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[Algorithms I] Week 3-1 Mergesort
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-mergesort" title="1. Mergesort">1. Mergesort</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><li><a class="toc-href" href="#practical-improvements" title="Practical improvements">Practical improvements</a></li></ul></li><li><a class="toc-href" href="#2-bottom-up-mergesort_1" title="2. Bottom-up Mergesort">2. Bottom-up Mergesort</a><ul><li><a class="toc-href" href="#implementation_1" title="implementation">implementation</a></li></ul></li><li><a class="toc-href" href="#3-sorting-complexity_1" title="3. Sorting Complexity">3. Sorting Complexity</a><ul><li><a class="toc-href" href="#lower-bound-for-sorting" title="lower bound for sorting">lower bound for sorting</a></li></ul></li><li><a class="toc-href" href="#4-comparators_1" title="4. Comparators">4. Comparators</a><ul><li><a class="toc-href" href="#comparable-interface" title="Comparable interface">Comparable interface</a></li><li><a class="toc-href" href="#comparator-interface" title="Comparator interface">Comparator interface</a></li></ul></li><li><a class="toc-href" href="#5-stability_1" title="5. Stability">5. Stability</a></li></ul></div>
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<p>Two classical sorting algorithms: mergesort, quicksort. </p>
<h1 id="1-mergesort">1. Mergesort</h1>
<p>Divide and conquer: top 10 algorithms of the 20th century, invented by von Neumann. </p>
<p><strong>Idea</strong>: </p>
<ul>
<li>divide array into 2 halves </li>
<li>recursively sort each half </li>
<li>merge two sorted halves </li>
</ul>
<h2 id="implementation">Implementation</h2>
<p><strong>Merge</strong>: <br/>
Goal: a[lo] to a[mid] and a[mid+1] to a[hi] are sorted ⇒ get a[lo] to a[hi] sorted. <br/>
→<em>use an auxiliary array to copy data</em>: using 3 indices i,j,k. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image.png"/> </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">void</span><span class="w"> </span><span class="k">merge</span><span class="p">(</span><span class="n">Comparable</span><span class="err">[]</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">Comparable</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">hi</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">mid</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="w"> </span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="k">for</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">k</span><span class="o">=</span><span class="n">lo</span><span class="p">;</span><span class="n">k</span><span class="o"><=</span><span class="n">hi</span><span class="p">;</span><span class="n">k</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">k</span><span class="o">]=</span><span class="n">a</span><span class="o">[</span><span class="n">k</span><span class="o">]</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="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="w"> </span><span class="n">j</span><span class="o">=</span><span class="n">mid</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="o">=</span><span class="n">lo</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="k">while</span><span class="p">(</span><span class="n">i</span><span class="o"><=</span><span class="n">mid</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="n">j</span><span class="o"><=</span><span class="n">hi</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="k">less</span><span class="p">(</span><span class="n">aux</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">,</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 class="n">a</span><span class="o">[</span><span class="n">k++</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">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">else</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">k++</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++</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>
<span class="code-line"><span class="k">while</span><span class="p">(</span><span class="n">i</span><span class="o"><=</span><span class="n">mid</span><span class="p">)</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">k++</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++</span><span class="o">]</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="k">while</span><span class="p">(</span><span class="n">j</span><span class="o"><=</span><span class="n">hi</span><span class="p">)</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">k++</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">j++</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>
<p>note: we use <code>if(less(aux[j],aux[i]))</code>instead of<code>if(less(aux[j],aux[i]))</code>, because less(a,b) == true iff a<b (strict), and we want to make mergesort <strong>stable</strong>. See section (5) below. </p>
<p><strong>Assertion</strong> <br/>
We can also add <em>assertions</em>: <code>assert isSorted(a, lo, mid); assert isSorted(a,mid+1, hi);</code> <br/>
Enable/disable assertion at runtime: <br/>
<code>java -ea MyProgram</code> //enable assertion <br/>
<code>java -da MyProgram</code> //disable assertion: <strong>default</strong> <br/>
Best practice: <br/>
use assertions to check interval invariants; <br/>
do NOT use assert for external argument checking ! </p>
<p><strong>Mergesort </strong> <br/>
为了mergesort需要写两个辅助函数: merge和sort(recursive): </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span> <span class="k">class</span> <span class="n">MergeSort</span> <span class="n">extends</span> <span class="n">AbstractSort</span><span class="err">{</span> </span>
<span class="code-line"> <span class="k">public</span> <span class="k">static</span> <span class="n">void</span> <span class="n">sort</span><span class="p">(</span><span class="n">Comparable</span><span class="p">[]</span> <span class="n">a</span><span class="p">)</span><span class="err">{</span> </span>
<span class="code-line"> <span class="n">Comparable</span><span class="p">[]</span> <span class="n">aux</span> <span class="o">=</span> <span class="k">new</span> <span class="n">Comparable</span><span class="p">[</span><span class="n">a</span><span class="p">.</span><span class="k">length</span><span class="p">];</span> </span>
<span class="code-line"> <span class="n">sort</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">aux</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">a</span><span class="p">.</span><span class="k">length</span><span class="o">-</span><span class="mi">1</span><span class="p">);</span> </span>
<span class="code-line"> <span class="err">}</span> </span>
<span class="code-line"> <span class="n">private</span> <span class="k">static</span> <span class="n">void</span> <span class="n">sort</span><span class="p">(</span><span class="n">Comparable</span><span class="p">[]</span> <span class="n">a</span><span class="p">,</span> <span class="n">Comparable</span><span class="p">[]</span> <span class="n">aux</span><span class="p">,</span> <span class="nb">int</span> <span class="n">lo</span><span class="p">,</span> <span class="nb">int</span> <span class="n">hi</span><span class="p">)</span><span class="err">{</span> </span>
<span class="code-line"> <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="k">return</span><span class="p">;</span> </span>
<span class="code-line"> <span class="nb">int</span> <span class="n">mid</span> <span class="o">=</span> <span class="p">(</span><span class="n">hi</span><span class="o">+</span><span class="n">lo</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">;</span> </span>
<span class="code-line"> <span class="n">sort</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">aux</span><span class="p">,</span><span class="n">mid</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">hi</span><span class="p">);</span> </span>
<span class="code-line"> <span class="n">sort</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">aux</span><span class="p">,</span><span class="n">lo</span><span class="p">,</span><span class="n">mid</span><span class="p">);</span> </span>
<span class="code-line"> <span class="n">merge</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">aux</span><span class="p">,</span><span class="n">hi</span><span class="p">,</span><span class="n">mid</span><span class="p">,</span><span class="n">lo</span><span class="p">);</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="err">}</span> </span>
<span class="code-line"> <span class="n">private</span> <span class="k">static</span> <span class="n">void</span> <span class="n">merge</span><span class="p">(</span><span class="n">Comparable</span><span class="p">[]</span><span class="n">a</span><span class="p">,</span> <span class="n">Comparable</span><span class="p">[]</span> <span class="n">aux</span><span class="p">,</span> <span class="nb">int</span> <span class="n">hi</span><span class="p">,</span> <span class="nb">int</span> <span class="n">mid</span><span class="p">,</span> <span class="nb">int</span> <span class="n">lo</span> <span class="p">)</span><span class="err">{</span> </span>
<span class="code-line"> <span class="o">//</span> <span class="k">as</span> <span class="k">before</span><span class="p">...</span> </span>
<span class="code-line"> <span class="err">}</span> </span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>有几点注意的: </p>
<ol>
<li>注意递归终止条件是<code>hi<=lo</code>, 不是<code>==</code>. </li>
<li>实现的代码不难, 但是定义好辅助函数的参数并不简单... 比如在merge里使用了mid作为参数. </li>
<li>aux的数组直接写在了函数参数里面, 好处是可以防止反复声明数组带来的开销. </li>
<li>发现他们写的时候mid都是写成: <code>mid = lo+(hi-lo)/2</code>, <a href="http://stackoverflow.com/questions/25571359/why-we-write-lohi-lo-2-in-binary-search">查了一下</a>发现这样的原因是为了防止hi+lo整数溢出...嗯有道理. </li>
</ol>
<h2 id="analysis">Analysis</h2>
<p><strong>proposition (time)</strong> <br/>
Mergesort takes at most NlgN compares and 6NlgN array access. <br/>
<em>Proof.</em> <br/>
def: C(N)=#compares for N elements, A(N)=#array access for N elements, the recurrence eq: <br/>
C(N) <= 2<em>C(N/2) + N; C(1) = 0 (最多N次比较: 每个a[]的数都由比较得到) <br/>
A(N) <= 2</em>A(N/2) + 6N; A(1) = 0 (~~这里不太理解为什么是6N, 怎么数都是4N或者5N啊?......~~2N复制到aux, 2N复制回去, 2N比较) <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image001.png"/> <br/>
(上图蓝色的一列是extra cost) <br/>
或者用递推公式, 发现D(N)/N的递推关系: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image002.png"/> <br/>
或者用数学归纳法: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image003.png"/> </p>
<p><strong>Proposition(memory)</strong> <br/>
Mergesort takes N extra memory. <br/>
⇒ mergesort is NOT <em>in-place</em> sorting method. </p>
<p><strong>def</strong>. <em>"in-place"</em> <br/>
A sorting algo is in-place if it uses <C*lgN extra memory. <br/>
ex. insertion sort, shellsort, selection sort... </p>
<h2 id="practical-improvements">Practical improvements</h2>
<ul>
<li>use insertion sort for small arrays: </li>
</ul>
<p>cutoff to insertion sort for <7 items. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image004.png"/> <br/>
⇒ lead to 20% improvement! </p>
<ul>
<li>stop if alread sorted: *a[mid]<a[mid+1]! * </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image005.png"/> </p>
<ul>
<li>switch the role of <code>a[]</code> and <code>aux[]</code> </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image006.png"/> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image007.png"/> </p>
<h1 id="2-bottom-up-mergesort_1">2. Bottom-up Mergesort</h1>
<p>bottom-up version of mergesort <em>without recursion</em>. <br/>
Idea: </p>
<ul>
<li>pass and merge subarrays of size 1 in pairs </li>
<li>repeat for subarrays of size 2,4,8,.... </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image008.png"/> </p>
<h2 id="implementation_1">implementation</h2>
<p>看着上面那个图好写一点... 就是每次用更大的size来两两merge一遍数组 </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public static void sort(Comparable[] a){ </span></span>
<span class="code-line"><span class="err"> Comparable[] aux = new Comparable[a.length]; </span></span>
<span class="code-line"><span class="err"> for(int sz=1;sz<a.length;sz*=2){ </span></span>
<span class="code-line"><span class="err"> for(int i=0;i<a.length-sz;i+=sz*2) </span></span>
<span class="code-line"><span class="err"> merge( a,aux,i,i+sz-1,Math.min(i+sz*2-1) );//<-注意mid和hi的计算方法 </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h1 id="3-sorting-complexity_1">3. Sorting Complexity</h1>
<p>some items: </p>
<ul>
<li>computational model: allowed operations, (ex. <em>decision tree for compare based sorting</em>). </li>
<li>cost model: operation counts. </li>
<li>upper bound: cost guarantee (ex. NlgN for mergesort). . </li>
<li>lower bound: limit on cost guarantee for all algorithms (no algorithm can do better). </li>
<li>Optimal algorithm: algo with best cost guarantee. (upper bound=lower bound) </li>
</ul>
<p>证明lower bound 的方法很有意思: </p>
<h2 id="lower-bound-for-sorting">lower bound for sorting</h2>
<p>(binary) decision tree for the case of 3 distinct items: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image009.png"/> </p>
<p>each possible ordering is a leaf of the decision tree → there are N! possible orderings → tree height should be lg(N!) = NlgN (cf. <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation">https://en.wikipedia.org/wiki/Stirling%27s_approximation</a>) <br/>
<strong>proposition. </strong> <br/>
Any compare-based sorting algorithm must have at least lg(N!)~NlgN compares in the worst case. (for N distinct keys) <br/>
<em>proof.</em> </p>
<ul>
<li>binary tree of height h has at most 2^h leaves </li>
<li>N! possible orderings → at least N! leaves </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image010.png"/> <br/>
So, lower bound for sorting = ~NlgN ⇒ <em>mergesort <strong><em>is</em></strong> an asymptotical optimal algorithm</em>. (In terms of time complexity, the shortcoming of mergesort is always the extra space usage) <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image011.png"/> <br/>
但是并不是说NlgN是<em>所有</em>排序问题的下界: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image012.png"/> </p>
<h1 id="4-comparators_1">4. Comparators</h1>
<p>Java mechenism for comparing same data on different ways. </p>
<h2 id="comparable-interface">Comparable interface</h2>
<p>→Sorting using <em>natural order</em> for a data type. <br/>
public class Data implements Comparable<date>{ <br/>
public int compareTo(Date that){ <br/>
//...} <br/>
} </date></p>
<h2 id="comparator-interface">Comparator interface</h2>
<p>→Sorting using an <em>alternative order</em>. (total order property is required...) </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public interface Comparator<Key>{ </span></span>
<span class="code-line"><span class="err"> int compare(Key v, Key w); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>Comparator can be passed as arguments in java system sort: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image013.png"/> </p>
<p><strong>Using Comparators in our sorting algos </strong> <br/>
Use another signature: </p>
<ul>
<li>change Comparable to Object </li>
<li>
<p>add Comparator in arguments </p>
<p>public static void sort(Object[] a, Comparator comparator); <br/>
public static boolean less(Comparator c, Object v, Object w); </p>
<p>public static void exch(Object[] a, int i, int j); </p>
</li>
</ul>
<p><strong>Implementing a comparator </strong> <br/>
Add static comparator to a class: </p>
<ul>
<li>In the declaration of a class, define an inner class that implements the Comparator interface, </li>
<li>Then declare an instance of this inner class as a static final variable... </li>
<li>note: the inner class should be <strong>static</strong> also. </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image014.png"/> <br/>
以上的方法目的是为某个类提供预先定义好的comparator(比如<code>String.CASE_INSENSITIVE_ORDER</code>), 另外也可以直接再声明一个类作为comparator, 然后实例化这个类传进去, 就像之前做过的那样. </p>
<p><strong>Application: Graham scan algo for convex hull </strong> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image015.png"/> <br/>
We can get the result of compare by calling <code>ccw(a,b,c)</code> : <br/>
*ccw(p,q1,q2)=true ⇔ q2>q1 in terms of polar angle wrt p. (别忘了p是y坐标最小的点, 否则还要讨论角坐标为负数的情况, 见下图). * <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image016.png"/> </p>
<h1 id="5-stability_1">5. Stability</h1>
<p>Typical application: first sort by student name then by section. </p>
<p><strong>def. :stable"</strong> <br/>
<em>A stable sort preserves the relative order for items with equal keys.</em> </p>
<p><em>Insertion sort and mergesort are stable, whereas selection sort and shell sort are not.</em> (And always need to carefully check code: "less than" vs. "less than or equal to"). </p>
<p><strong>insertion sort:</strong> <br/>
stable <br/>
proof: we never move equal items pass each other: <code>if(less(a[j], a[j-1])) exch(...)</code> </p>
<p><strong>selection sort: </strong> <br/>
not stable <br/>
counter example: <br/>
when exchanging A1 and B1, we move B1 behind B2 <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image017.png"/> </p>
<p><strong>shell sort: </strong> <br/>
counter example: long-dist exchanges <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_1/pasted_image018.png"/> </p>
<p><strong>merge sort: </strong> <br/>
stable <br/>
proof: suffices to verify that merge operation is stable, if keys are equal, always take element from left subarray. </p>
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