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[Algorithms I] Week1-Lab: Percolation
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#model-problem" title="model & problem">model & problem</a></li><li><a class="toc-href" href="#method" title="Method">Method</a></li><li><a class="toc-href" href="#code" title="Code">Code</a><ul><li><a class="toc-href" href="#backwashwen-ti" title="backwash问题">backwash问题</a></li></ul></li></ul></div>
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<h1 id="model-problem">model & problem</h1>
<p>(原文描述太啰嗦了)<br/>
A system using an N-by-N grid of sites. <br/>
→ Each site is either open or blocked. <br/>
→ A <strong>full</strong> site is an open site that can be connected to an open site in the top row via a chain of neighboring open sites. (这个full的定义有玄机 而且导致后面写程序时有个问题, 看论坛想了半天才想出来, 见后文.)<br/>
→ We say the system <strong>percolates</strong> if there is a path of connected open sites form the top row to the bottom row. </p>
<p><img alt="" class="img-responsive" src="../images/file:///home/wx/Dropbox/ZIM_NOTES/0._TmpNotes/Algorithms%2C_4th_ed/Week1-Assignment-Percolation/pasted_image.png"/> </p>
<p>⇒ pb: if sites are independently set to be open with probability <strong>p</strong>, what is the probability that the system percolates? <br/>
<img alt="" class="img-responsive" src="../images/file:///home/wx/Dropbox/ZIM_NOTES/0._TmpNotes/Algorithms%2C_4th_ed/Week1-Assignment-Percolation/pasted_image001.png"/> <br/>
→ When N is sufficiently large, there is a threshold value **p<strong><em> such that when p < p</em> a random N-by-N grid almost never percolates, and when p > p<em>, a random N-by-N grid almost always percolates. <br/>
→ No mathematical solution for determining the percolation threshold p</em> has yet been derived. <br/>
⇒ Your task is to *write a computer program to estimate p</strong>. </p>
<h1 id="method">Method</h1>
<ul>
<li>
<p>API:</p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class Percolation {</span></span>
<span class="code-line"><span class="err"> public Percolation(int N) // create N-by-N grid, with all sites blocked</span></span>
<span class="code-line"><span class="err"> public void open(int i, int j) // open site (row i, column j) if it is not open already</span></span>
<span class="code-line"><span class="err"> public boolean isOpen(int i, int j) // is site (row i, column j) open?</span></span>
<span class="code-line"><span class="err"> public boolean isFull(int i, int j) // is site (row i, column j) full?</span></span>
<span class="code-line"><span class="err"> public boolean percolates() // does the system percolate?</span></span>
<span class="code-line"><span class="err"> public static void main(String[] args // test client (optional)</span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
</li>
<li>
<p>Corner cases: the row and column indices i and j are integers between 1 and N. <strong>1≤i,j≤N</strong></p>
</li>
</ul>
<p>if i/j out of range: <code>java.lang.IndexOutOfBoundsException</code>
if N<=0 in constructor: <code>java.lang.IllegalArgumentException</code></p>
<ul>
<li>
<p>Performance requirements: N2 for constructor, const for other operations</p>
</li>
<li>
<p><strong>Monte Carlo simulation</strong></p>
<ul>
<li>all sites init to be closed</li>
</ul>
<p>→ randomly choose a blocked site (i,j) and open it
→ <em>repeat until percolates</em> ⇒ the fraction of opened sites is an estimation of p*</p>
</li>
<li>
<p>ex. 20*20 grid, when percolated: </p>
</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/file:///home/wx/Dropbox/ZIM_NOTES/0._TmpNotes/Algorithms%2C_4th_ed/Week1-Assignment-Percolation/pasted_image002.png"/> <br/>
⇒ estimated p* = 204/400=0.51</p>
<ul>
<li>
<p>repeat the estimation for T times, get T estimations </p>
<p>→ get mean and std: <br/>
<img alt="" class="img-responsive" src="../images/file:///home/wx/Dropbox/ZIM_NOTES/0._TmpNotes/Algorithms%2C_4th_ed/Week1-Assignment-Percolation/pasted_image003.png"/> <br/>
→ 95% 置信区间: <br/>
<img alt="" class="img-responsive" src="../images/file:///home/wx/Dropbox/ZIM_NOTES/0._TmpNotes/Algorithms%2C_4th_ed/Week1-Assignment-Percolation/pasted_image004.png"/> </p>
</li>
<li>
<p>create API for this simulation: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class PercolationStats {</span></span>
<span class="code-line"><span class="err"> public PercolationStats(int N, int T) // perform T independent experiments on an N-by-N grid</span></span>
<span class="code-line"><span class="err"> public double mean() // sample mean of percolation threshold</span></span>
<span class="code-line"><span class="err"> public double stddev() // sample standard deviation of percolation threshold</span></span>
<span class="code-line"><span class="err"> public double confidenceLo() // low endpoint of 95% confidence interval</span></span>
<span class="code-line"><span class="err"> public double confidenceHi() // high endpoint of 95% confidence interval</span></span>
<span class="code-line"><span class="err"> public static void main(String[] args) // test client (described below)</span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>-if N ≤ 0 or T ≤ 0: <code>java.lang.IllegalArgumentException</code><br/>
-<code>main()</code> : takes two command-line arguments N and T<br/>
⇒ performs T independent computational experiments on an N-by-N grid, and prints out the mean, standard deviation, and the 95% confidence interval for p*. <br/>
(Use <a href="http://introcs.cs.princeton.edu/java/stdlib/javadoc/StdRandom.html">standard random</a> from our standard libraries to generate random numbers; use <a href="http://introcs.cs.princeton.edu/java/stdlib/javadoc/StdStats.html">standard statistics</a> to compute the sample mean and standard deviation. <br/>
Here is the algo API: <a href="http://algs4.cs.princeton.edu/code/index.php">http://algs4.cs.princeton.edu/code/index.php</a>) </p>
</li>
</ul>
<h1 id="code">Code</h1>
<p>注意一定要用它们提供的那些库, 否则自己写的话代码就长了.... <br/>
shuffle, mean, stddev什么的直接用他们的函数库就可以做到. <br/>
<a href="http://algs4.cs.princeton.edu/code/index.php">http://algs4.cs.princeton.edu/code/index.php</a> <br/>
另外UF也是用他们写好的, WeightedQuickUnionUF. </p>
<p>按照提示, 除了格子的N^2个节点以外再增加两个节点: 顶部和底部的虚拟节点. 这里写的时候注意一开始也是不恩能够把它们与第一行/最后一行相连的 — 要在一个格子open以后再相连. </p>
<h3 id="backwashwen-ti">backwash问题</h3>
<p>这次题目有一点比较困难就是, 需要实现isFull()函数, 这个函数判断一个格子(i,j)是否和顶部相连. 这里如果直接用UF的connected()判断是否和顶部虚拟节点相连的话是有问题的, 如下图: <br/>
<img alt="" class="img-responsive" src="../images/file:///home/wx/Dropbox/ZIM_NOTES/0._TmpNotes/Algorithms%2C_4th_ed/Week1-Assignment-Percolation/pasted_image005.png"/> <br/>
白色格子表示格子是open的, 蓝色格子表示格子是open并且是<em>full</em>的(i.e. 和顶部相连的), 左边图片里的状态是对的, 右边图片里底下部分的格子状态则不对: 如左下角的格子, 其实是没有和顶部联通的, 如果我们用两个虚拟节点的话, 由于底部虚拟节点和顶部虚拟节点相连, 所以和底部虚拟节点相连的左下角部分就被判断成了full的. </p>
<p>这个问题一开始我以为可以很简单解决, 后来发现没那么容易... (注意题目还要求isFull()也要在常数时间给出结果).<br/>
一个不优雅的办法是, 建立两个UF, 一个用来判断percolation, 另一个UF里没有底部虚拟节点所以可以专门用来判断isFull(). </p>
<p>这样解决的话使可以通过测试, 不过非常不好看, 另外一个UF的内存占用是8N^2(内部有size[]和id[]两个int数组), 比较大. </p>
<p>在论坛上找了半天, 看了一些人的分享终于想到了这个非常妙的办法: </p>
<ol>
<li>UF只建立顶部虚拟节点, 不建立底部虚拟节点. </li>
<li>判断isFull只需要用UF的connected()一下就好了</li>
<li>问题是怎么判断percolation: <br/>
a. 建立一个数组 <code>boolean connectedToBottom[]</code>, 指示某一点是否和底部相连 <br/>
b. trick在这里: 不必修改一个联通分支的所有点的<code>connectedToBottom</code>的值, <strong>只需要修改联通分支的root(UF的find)即可</strong>. 在进行union的时候先查看两个component的root是不是连到底部, 然后有一个连到底部的话, 在union以后把合并后的联通分支的<code>connectedToBottom</code>状态改为true即可 <br/>
c. 然后判断percolate: 先找到顶部虚拟节点锁在component的root, 然后看这个root是否连到底部即可! </li>
</ol>
<p>这样用一个boolean数组(N^2内存)代替了一个新的UF(8N^2内存), 而且实现也更加优雅. <br/>
非常有意思的练习...</p>
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