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[Algorithms I] Week 2-2 Elementary Sorts
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-introduction" title="1. Introduction">1. Introduction</a><ul><li><a class="toc-href" href="#callback" title="callback">callback</a></li><li><a class="toc-href" href="#helper-functions" title="helper functions">helper functions</a></li></ul></li><li><a class="toc-href" href="#2-selection-sort_1" title="2. Selection Sort">2. Selection Sort</a><ul><li><a class="toc-href" href="#invariants" title="invariants">invariants</a></li><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-insertion-sort_1" title="3. Insertion sort">3. Insertion sort</a><ul><li><a class="toc-href" href="#invariants_1" title="invariants">invariants</a></li><li><a class="toc-href" href="#implementation_1" title="implementation">implementation</a></li><li><a class="toc-href" href="#analysis_1" title="analysis">analysis</a><ul><li><a class="toc-href" href="#best-case-and-worst-case" title="best case and worst case">best case and worst case</a></li><li><a class="toc-href" href="#partially-sorted-arrays" title="partially sorted arrays">partially sorted arrays</a></li></ul></li></ul></li><li><a class="toc-href" href="#4-shell-sort_2" title="4. Shell Sort">4. Shell Sort</a><ul><li><a class="toc-href" href="#implementation_2" title="implementation">implementation</a></li><li><a class="toc-href" href="#analysis_2" title="analysis">analysis</a></li></ul></li><li><a class="toc-href" href="#of-compares-cte-n-of-h-used_1" title="of compares < Cte * N * (# of h used )">of compares < Cte * N * (# of h used )</a></li><li><a class="toc-href" href="#5-shuffling" title="5. shuffling">5. shuffling</a><ul><li><a class="toc-href" href="#shuffle-array-using-sort" title="shuffle array using sort">shuffle array using sort</a></li><li><a class="toc-href" href="#knuth-shuffle" title="Knuth shuffle">Knuth shuffle</a></li><li><a class="toc-href" href="#example-online-poker" title="example: online poker">example: online poker</a></li></ul></li><li><a class="toc-href" href="#6-convex-hull_1" title="6. Convex Hull">6. Convex Hull</a><ul><li><a class="toc-href" href="#convex-hull" title="convex hull">convex hull</a></li><li><a class="toc-href" href="#algo" title="Algo">Algo</a></li><li><a class="toc-href" href="#implementation_3" title="implementation">implementation</a></li></ul></li></ul></div>
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</div>
<h1 id="1-introduction">1. Introduction</h1>
<p>rearanging array of size N into ascending order <br/>
test client code: <code>Insertion.sort(a);</code> </p>
<p>sort <em>any</em> datatype </p>
<h3 id="callback">callback</h3>
<p><em>callback = reference to executable code</em> <br/>
i.e. passing functions as argument to sort() method<br/>
sort() function calls object's <code>compareTo()</code> method <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image.png"/> <br/>
→ implement the <code>Comparable</code> interface: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err"> public class XX implements Comparable<XX>{</span></span>
<span class="code-line"><span class="err"> ...</span></span>
<span class="code-line"><span class="err"> }</span></span>
</pre></div>
<p>the interface: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err"> public interface Comparable<Item>{</span></span>
<span class="code-line"><span class="err"> public int compareTo(Item that);</span></span>
<span class="code-line"><span class="err"> }</span></span>
</pre></div>
<p>compareTo():<br/>
return -1 (if this<that)/+1/0;
needs a <em>total order.</em>
→ in the sort() implementation:<br/>
has not dependencies on type of data. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="w"> </span><span class="k">public</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">Comparable</span><span class="err">[]</span><span class="w"> </span><span class="n">a</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">a</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">.</span><span class="n">compareTo</span><span class="p">(</span><span class="n">a</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">)</span><span class="o">></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="err">}</span><span class="w"></span></span>
</pre></div>
<h3 id="helper-functions">helper functions</h3>
<ul>
<li>
<p>less</p>
<p>private static boolean less(Comparable v, Comparable u){
returnv.compareTo(u)<0;
}</p>
</li>
<li>
<p>exch</p>
<p>private void exch(Comparable[] a, int i, int j) {
Comparable swap = a[i];
a[i] = a[j];
a[j] = swap;
}</p>
</li>
<li>
<p>isSorted</p>
</li>
</ul>
<p>test if sorted
if algo passes the test using only less ant swap, then it's correct. </p>
<h1 id="2-selection-sort_1">2. Selection Sort</h1>
<p>Idea: each time<em>: find the minimum from the remaining items.</em>
<strong>a[min] is the smallest element to right of a[i] ⇒ swap a[i] and a[min]</strong> (elements to left of i are sorted) <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image002.png"/> </p>
<h2 id="invariants">invariants</h2>
<ul>
<li>entries to the left of i are in sorted order, and are fixed (<em>in final position</em>) ever since</li>
<li>no entry to the right of i is smaller than any entry to the left of i</li>
</ul>
<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">SelectionSort</span><span class="w"> </span><span class="n">extends</span><span class="w"> </span><span class="n">AbstractSorting</span><span class="err">{</span><span class="w"></span></span>
<span class="code-line"><span class="o">//</span><span class="p">...</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">void</span><span class="w"> </span><span class="n">sort</span><span class="p">(</span><span class="n">Comparable</span><span class="err">[]</span><span class="w"> </span><span class="n">a</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">for</span><span class="p">(</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="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o"><</span><span class="n">a</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </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="nf">min</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</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">j</span><span class="w"> </span><span class="o">=</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="n">j</span><span class="o"><</span><span class="n">a</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">j</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="k">less</span><span class="p">(</span><span class="n">a</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">,</span><span class="n">a</span><span class="o">[</span><span class="n">min</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">min</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="n">exch</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="nf">min</span><span class="p">,</span><span class="n">i</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>
<span class="code-line"><span class="err">}</span><span class="w"></span></span>
</pre></div>
<h2 id="analysis">analysis</h2>
<p><strong>proposition:</strong>
selection sort uses <em>N-1 + N-2 + ... + 1 = ~N^2/2</em> compares, and <em>N</em> exchanges.
→ quadratic time</p>
<p><img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image003.png"/></p>
<ul>
<li><strong>insensitive</strong> to input: quadratic time <em>even if input is already sorted.</em> </li>
<li>data movement is <em>minimum</em>: linear time of exchanges (every exchange puts an item to its final position)</li>
</ul>
<h1 id="3-insertion-sort_1">3. Insertion sort</h1>
<p>quite different performance characteritics than selection sort. </p>
<p>Idea: <strong>In iteration i: move all entries larger than a[i] to its left.</strong> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image004.png"/> </p>
<h2 id="invariants_1">invariants</h2>
<ul>
<li>entries to the left of i are in ascending order (but <em>not</em> in final position)</li>
<li>entries to the right of i are not yet been seen </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image005.png"/> </p>
<h2 id="implementation_1">implementation</h2>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">publc</span><span class="w"> </span><span class="k">class</span><span class="w"> </span><span class="n">InsertionSorting</span><span class="w"> </span><span class="n">extends</span><span class="w"> </span><span class="n">AbstractSorting</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">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">Comparable</span><span class="err">[]</span><span class="w"> </span><span class="n">a</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">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">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o"><</span><span class="n">a</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="k">for</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">j</span><span class="o">=</span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">></span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="c1">--){</span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="k">less</span><span class="p">(</span><span class="n">a</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">,</span><span class="n">a</span><span class="o">[</span><span class="n">j-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="n">exch</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">j</span><span class="p">,</span><span class="n">j</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="k">break</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="err">}</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="o">//</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="analysis_1">analysis</h2>
<p><strong>proposition </strong>(average case): <br/>
(the performance <em>on average</em> — <em>for randomly sorted array</em> ) <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image007.png"/> <br/>
<em>proof:</em>
<em>expect each entry to move halfway back</em> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image008.png"/> </p>
<h3 id="best-case-and-worst-case">best case and worst case</h3>
<p><strong>best case</strong>
if array already sorted min ascending order:
<em>N-1 compares, 0 exchanges.</em> </p>
<p><strong>worst case</strong>
if array sorted in descending order:
every element goes all the way back → 1/2<em>N^2 compares, 1/2</em>N^2 exchanges</p>
<h3 id="partially-sorted-arrays">partially sorted arrays</h3>
<p>def.<strong>" inversion"</strong>
an inversion is a pair of entries that are out of order. </p>
<p>def. <strong>"partially sorted"</strong><br/>
An array is called partially sorted if the <em>number of inversions is <= c</em>N. *</p>
<p><strong>proposition. </strong><br/>
<strong>Insertion sort runs in linear time for partially sorted array.</strong>
<em>proof.</em>
<em>number of exchanges = number of inversions. </em>
<em>number of compares = number of exchanges + N-1</em></p>
<h1 id="4-shell-sort_2">4. Shell Sort</h1>
<p>First non-trival sorting methode: an improvement of insertion sort. </p>
<p>def. <strong>"h-sorted array"</strong><br/>
an array is h-sorted if every h-interleaved subarray is sorted. (h=1: just a sorted array) <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image009.png"/> </p>
<p>Idea: move entries >1 position at a time by <em>h-sorting</em> the array, then decrease h. </p>
<p>use decreasing sequences of value h: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image010.png"/> </p>
<h2 id="implementation_2">implementation</h2>
<p><strong>How to h -sort </strong><br/>
simply insertion sort with <em>stride length=h</em>.<br/>
why insertion sort: <br/>
for big h: small subarray <br/>
for small h: nearly in order </p>
<p><strong>proposition</strong><br/>
A g-sorted array <em>remains g-sorted</em> after h-sorting it. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image011.png"/> <br/>
(subtle to prove...) </p>
<p><strong>which sequence of h to use</strong>
<code>3x+1</code> sequence proposed by Knuth. <em>1,4,13,40....</em></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">ShellSort</span><span class="w"> </span><span class="n">extends</span><span class="w"> </span><span class="n">AbstractSort</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">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">Comparable</span><span class="err">[]</span><span class="w"> </span><span class="n">a</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="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">N</span><span class="o">=</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="k">while</span><span class="p">(</span><span class="n">h</span><span class="o"><</span><span class="n">N</span><span class="o">/</span><span class="mi">3</span><span class="p">)</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">h</span><span class="o">*</span><span class="mi">3</span><span class="o">+</span><span class="mi">1</span><span class="p">;</span><span class="o">//</span><span class="n">find</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="n">beginning</span><span class="w"> </span><span class="n">h</span><span class="w"> </span><span class="p">(</span><span class="n">N</span><span class="o">></span><span class="n">h</span><span class="o">></span><span class="n">N</span><span class="o">/</span><span class="mi">3</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">h</span><span class="o">>=</span><span class="mi">1</span><span class="p">)</span><span class="err">{</span><span class="o">//</span><span class="n">performs</span><span class="w"> </span><span class="n">h</span><span class="o">-</span><span class="n">sort</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="w"> </span><span class="n">h</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="n">h</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">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="n">j</span><span class="o">-</span><span class="n">h</span><span class="o">>=</span><span class="mi">0</span><span class="p">;</span><span class="n">j</span><span class="o">-=</span><span class="n">h</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="w"> </span><span class="k">less</span><span class="p">(</span><span class="n">a</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">,</span><span class="n">a</span><span class="o">[</span><span class="n">j-h</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">)</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="n">exch</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">j</span><span class="p">,</span><span class="n">j</span><span class="o">-</span><span class="n">h</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">break</span><span class="p">;</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="n">h</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">h</span><span class="o">/</span><span class="mi">3</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="err">}</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </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">boolean</span><span class="w"> </span><span class="n">isHsorted</span><span class="p">(</span><span class="n">Comparable</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">h</span><span class="p">)</span><span class="w"> </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="w"> </span><span class="p">(</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">h</span><span class="p">;</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">a</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">less</span><span class="p">(</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 class="n">a</span><span class="o">[</span><span class="n">i-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="k">false</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">true</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>每次hsort, 外围的循环是<code>for(int i= h;i<N;i+=h)</code>, 需要理解一下: i移动一次以后, 进行的是另一个subarray 的插入排序, 当移动到N-1的时候所有subarray的插入排序才结束. (也就是说不是先完成一个subarray的插入排序再完成另一个, 这些是插入排序是同步进行的)</p>
<h2 id="analysis_2">analysis</h2>
<p><strong>proposition (for worst case )</strong><br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image012.png"/> <br/>
→ better than quadratic time ! </p>
<p><strong>property (found in practice)</strong></p>
<h1 id="of-compares-cte-n-of-h-used_1">of compares < Cte * N * (# of <em>h</em> used )</h1>
<p>→ #compares < NlgN * Cte <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image013.png"/> <br/>
*accurate model has not been discovered *</p>
<p>(所以shellsort在实际使用中几乎和快速排序一样快! — 尽管没有数学证明来保证)</p>
<p><strong>why we are interested in shell sort</strong>
useful in practice: </p>
<ul>
<li>fast for medium sized arrays (beat even the classical sophistiated algorithms) </li>
<li>tiny code volumn (used in embeded systems) </li>
</ul>
<p>lead to interesting questions for 50 years: </p>
<ul>
<li>asymptotic growth rate ?</li>
<li>best sequence of h ?</li>
<li>average case performance ?</li>
</ul>
<h1 id="5-shuffling">5. shuffling</h1>
<h2 id="shuffle-array-using-sort">shuffle array using sort</h2>
<p>one way to shuffle an array: </p>
<ol>
<li>for each array entry, generate a random real number</li>
<li>sort the array of real numbers</li>
<li>⇒ the original array is shuffled !</li>
</ol>
<p><strong>proposition </strong><br/>
this shuffle sort produces a uniformly random permutation of input array <br/>
<em>drawback: cost for sorting...</em></p>
<p>Goal: get uniformly random permutation in <em>linear time</em>. </p>
<h2 id="knuth-shuffle">Knuth shuffle</h2>
<p>algo: </p>
<blockquote>
<p>for i = [0,N):<br/>
- r = rand( [0~i<strong>]</strong> ) or rand( <strong>[</strong>i, N-1] ) <br/>
- swap a[r] and a[i] </p>
</blockquote>
<p><img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image014.png"/></p>
<p>implementation: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public static void shuffle(Object[] a){</span></span>
<span class="code-line"><span class="err"> for(int i=0;i<a.length;i++){</span></span>
<span class="code-line"><span class="err"> int r = StdRandom.uniform(i+1);</span></span>
<span class="code-line"><span class="err"> exch(a,r,i);</span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p><strong>proposition</strong><br/>
Knuth algo produces an uniformly random permutation of input array. </p>
<p><em>proof.</em><br/>
Sufficient to prove that, for card i and position j, the proba(card i comes to position j) = 1/N. </p>
<ol>
<li>if i<=j, P = 1/j * j/(j+1) * (j+1)/(j+2) * ... * (N-1)/N</li>
<li>if j<i, P = 1/i * i/(i+1) * (i+1)/(i+2) * ... * (N-1)/N</li>
</ol>
<p>CQFD.</p>
<h2 id="example-online-poker">example: online poker</h2>
<p><a href="https://www.cigital.com/papers/download/developer_gambling.php">https://www.cigital.com/papers/download/developer_gambling.php</a> ←那个扑克网站已经被黑出翔了... <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image015.png"/> <br/>
bugs: </p>
<ol>
<li>r never get 52 (52th card never moved)</li>
<li>r = rand(N) instead of rand(0~i), → shuffle not uniform</li>
<li>random() uses 32bit seed: only 2^32 possible shuffles, <em>2^32<52!</em></li>
<li>seed = millisec from midnight, ~86*10^6 suffles</li>
</ol>
<h1 id="6-convex-hull_1">6. Convex Hull</h1>
<p>application of sorting for the field of computational geometry. </p>
<h2 id="convex-hull">convex hull</h2>
<p>smallest polygoneenclosing all N points. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image017.png"/> </p>
<ul>
<li>input: N points</li>
<li>output: sequence of <em>vertices</em> in counterclockwise (<em>ccw</em>) order. </li>
</ul>
<p>application: robot motion planning; farest pair. </p>
<p><strong>geometric properties</strong>: </p>
<ul>
<li>can traverse convex hull by making only ccw turns</li>
<li>let p be the point with lowest y-coord, wrt p, vertices appear in increasing order of polar angle. </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image018.png"/> </p>
<h2 id="algo">Algo</h2>
<blockquote>
<p><strong>Graham scan algorithm:</strong><br/>
* choose p with smallest y coord <br/>
* <em>sort points by polar angle</em> with p <br/>
* consider points in order (stack is used), <em>discard unless creates a ccw turn</em>. </p>
</blockquote>
<p><img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image019.png"/> <img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image020.png"/> <img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image021.png"/> </p>
<p><strong>CCW</strong> <br/>
given three points a b c, returns if a→b→c is a CCW turn. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image023.png"/> <br/>
(assumption: no 3 points on a line)<br/>
⇒ calculate <em>cross product of ab and bc ⇒ determinants!</em> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week2_2/pasted_image024.png"/> <br/>
area>0 ⇔ CCW </p>
<h2 id="implementation_3">implementation</h2>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span> <span class="k">class</span> <span class="n">Point2D</span><span class="err">{</span></span>
<span class="code-line"><span class="n">private</span> <span class="n">double</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">;</span></span>
<span class="code-line"><span class="k">public</span> <span class="k">static</span> <span class="nb">boolean</span> <span class="n">ccw</span><span class="p">(</span><span class="n">Point2D</span> <span class="n">a</span><span class="p">,</span><span class="n">Point2D</span> <span class="n">b</span><span class="p">,</span><span class="n">Point2D</span> <span class="k">c</span><span class="p">)</span><span class="err">{</span></span>
<span class="code-line"> <span class="n">double</span> <span class="n">area2</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="p">.</span><span class="n">x</span><span class="o">-</span><span class="n">a</span><span class="p">.</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="k">c</span><span class="p">.</span><span class="n">y</span><span class="o">-</span><span class="n">a</span><span class="p">.</span><span class="n">y</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">b</span><span class="p">.</span><span class="n">y</span><span class="o">-</span><span class="n">a</span><span class="p">.</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="k">c</span><span class="p">.</span><span class="n">x</span><span class="o">-</span><span class="n">a</span><span class="p">.</span><span class="n">x</span><span class="p">);</span></span>
<span class="code-line"> <span class="k">return</span> <span class="n">area2</span><span class="o">></span><span class="mi">0</span><span class="p">;</span></span>
<span class="code-line"><span class="err">}</span></span>
<span class="code-line"></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>convex hull: </p>
<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">Stack</span><span class="o"><</span><span class="n">Point2D</span><span class="o">></span><span class="w"> </span><span class="n">GrahamScan</span><span class="p">(</span><span class="n">Point2D</span><span class="err">[]</span><span class="w"> </span><span class="n">p</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="cm">/* assumes that points are sorted by polar angle in p[]</span></span>
<span class="code-line"><span class="cm"> Stack<Point2D> hull = new Stack<Point2D>();</span></span>
<span class="code-line"><span class="cm"> hull.push(p[0]);</span></span>
<span class="code-line"><span class="cm"> hull.push(p[1]);</span></span>
<span class="code-line"><span class="cm"> for(int i=2;i<p.length;i++){</span></span>
<span class="code-line"><span class="cm"> Point2D b = hull.pop(), a = hull.peek(), c = p[i];</span></span>
<span class="code-line"><span class="cm"> while(!Point2D.ccw(a,b,c)){</span></span>
<span class="code-line"><span class="cm"> b = hull.pop();</span></span>
<span class="code-line"><span class="cm"> a = hull.peek();</span></span>
<span class="code-line"><span class="cm"> }</span></span>
<span class="code-line"><span class="cm"> //now a,b,c makes a ccw turn:</span></span>
<span class="code-line"><span class="cm"> hull.push(b);</span></span>
<span class="code-line"><span class="cm"> hull.push(c);</span></span>
<span class="code-line"><span class="cm"> }</span></span>
<span class="code-line"><span class="cm">}</span></span>
</pre></div>
<p>running time: <em>NlgN for sorting and linear for the rest.</em> </p>
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目录</h4>
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<div id="toc"><ul><li><a class="toc-href" href="#1-introduction" title="1. Introduction">1. Introduction</a><ul><li><a class="toc-href" href="#callback" title="callback">callback</a></li><li><a class="toc-href" href="#helper-functions" title="helper functions">helper functions</a></li></ul></li><li><a class="toc-href" href="#2-selection-sort_1" title="2. Selection Sort">2. Selection Sort</a><ul><li><a class="toc-href" href="#invariants" title="invariants">invariants</a></li><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-insertion-sort_1" title="3. Insertion sort">3. Insertion sort</a><ul><li><a class="toc-href" href="#invariants_1" title="invariants">invariants</a></li><li><a class="toc-href" href="#implementation_1" title="implementation">implementation</a></li><li><a class="toc-href" href="#analysis_1" title="analysis">analysis</a><ul><li><a class="toc-href" href="#best-case-and-worst-case" title="best case and worst case">best case and worst case</a></li><li><a class="toc-href" href="#partially-sorted-arrays" title="partially sorted arrays">partially sorted arrays</a></li></ul></li></ul></li><li><a class="toc-href" href="#4-shell-sort_2" title="4. Shell Sort">4. Shell Sort</a><ul><li><a class="toc-href" href="#implementation_2" title="implementation">implementation</a></li><li><a class="toc-href" href="#analysis_2" title="analysis">analysis</a></li></ul></li><li><a class="toc-href" href="#of-compares-cte-n-of-h-used_1" title="of compares < Cte * N * (# of h used )">of compares < Cte * N * (# of h used )</a></li><li><a class="toc-href" href="#5-shuffling" title="5. shuffling">5. shuffling</a><ul><li><a class="toc-href" href="#shuffle-array-using-sort" title="shuffle array using sort">shuffle array using sort</a></li><li><a class="toc-href" href="#knuth-shuffle" title="Knuth shuffle">Knuth shuffle</a></li><li><a class="toc-href" href="#example-online-poker" title="example: online poker">example: online poker</a></li></ul></li><li><a class="toc-href" href="#6-convex-hull_1" title="6. Convex Hull">6. Convex Hull</a><ul><li><a class="toc-href" href="#convex-hull" title="convex hull">convex hull</a></li><li><a class="toc-href" href="#algo" title="Algo">Algo</a></li><li><a class="toc-href" href="#implementation_3" title="implementation">implementation</a></li></ul></li></ul></div>
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