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[Algorithms I] Week 1-1 Union-Find
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-dynamic-connectivity-pb" title="1. Dynamic Connectivity pb">1. Dynamic Connectivity pb</a><ul><li><a class="toc-href" href="#pb-statement" title="pb statement">pb statement</a></li><li><a class="toc-href" href="#uf-api" title="UF API">UF API</a></li></ul></li><li><a class="toc-href" href="#2-quick-find_1" title="2. Quick Find">2. Quick Find</a><ul><li><a class="toc-href" href="#data-structure" title="data structure">data structure</a></li><li><a class="toc-href" href="#uf-operations" title="UF operations">UF operations</a></li><li><a class="toc-href" href="#implementation" title="implementation">implementation</a></li></ul></li><li><a class="toc-href" href="#3-quick-union_1" title="3. Quick Union">3. Quick Union</a><ul><li><a class="toc-href" href="#data-structure_1" title="data structure">data structure</a></li><li><a class="toc-href" href="#uf-operations_1" title="UF operations">UF operations</a></li><li><a class="toc-href" href="#implementation_1" title="implementation">implementation</a></li><li><a class="toc-href" href="#complexity" title="complexity">complexity</a></li></ul></li><li><a class="toc-href" href="#4-quick-union-improvements_1" title="4. Quick Union Improvements">4. Quick Union Improvements</a><ul><li><a class="toc-href" href="#improvement1-weighting" title="improvement1: weighting">improvement1: weighting</a><ul><li><a class="toc-href" href="#implementation_2" title="implementation">implementation</a></li><li><a class="toc-href" href="#complexity_1" title="complexity">complexity</a></li></ul></li><li><a class="toc-href" href="#improvement-2-path-compression_1" title="improvement 2: path compression">improvement 2: path compression</a><ul><li><a class="toc-href" href="#complexity_2" title="complexity">complexity</a></li></ul></li><li><a class="toc-href" href="#summery_1" title="summery">summery</a></li></ul></li><li><a class="toc-href" href="#5-union-find-application_1" title="5. Union Find Application">5. Union Find Application</a><ul><li><a class="toc-href" href="#percolation" title="percolation">percolation</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-dynamic-connectivity-pb">1. Dynamic Connectivity pb</h1>
<h3 id="pb-statement">pb statement</h3>
<p>a set of N obj, <em>indexed by 0,1,...,N-1</em>
⇒
<em> UNION: connect objects <code>void union(int p, int q)</code>
</em> FIND: is there a path connecting 2 obj? <code>boolean connected(int p, int q)</code></p>
<p>ex: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image.png"/></p>
<p><strong>connect components</strong>(联通分支): max set of obj that are mutually connected. </p>
<h3 id="uf-api">UF API</h3>
<ul>
<li>union(p,q): connect 2 obj</li>
<li>connected(p,q): test if p and q are connected </li>
<li><strong>find(p)</strong>: find the <em>component id</em> of p</li>
<li>count(): nb of components</li>
</ul>
<p>注意:
命名不是很好, 这里的<code>find()</code>函数不对应FIND query, <code>connected()</code>函数才是真正的FIND query, find()函数是为了connected()函数而做的一个辅助函数(find(p): return the root of the node p )
在connected里就可以调用find: <code>return find(p)==find(q)</code></p>
<p>应该是interface更好一些... </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public interface UF{</span></span>
<span class="code-line"><span class="err">void union(int p, int q);</span></span>
<span class="code-line"><span class="err">boolean connected(int p, int q);</span></span>
<span class="code-line"><span class="err">//int find(int p);</span></span>
<span class="code-line"><span class="err">//int count();</span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>测试client: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image003.png"/></p>
<h1 id="2-quick-find_1">2. Quick Find</h1>
<p><em>"eager approach"</em></p>
<h3 id="data-structure">data structure</h3>
<p>⇒ an int array <code>id[]</code>
initialized to <em>id[p]=p for all p</em>
interpretation: <em>id[p] = <strong><em>component id of obj p</em></strong>
⇒ p and q are connected </em>iff* id[p]==id[q] (ie. find very fast)
没有用find()函数 <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image002.png"/> </p>
<h3 id="uf-operations">UF operations</h3>
<ul>
<li>FIND:</li>
</ul>
<p><code>connected(p,q):</code> very fast, just check id[p] and id[q]</p>
<ul>
<li>UNION:</li>
</ul>
<p>when merging 2 components :
<code>union(p,q)</code>: <code>id[p]=id[q]</code>(总是让第一个参数p的id变为第二个参数q的id), <br/>
⇒ then have to modify <em>all entries</em> with id equal to <code>id[p]</code> !<br/>
⇒ too many entries to change </p>
<h3 id="implementation">implementation</h3>
<p><em>(class QuickFindUF implements UF)</em><br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image004.png"/></p>
<p>complexity: </p>
<ul>
<li>FIND: cte</li>
<li>UNION: lin...</li>
</ul>
<p>if N obj + N unions ⇒ <em>quad time !</em><br/>
btw, 程序运行速度: <strong>~10^9/s</strong></p>
<h1 id="3-quick-union_1">3. Quick Union</h1>
<p><em>"lazy approach"</em></p>
<h3 id="data-structure_1">data structure</h3>
<p>⇒ also an int array <code>id[]</code>
considering a set of <em>trees, </em>此时每个联通分支都是一个tree<br/>
interpretation: <em>id[p] = <em><strong>parent index of obj p </strong>(觉得这个数组叫做father更好....)<br/>
⇒ p is a root node </em>iff</em> id[p]==p</p>
<p><img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image005.png"/></p>
<h3 id="uf-operations_1">UF operations</h3>
<ul>
<li>FIND:</li>
</ul>
<p><code>connected(p,q):</code>check if <em>root of p == root of q</em></p>
<ul>
<li>UNION: </li>
</ul>
<p><code>union(p,q):</code>just set p's root to be <em>child</em> of q's root (把第一个参数p的那棵树放入第二个参数q的树的根节点作为子树)<br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image006.png"/></p>
<ul>
<li>root():</li>
</ul>
<p>前两个的操作都需要一个函数查找一个节点的root, 需要写一个函数实现, 也很简单, 一路找parent即可: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">private</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">root</span><span class="p">(</span><span class="nc">int</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="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="o">!=</span><span class="n">id</span><span class="o">[</span><span class="n">p</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="n">p</span><span class="o">=</span><span class="n">id</span><span class="o">[</span><span class="n">p</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">return</span><span class="w"> </span><span class="n">p</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="implementation_1">implementation</h3>
<p><img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image007.png"/></p>
<h3 id="complexity">complexity</h3>
<p>in the worst case (all elements is in a list form), root() is ~N, so:</p>
<ul>
<li>FIND: lin</li>
<li>UNION: lin</li>
</ul>
<p>quick find和quick union的问题: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image009.png"/></p>
<h1 id="4-quick-union-improvements_1">4. Quick Union Improvements</h1>
<h2 id="improvement1-weighting">improvement1: weighting</h2>
<p><em>keep track of tree size</em> ⇒ balance by taking the small tree be a child of the large tree <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image010.png"/></p>
<p><img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image011.png"/></p>
<p>⇒ add an extra array: <code>sz[]</code> sz[i] is the size of the tree with root i</p>
<h3 id="implementation_2">implementation</h3>
<p>(数组<code>int sz[]</code> 初始全部为1)</p>
<ul>
<li>
<p>依然需要<code>root()</code>函数.</p>
<p>private int root(int p){
while(p!=id[p]) p=id[p];
return p;
}</p>
</li>
<li>
<p>FIND </p>
<p>public boolean connected(int p, int q){
return root(p)==root(q);
}</p>
</li>
<li>
<p>UNION</p>
<p>public void union(int p, int q){
int rp = root(p), rq=root(q);
if(rp==rq) return; //
if (sz[rp]<sz[rq]){
id[rp]=rq;
sz[rq]+=sz[rp];
}
else{...}
}</p>
</li>
</ul>
<h3 id="complexity_1">complexity</h3>
<p>FIND: proportional to <em>depth of p and</em> q in their tree
UNION: const if p and q are root</p>
<ul>
<li><strong>proposition</strong></li>
</ul>
<p>the max depth of weightedQuickUnion is <strong>lgN</strong></p>
<p>[pf] considering a node <code>x</code>, in tree <code>T1</code>, <code>dep(x)</code> is x's depth in its tree. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image012.png"/> <br/>
→ <code>dep(x)</code> will increase by 1, iff <code>T1</code> is merged into another tree <code>T2</code> (and by the algo, shoud have |T1|<=|T2| )<br/>
→ x's tree's size become |T1|+|T2| >= 2<em>|T1|
⇒ everytime dep(x) increased by 1, x's tree's size will </em>at least double*<br/>
at first dep(x)=1, if dep(x) increases lgN times, the size of the tree will be >= N <br/>
CQFD.</p>
<p>so the <code>root()</code> function takes only lgN time.
<strong>conclusion</strong>: both UNION and FIND will be in <strong>lgN </strong>time. </p>
<h2 id="improvement-2-path-compression_1">improvement 2: path compression</h2>
<ul>
<li>imporve the root() function: </li>
</ul>
<p>when looking for root of a node ⇒ link <em>all nodes in the path </em>up to the root. <br/>
⇒ just a constant extra time compared to old implementation. </p>
<ul>
<li>
<p>2 pass implementation: </p>
<p>private int root(int p){
int r = p;
while(r!=id[r]) r=id[r];
while(p!=r){
int t = p;
p=id[p];
id[t]=r;
}
return r;
}</p>
</li>
</ul>
<p>flatens the tree greatly. </p>
<ul>
<li>single pass implementation: </li>
</ul>
<p>just make all other node <em>point to its grandparent</em> (halving the path length)
⇒ not as flatening as before, but in practice will almost be the same.
just one extra line of code:
private int root(int p){
while(p!=id[p]){
id[p] = id[ id[p] ];
p=id[p]; <br/>
}
return p;
}</p>
<h3 id="complexity_2">complexity</h3>
<p>(for weighet quick union with path compression — <em>WQUPC</em>)<br/>
very very small: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image013.png"/></p>
<p><strong>lg*()</strong> function: "<em>iterated log function</em>", lg<em>(N) = the number of time to take log to get to 1
lg</em>()几乎可以看成常数了: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image014.png"/> <br/>
ex. <em>lg</em>(65536) = 4* (x^16=65536)<br/>
because: lg(65536)=16 ; lg(16) = 4; lg(4)=2; lg(2)=1. <br/>
<strong>⇒ N obj, M unions will take (almost) linear time</strong> </p>
<p><img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image015.png"/> <br/>
(有人证明了不存在<em>理论上</em>linear的算法. )</p>
<p><strong>conclusion</strong>: both UNION and FIND will be in <strong>constant time</strong>.</p>
<h2 id="summery_1">summery</h2>
<p><img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image017.png"/> <br/>
上面这个表格好像quick union的部分有问题? 最坏情况下应该是N+MN吧??<br/>
书上是这么写的: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image016.png"/></p>
<p>WQUCF reduce 30 years to 6 seconds. </p>
<h1 id="5-union-find-application_1">5. Union Find Application</h1>
<p><img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image018.png"/></p>
<ul>
<li>percolation</li>
<li>dynamic connectivity</li>
<li>Kruskal MST algo</li>
<li>Games (GO)</li>
<li>.......</li>
</ul>
<h3 id="percolation">percolation</h3>
<ul>
<li>model: </li>
</ul>
<p>N<em>N grid of sites<br/>
⇒ each site is open with proba=</em>p*<br/>
⇒ sys <strong>percolate</strong> iff bottom and top are connected by open sites. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image019.png"/></p>
<ul>
<li>question: the <em>percolation probability</em> as a function of <em>p </em>(<strong>phase transition</strong>)</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image020.png"/> <br/>
nobody knows how to get the threshold mathematically<br/>
⇒ run <em>simulations</em> to find out the phase transition <em>threshold</em>. </p>
<ul>
<li>Monte Carlo simulation </li>
</ul>
<p>→ all sites initilized to be <em>closed</em><br/>
→ randomly open sites <em>one by one</em> <br/>
→ when the sys percolates, the <em>vacancy percentage</em> is an estimate of <em>p</em> <br/>
*(run above simulation for millions of times) </p>
<h3 id="implementation_3">implementation</h3>
<ul>
<li>N^2 sites, named 0 to N^2-1 </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image022.png"/> </p>
<ul>
<li>add 2 more vertual sites: one on top, one on bottom </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week1_1/pasted_image021.png"/> </p>
<ul>
<li>openning a site: union to adjcent open sites (at most 4 unions)</li>
</ul>
</div>
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<div id="toc"><ul><li><a class="toc-href" href="#1-dynamic-connectivity-pb" title="1. Dynamic Connectivity pb">1. Dynamic Connectivity pb</a><ul><li><a class="toc-href" href="#pb-statement" title="pb statement">pb statement</a></li><li><a class="toc-href" href="#uf-api" title="UF API">UF API</a></li></ul></li><li><a class="toc-href" href="#2-quick-find_1" title="2. Quick Find">2. Quick Find</a><ul><li><a class="toc-href" href="#data-structure" title="data structure">data structure</a></li><li><a class="toc-href" href="#uf-operations" title="UF operations">UF operations</a></li><li><a class="toc-href" href="#implementation" title="implementation">implementation</a></li></ul></li><li><a class="toc-href" href="#3-quick-union_1" title="3. Quick Union">3. Quick Union</a><ul><li><a class="toc-href" href="#data-structure_1" title="data structure">data structure</a></li><li><a class="toc-href" href="#uf-operations_1" title="UF operations">UF operations</a></li><li><a class="toc-href" href="#implementation_1" title="implementation">implementation</a></li><li><a class="toc-href" href="#complexity" title="complexity">complexity</a></li></ul></li><li><a class="toc-href" href="#4-quick-union-improvements_1" title="4. Quick Union Improvements">4. Quick Union Improvements</a><ul><li><a class="toc-href" href="#improvement1-weighting" title="improvement1: weighting">improvement1: weighting</a><ul><li><a class="toc-href" href="#implementation_2" title="implementation">implementation</a></li><li><a class="toc-href" href="#complexity_1" title="complexity">complexity</a></li></ul></li><li><a class="toc-href" href="#improvement-2-path-compression_1" title="improvement 2: path compression">improvement 2: path compression</a><ul><li><a class="toc-href" href="#complexity_2" title="complexity">complexity</a></li></ul></li><li><a class="toc-href" href="#summery_1" title="summery">summery</a></li></ul></li><li><a class="toc-href" href="#5-union-find-application_1" title="5. Union Find Application">5. Union Find Application</a><ul><li><a class="toc-href" href="#percolation" title="percolation">percolation</a></li><li><a class="toc-href" href="#implementation_3" title="implementation">implementation</a></li></ul></li></ul></div>
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