MSD.</em> <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image015.png"/><br/>
idea: do <strong>3-way partition by the dth character</strong>. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image013.png"/> </p>
<ul>
<li>less overhead than <em>R-way</em> partitioning for MSD </li>
<li>do not re-examine chars equal to the partitioning char </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image014.png"/> </p>
<h3 id="implementation_2">Implementation</h3>
<p>modification of the 3-way qsort. </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">sort</span><span class="p">(</span><span class="n">String</span><span class="err">[]</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">lo</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">hi</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="k">if</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="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="nc">int</span><span class="w"> </span><span class="n">lt</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">lo</span><span class="p">,</span><span class="w"> </span><span class="n">gt</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hi</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">=</span><span class="n">lo</span><span class="o">+</span><span class="mi">1</span><span class="p">;</span><span class="o">//</span><span class="w"> </span><span class="k">use</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="nl">pointers</span><span class="p">:</span><span class="w"> </span><span class="n">lt</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">gt</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="k">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">lo</span><span class="o">]</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">while</span><span class="p">(</span><span class="n">i</span><span class="o"><=</span><span class="n">gt</span><span class="p">)</span><span class="err">{</span><span class="o">//</span><span class="nl">invariant</span><span class="p">:</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">lo,lt)<pivot, a(gt,hi</span><span class="o">]></span><span class="k">pivot</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">lt,i</span><span class="o">]=</span><span class="k">pivot</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">a</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">.</span><span class="n">charAt</span><span class="o">[</span><span class="n">d</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">c</span><span class="o"><</span><span class="k">pivot</span><span class="p">)</span><span class="w"> </span><span class="n">exch</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">lt</span><span class="o">++</span><span class="p">,</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="k">else</span><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">c</span><span class="o">></span><span class="k">pivot</span><span class="p">)</span><span class="w"> </span><span class="n">exch</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">i</span><span class="p">,</span><span class="n">gt</span><span class="c1">--); </span></span>
<span class="code-line"><span class="w"> </span><span class="k">else</span><span class="err">:</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="err">}</span><span class="o">//</span><span class="mi">3</span><span class="o">-</span><span class="n">way</span><span class="w"> </span><span class="k">partition</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">sort</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">lo</span><span class="p">,</span><span class="n">lt</span><span class="o">-</span><span class="mi">1</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">if</span><span class="p">(</span><span class="k">pivot</span><span class="o">>=</span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="n">sort</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">lt</span><span class="p">,</span><span class="n">gt</span><span class="p">,</span><span class="n">d</span><span class="o">+</span><span class="mi">1</span><span class="p">);</span><span class="o">//</span><span class="k">pivot</span><span class="o"><</span><span class="mi">0</span><span class="w"> </span><span class="n">means</span><span class="w"> </span><span class="k">end</span><span class="o">-</span><span class="k">of</span><span class="o">-</span><span class="n">string</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">sort</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">gt</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">hi</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="err">}</span><span class="w"></span></span>
</pre></div>
<h3 id="analysis_3">analysis</h3>
<p><strong>wrt. qsort:</strong> </p>
<ul>
<li>from NlgN <strong>string</strong> compares to NlgN <strong>char</strong> compares. </li>
<li>avoids re-comparing long common prefixes. </li>
</ul>
<p><strong>wrt. MSD:</strong> </p>
<ul>
<li>short inner loop </li>
<li>cache friendly </li>
<li>in-place </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image016.png"/> </p>
<h1 id="6-suffix-arrays_1">6. Suffix Arrays</h1>
<p>(<em>some applications of suffix array)</em> </p>
<h3 id="keyword-in-context-search">keyword-in-context search</h3>
<p>given N chars (N huge), <em>preprocess</em> it to enable fast substring search. <br/>
⇒ suffix sort </p>
<ul>
<li>generate suffix array (linear time & space ) </li>
<li>sort on the suffix ⇒ brings repeated suffixes together </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image017.png"/> </p>
<h3 id="longest-repeated-substring">longest repeated substring</h3>
<ul>
<li>brute force algorithm </li>
</ul>
<p>try all i, j as starts of indices, then compute longest common prefix (LCP)<br/>
→ O(D*N2) where D is length of longest repeated substring. </p>
<ul>
<li>⇒ use suffix array </li>
</ul>
<p>sorting suffix array will bring repeated substrings together<br/>
java code: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="o">//</span><span class="nc">int</span><span class="w"> </span><span class="n">lcp</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="n">s1</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">s2</span><span class="p">)</span><span class="w"> </span><span class="n">defined</span><span class="w"> </span></span>
<span class="code-line"><span class="k">public</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="n">lrs</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="n">N</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">length</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">String</span><span class="err">[]</span><span class="w"> </span><span class="n">suffix</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">String</span><span class="o">[</span><span class="n">N</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">int</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="n">i</span><span class="o"><</span><span class="n">N</span><span class="p">;</span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="n">suffix</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</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="n">i</span><span class="p">);</span><span class="o">//</span><span class="n">construct</span><span class="w"> </span><span class="n">suffix</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">Arrays</span><span class="p">.</span><span class="n">sort</span><span class="p">(</span><span class="n">suffix</span><span class="p">);</span><span class="o">//</span><span class="w"> </span><span class="n">sort</span><span class="w"> </span><span class="n">suffix</span><span class="w"> </span><span class="k">array</span><span class="w"> </span><span class="o">==></span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="mi">3</span><span class="o">-</span><span class="n">way</span><span class="w"> </span><span class="n">radix</span><span class="w"> </span><span class="n">sort</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="n">one</span><span class="w"> </span><span class="n">pass</span><span class="w"> </span><span class="k">to</span><span class="w"> </span><span class="k">get</span><span class="w"> </span><span class="n">longest</span><span class="w"> </span><span class="n">repeated</span><span class="w"> </span><span class="nf">substring</span><span class="w"> </span><span class="p">(</span><span class="n">bigest</span><span class="w"> </span><span class="n">lcp</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">lrs</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ss">""</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">int</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="n">i</span><span class="o"><</span><span class="n">N</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="n">i</span><span class="o">++</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">lcp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">lcp</span><span class="p">(</span><span class="n">suffix</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">suffix</span><span class="o">[</span><span class="n">i+1</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">lcp</span><span class="o">></span><span class="n">lrs</span><span class="p">.</span><span class="n">length</span><span class="p">)</span><span class="w"> </span><span class="n">lrs</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">suffix</span><span class="o">[</span><span class="n">i</span><span class="o">]</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="n">lcp</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">lrs</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>lrs worst-case input</strong>: <em>lrs very long</em> (say N/2). <br/>
⇒ <em>quadratic</em> for lrs and for sorting. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week3_2/pasted_image018.png"/> </p>
<p>→ <em>improvement of lrs for worst-case performance:</em><br/>
<strong>Manber-Myers algo</strong><br/>
sketch: </p>
<ul>
<li>phase 0: sort suffix[] on 1st char </li>
<li>phase i: given suffix[] is sorted based on first 2^(i-1) chars → create suffix[] sorted on first 2^i chars. </li>
</ul>
<p>each phase: double the nb of chars sorted on. <br/>
maintain an inverse[] array, to make comparisons constant time. <br/>
performance: NlgN </p>
<h2 id="string-sorting-summery_1">String sorting summery</h2>
<ul>
<li>can have linear sort: use chars as array index </li>
<li>sublinear sort: not all date need to be examined </li>
<li>3-way radix qsort is <em>asymptotically optimal</em> </li>
</ul>
</div>
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Part 6 of series «Algorithms Princeton MOOC II»：
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<ul class="list-group">
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week1_1.html' style="text-align:left">[Algorithms II] Week 1-1 Undirected Graphs</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week1_2.html' style="text-align:left">[Algorithms II] Week 1-2 Directed Graphs</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week2_1.html' style="text-align:left">[Algorithms II] Week 2-1 Minimum Spanning Trees</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week2_2.html' style="text-align:left">[Algorithms II] Week 2-2 Shortest Paths</a></li>
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<div id="toc"><ul><li><a class="toc-href" href="#1-strings-in-java" title="1. Strings in Java">1. Strings in Java</a><ul><li><a class="toc-href" href="#char-data-type" title="char data type">char data type</a></li><li><a class="toc-href" href="#string-data-type" title="String data type">String data type</a></li><li><a class="toc-href" href="#stringbuilder-data-type" title="StringBuilder data type">StringBuilder data type</a></li><li><a class="toc-href" href="#alphabet" title="Alphabet">Alphabet</a></li></ul></li><li><a class="toc-href" href="#2-key-indexed-counting_1" title="2. Key-Indexed Counting">2. Key-Indexed Counting</a><ul><li><a class="toc-href" href="#key-indexed-counting" title="Key-indexed counting">Key-indexed counting</a></li><li><a class="toc-href" href="#analysis" title="analysis">analysis</a></li></ul></li><li><a class="toc-href" href="#3-lsd-radix-sort_1" title="3. LSD Radix Sort">3. LSD Radix Sort</a><ul><li><a class="toc-href" href="#analysis_1" title="analysis">analysis</a></li><li><a class="toc-href" href="#implementation" title="implementation">implementation</a></li></ul></li><li><a class="toc-href" href="#4-msd-radix-sort_1" title="4. MSD Radix Sort">4. MSD Radix Sort</a><ul><li><a class="toc-href" href="#implementation_1" title="implementation">implementation</a></li><li><a class="toc-href" href="#analysis_2" title="analysis">analysis</a></li><li><a class="toc-href" href="#msd-vs-quicksort" title="MSD vs. quicksort">MSD vs. quicksort</a></li></ul></li><li><a class="toc-href" href="#5-3-way-radix-quicksort_1" title="5. 3-way Radix Quicksort">5. 3-way Radix Quicksort</a><ul><li><a class="toc-href" href="#implementation_2" title="Implementation">Implementation</a></li><li><a class="toc-href" href="#analysis_3" title="analysis">analysis</a></li></ul></li><li><a class="toc-href" href="#6-suffix-arrays_1" title="6. Suffix Arrays">6. Suffix Arrays</a><ul><li><a class="toc-href" href="#keyword-in-context-search" title="keyword-in-context search">keyword-in-context search</a></li><li><a class="toc-href" href="#longest-repeated-substring" title="longest repeated substring">longest repeated substring</a></li><li><a class="toc-href" href="#string-sorting-summery_1" title="String sorting summery">String sorting summery</a></li></ul></li></ul></div>
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