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[Algorithms I] Week 3-2 Quicksort
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-quicksort" title="1. Quicksort">1. Quicksort</a><ul><li><a class="toc-href" href="#implemetation" title="Implemetation">Implemetation</a></li><li><a class="toc-href" href="#analysis" title="Analysis">Analysis</a></li></ul></li><li><a class="toc-href" href="#compares-nlgn_1" title="compares = NlgN">compares = NlgN</a></li><li><a class="toc-href" href="#compares-12n2" title="compares = 1/2*N^2">compares = 1/2*N^2</a><ul><li><a class="toc-href" href="#practical-improvements" title="Practical improvements">Practical improvements</a></li></ul></li><li><a class="toc-href" href="#2-selection_1" title="2. Selection">2. Selection</a><ul><li><a class="toc-href" href="#quick-select" title="Quick select">Quick select</a></li><li><a class="toc-href" href="#implementation" title="Implementation">Implementation</a></li><li><a class="toc-href" href="#analysis_1" title="Analysis">Analysis</a></li></ul></li><li><a class="toc-href" href="#3-duplicate-keys_1" title="3. Duplicate keys">3. Duplicate keys</a><ul><li><a class="toc-href" href="#3-way-partitionning" title="3-way partitionning">3-way partitionning</a></li><li><a class="toc-href" href="#implementation-3-way-quick-sort" title="Implementation: 3-way quick sort">Implementation: 3-way quick sort</a></li></ul></li><li><a class="toc-href" href="#4-system-sorts_1" title="4. System Sorts">4. System Sorts</a></li></ul></div>
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<p>(maybe best algorithm for sorting.) </p>
<h1 id="1-quicksort">1. Quicksort</h1>
<p><strong>Idea:</strong> </p>
<ol>
<li>shuffle the array </li>
<li><em>Partition</em> the array into two subarrays to left and right of pivot (*now pivot is <strong>*in its final position</strong>) </li>
</ol>
<p>no larger entry to the left of pivot <br/>
no smaller entry to the right of pivot </p>
<ol>
<li>sort each subarray recursively </li>
</ol>
<p><img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image.png"/> </p>
<h2 id="implemetation">Implemetation</h2>
<p><strong>The partition process:</strong> <br/>
这个方法也比较巧妙. <br/>
Use 2 pointers i and j (个人觉得用hi, lo, pivot更好...) : <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image001.png"/> <br/>
→ a[i]>=a[lo], a[j]<<strong>=</strong>a[lo] (注意是大于<strong>等于</strong>/小于<strong>等于</strong>) <br/>
⇒ exchange i and j <br/>
→ Scan until i and j cross (ie. j<=i) <br/>
⇒ finally exchange lo with <strong>j</strong> <br/>
函数的签名定义的好: 把lo到hi部分的数组分成两部分, 并返回分割点的index. </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="nc">int</span><span class="w"> </span><span class="k">partition</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">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="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">i</span><span class="o">=</span><span class="n">lo</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">hi</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">j</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">while</span><span class="p">(</span><span class="w"> </span><span class="n">i</span><span class="o"><=</span><span class="n">hi</span><span class="w"> </span><span class="o">&&</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">i</span><span class="o">]</span><span class="p">,</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="w"> </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 class="o">//</span><span class="n">a</span><span class="o">[</span><span class="n">i</span><span class="o">]>=</span><span class="n">a</span><span class="o">[</span><span class="n">lo</span><span class="o">]</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="w"> </span><span class="n">j</span><span class="o">>=</span><span class="n">lo</span><span class="w"> </span><span class="o">&&</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">lo</span><span class="o">]</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="w"> </span><span class="p">)</span><span class="w"> </span><span class="n">j</span><span class="c1">--; //a[j]<=a[lo] </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">i</span><span class="o"><</span><span class="n">j</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="o">++</span><span class="p">,</span><span class="n">j</span><span class="c1">--); </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="n">exch</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">j</span><span class="p">);</span><span class="w"> </span><span class="o">//</span><span class="n">exchange</span><span class="w"> </span><span class="k">pivot</span><span class="w"> </span><span class="k">with</span><span class="w"> </span><span class="n">j</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">j</span><span class="p">;</span><span class="w"> </span><span class="o">//</span><span class="n">j</span><span class="w"> </span><span class="ow">in</span><span class="w"> </span><span class="n">its</span><span class="w"> </span><span class="n">final</span><span class="w"> </span><span class="k">position</span><span class="w"> </span></span>
<span class="code-line"><span class="err">}</span><span class="w"></span></span>
</pre></div>
<p>这个函数其实并不好写对: </p>
<ul>
<li>test for cross pointers is not trival (ex. edge case: the pivot is the smallest/largest entry in the range) </li>
<li>i<=hi is necessary ! </li>
<li>for keys equal to a[lo]: better to stop at them </li>
</ul>
<p>invariance: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image002.png"/> </p>
<p><strong>Quicksort: </strong> <br/>
使用partition函数和辅助sort函数(recursive). 注意在整个流程开始以前先shuffle一下. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private static void sort(Comparable[] a, int lo, int hi){ </span></span>
<span class="code-line"><span class="err"> if(hi<=lo) return; </span></span>
<span class="code-line"><span class="err"> int pivot = partition(a, lo, hi); </span></span>
<span class="code-line"><span class="err"> sort(a,lo,pivot-1); </span></span>
<span class="code-line"><span class="err"> sort(a,pivot+1,hi); </span></span>
<span class="code-line"><span class="err"> return;</span></span>
<span class="code-line"><span class="err">} </span></span>
<span class="code-line"><span class="err">public static void sort(Comparable[] a){ </span></span>
<span class="code-line"><span class="err"> StdRandom.shuffle(a); // don't forget to shuffle the array </span></span>
<span class="code-line"><span class="err"> sort(a,0,a.length-1); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>The randomness is preserved: the subarrays after partitionning is still randomly ordered. </p>
<h2 id="analysis">Analysis</h2>
<p>Performance: ~40% faster than mergesort. </p>
<p><strong>Best case</strong> </p>
<h1 id="compares-nlgn_1">compares = NlgN</h1>
<p>(each partition will divide the array in half) </p>
<p><strong>Worst case </strong> </p>
<h1 id="compares-12n2">compares = 1/2*N^2</h1>
<p>N+(N-1)+...+1 <br/>
if the array is already in order, <em>each partition will have one subarray of length=0</em> </p>
<p><strong>Average case </strong> </p>
<blockquote>
<p><strong>proposition </strong> <br/>
On average, for array with N distinct keys, the #compares = ~2NlnN, #exchanges = ~1/2<em>NlnN. <br/>
</em>Proof.* <br/>
C(N) := # compares for N entries <br/>
pivot 在N个数离的排名是uniform的 <br/>
接下来的数学推到很漂亮(不过可能没啥用..) <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image003.png"/> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image004.png"/> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image006.png"/> </p>
</blockquote>
<p>(上面最后一行写错了... 是2NlnN...orz)
random shuffle: probalistic guarantee against worst case. </p>
<p><strong>Pitfalls </strong> <br/>
implementations will get <em>quadratic</em> performance if array: </p>
<ul>
<li>is sorted or reverse sorted </li>
<li>has many duplicates (even if randomized) </li>
</ul>
<p><strong>Staility </strong> <br/>
Quicksort is <strong>NOT</strong> stable. <br/>
partitionning can make long range exchanges <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image007.png"/> </p>
<h2 id="practical-improvements">Practical improvements</h2>
<ul>
<li>cutoff to insertion sort for <10 items </li>
</ul>
<p>→ ~20% improvement <br/>
Or we can leave the small subarrays unsorted and sort them at last using insertion sort </p>
<ul>
<li>estimate median by sampling 3 items </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image008.png"/> <br/>
→ 10% improvement </p>
<h1 id="2-selection_1">2. Selection</h1>
<p>Goal: <em>given un array, find the kth largest item.</em> </p>
<ul>
<li>Upper bound for this problem: NlgN (just sort the array) </li>
<li>for small k (ex k=1,2,3), the upper bound is N (one-pass/two-pass) </li>
<li>Lower bound is N: at least have to look at everything </li>
</ul>
<h2 id="quick-select">Quick select</h2>
<p>Algo proposed also by Hoare: </p>
<ol>
<li>partition the array into two arrays left of pivot and right of pivot. </li>
<li>if pivot==k: return </li>
<li>continue the partition for just one of the subarrays </li>
</ol>
<p>类似于二分查找的过程.... <br/>
注意这里<strong>是不用递归的</strong>! 因为partition函数返回的直接就是pivot<em>在整个数组里的</em>位置! </p>
<h2 id="implementation">Implementation</h2>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">privater</span><span class="w"> </span><span class="k">static</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="k">partition</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">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="err">{</span><span class="p">...</span><span class="err">}</span><span class="w"></span></span>
<span class="code-line"></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">Comparable</span><span class="w"> </span><span class="k">select</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">k</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">StdRandom</span><span class="p">.</span><span class="n">shuffle</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="nc">int</span><span class="w"> </span><span class="n">lo</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="n">hi</span><span class="o">=</span><span class="n">a</span><span class="p">.</span><span class="n">length</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">while</span><span class="p">(</span><span class="k">true</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">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">partition</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">hi</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">j</span><span class="o"><</span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="n">lo</span><span class="o">=</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">if</span><span class="p">(</span><span class="n">j</span><span class="o">==</span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </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="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="n">hi</span><span class="o">=</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">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">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="k">return</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="p">;</span><span class="o">//</span><span class="n">这里不太理解为什么会在hi</span><span class="o"><</span><span class="n">lo的时候直接返回a</span><span class="o">[</span><span class="n">k</span><span class="o">]</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> <br/>
Quick selection takes <strong>linear</strong> time on average. <br/>
*proof * <br/>
intuitively, each partition will ct the subarray size in half: <br/>
N+N/2+N/4+... = 2N <br/>
formal analysis 略... </p>
<p><strong>worst case</strong>: quadratic (but very rare to happen) </p>
<p><strong>Theoretical results</strong> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image009.png"/> </p>
<h1 id="3-duplicate-keys_1">3. Duplicate keys</h1>
<p>if array contains many duplicate keys. </p>
<ul>
<li>huge array </li>
<li>small number of distinct keys </li>
</ul>
<p><strong>for mergesort </strong> <br/>
insensitive... always ~NlgN compares. </p>
<p><strong>for quicksort </strong> <br/>
Will get <em>quadratic</em> time if not stop on equal keys. (found in 1990s) <br/>
mistake: put all items equal to pivot *to just one side * <br/>
→ N^2 compares if all keys are equal from lo to hi. <br/>
correct: put all items equal to pivot in their final place. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image010.png"/> </p>
<h2 id="3-way-partitionning">3-way partitionning</h2>
<p>(by Dijkstra) <br/>
partition the array into 3 parts: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image011.png"/> </p>
<p>Dijkstra's 3-way partition algo: <br/>
使用3个指针: lt指向中间部分的左边界, gt指向右边界; i指针从左向右扫描, 算法很subtle: </p>
<ul>
<li>lt=lo, gt=hi, i=lo </li>
<li>if a[i]==v : i++ </li>
<li>if a[i]<v: exch(i,lt), i++, lt++ </li>
<li>if a[i]>v: exch(i,gt), gt-- </li>
<li>repeat until i and gt cross (i>gt) </li>
</ul>
<p><strong><em>invariance</em></strong>: </p>
<ul>
<li><code>[lo, lt)</code> <strong><em>all < v</em></strong></li>
<li><code>[lt, i)</code> <strong><em>all == v</em></strong></li>
<li><code>(gt, hi]</code> <strong><em>all >v</em></strong></li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image012.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image013.png"/> </p>
<h2 id="implementation-3-way-quick-sort">Implementation: 3-way quick sort</h2>
<p>不必再写partition函数, 直接在sort递归函数里面. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image014.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="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="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="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 class="o">//</span><span class="w"> </span><span class="n">递归函数别忘了先写终止条件</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="o">=</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>
<span class="code-line"><span class="w"> </span><span class="n">Comparable</span><span class="w"> </span><span class="n">v</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="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">gt</span><span class="p">;)</span><span class="err">{</span><span class="w"> </span><span class="o">//</span><span class="n">不能写</span><span class="w"> </span><span class="n">i</span><span class="o">++</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">i</span><span class="o">]</span><span class="p">,</span><span class="n">v</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="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">lt</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="w"> </span><span class="p">(</span><span class="w"> </span><span class="k">less</span><span class="p">(</span><span class="n">v</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="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">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="w"> </span><span class="o">//</span><span class="w"> </span><span class="n">v</span><span class="o">==</span><span class="n">a</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span></span>
<span class="code-line"><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="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">lo</span><span class="p">,</span><span class="w"> </span><span class="n">lt</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="n">sort</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">gt</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">hi</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><img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image015.png"/> <br/>
当N个数有很多重复的时候, lower bound可以变小于NlgN: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image016.png"/> <br/>
And Sedgewick proved that the 3-wy partition is propotional to the lower bound.... <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image017.png"/> </p>
<h1 id="4-system-sorts_1">4. System Sorts</h1>
<p>Arrays.sort() in java: </p>
<p><code>import java.util.Arrays;</code> </p>
<p><img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image018.png"/> <br/>
quicksort for primitive arrays, mergesort for objects: java设计者认为如果用obj array表示空间不是问题... </p>
<p>Pb in java's system sort: killer input exsit (havn't shuffle)... </p>
<p><img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image020.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image021.png"/> <br/>
总结一下学过的5/6种排序: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week3_2/pasted_image022.png"/> </p>
</div>
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var sr=$(this).attr('src');
$('#mimg').attr('src',sr);
$('#myModal').modal('show');
});
$('#affix-toc').affix({
offset: {
top: function(){
if($('#affix-toc').hasClass("affix"))
return $('#sidebar').height();
return $('#sidebar').height() - $('#affix-toc').height();
},
bottom: function (){
return $("#fcfooter").offset().top -
$("#article-content").offset().top -
$("#article-content").height() + 20;
}
}
});
$('#affix-toc').width($('#sidebar').width());
});
$(window).resize(function () {
$('#affix-toc').width($('#sidebar').width());
});
</script>
</body>