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[Algorithms I] Week 4-1 Priority Queue
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-api-and-elementary-implementations" title="1. API and elementary implementations">1. API and elementary implementations</a><ul><li><a class="toc-href" href="#pq-client-example" title="PQ client example">PQ client example</a></li><li><a class="toc-href" href="#elementarynaive-implementations" title="elementary(naive) implementations">elementary(naive) implementations</a></li></ul></li><li><a class="toc-href" href="#2-binary-heaps_1" title="2. Binary Heaps">2. Binary Heaps</a><ul><li><a class="toc-href" href="#array-representation-of-binary-heap" title="Array representation of binary heap">Array representation of binary heap</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-heapsort_1" title="3. Heapsort">3. Heapsort</a><ul><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></li></ul></li><li><a class="toc-href" href="#4-event-driven-simulation_1" title="4. Event-driven simulation">4. Event-driven simulation</a><ul><li><a class="toc-href" href="#model" title="model">model</a></li><li><a class="toc-href" href="#event-driven-simulation" title="Event driven simulation">Event driven simulation</a></li><li><a class="toc-href" href="#implementation_2" title="Implementation">Implementation</a></li></ul></li></ul></div>
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<h1 id="1-api-and-elementary-implementations">1. API and elementary implementations</h1>
<p>Collection: data struct for inserting and deleting items (ex. stack and queue). <br/>
Priority queue: a special kind of collection — remove <em>largest/smallest</em> element. </p>
<p>API: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span> <span class="k">class</span> <span class="k">Max</span><span class="o"><</span><span class="n">Kye</span> <span class="n">implements</span> <span class="n">Comparable</span><span class="o"><</span><span class="k">Key</span><span class="o">>></span><span class="err">{</span> </span>
<span class="code-line"> <span class="k">public</span> <span class="n">MaxPQ</span><span class="p">();</span> </span>
<span class="code-line"> <span class="k">public</span> <span class="n">void</span> <span class="k">insert</span><span class="p">(</span><span class="k">Key</span> <span class="n">k</span><span class="p">);</span> </span>
<span class="code-line"> <span class="k">public</span> <span class="k">Key</span> <span class="n">delMax</span><span class="p">();</span> </span>
<span class="code-line"> <span class="k">public</span> <span class="nb">boolean</span> <span class="n">isEmpty</span><span class="p">();</span> </span>
<span class="code-line"> <span class="k">public</span> <span class="k">Key</span> <span class="k">max</span><span class="p">();</span> </span>
<span class="code-line"> <span class="nb">int</span> <span class="k">size</span><span class="p">();</span></span>
<span class="code-line"></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h2 id="pq-client-example">PQ client example</h2>
<p>find <em>M largest</em> elements from N items. (N is too huge to store) <br/>
⇒ 思路: 用 <strong>Min</strong><em>PQ</em>, 当size>M时删掉最小元素 — 最后剩下的就是最大的M个元素了. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image.png"/> </p>
<h2 id="elementarynaive-implementations">elementary(naive) implementations</h2>
<p><strong>unordered array implementation</strong> <br/>
insert ~1, <br/>
remove ~N </p>
<p><strong>ordered array implementation</strong> <br/>
insert ~N — ~ insertion sort, <br/>
remove ~1 </p>
<p><strong>goal</strong>: <br/>
insert O(lgN) & remove max O(lgN) </p>
<h1 id="2-binary-heaps_1">2. Binary Heaps</h1>
<p>binary heap: a special kind of <em>complete binary tree</em>. </p>
<p>def. "<strong>complete binary tree"</strong> <br/>
All level except the last level are full, all nodes in the last level are as far left as possible. <br/>
完全二叉树大概长这个样子: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image001.png"/> </p>
<p>property: a complete binary tree with N items has height = lgN. </p>
<p>def. <strong>binary heap</strong> <br/>
Binary heap is a complete binary tree that satisfies: <em>each node >= any of its 2 children</em> ("heap ordering"). </p>
<p>property of binary heap: </p>
<ul>
<li>max element is root </li>
<li>height is lgN </li>
<li><em>each node still makes a heap</em> (use this property to construct heap from bottom up) </li>
</ul>
<h2 id="array-representation-of-binary-heap">Array representation of binary heap</h2>
<p>Use an array <code>a[]</code> to represent a complete binary tree: very easy to get children and parent. </p>
<ul>
<li><strong>index starting from 1</strong> (root = a[1], do not use a[0]) </li>
<li>for node at index <code>i</code>, its children are: <code>i*2</code> and <code>i*2+1</code> </li>
<li>for node at index <code>i</code> (i>1), its parent is: <code>i/2</code> </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image002.png"/> </p>
<h2 id="implementation">Implementation</h2>
<p>简言之就是: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image003.png"/> </p>
<p>内部成员变量: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class MaxPQ<Key implements Comparable<Key>>{ </span></span>
<span class="code-line"><span class="err"> private Key[] a; </span></span>
<span class="code-line"><span class="err"> private int N; </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p><strong>siftup (swim)</strong> <br/>
if a node is larger than its parent: <em>exchange with parent until heap ordering is restored.</em> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">private</span><span class="w"> </span><span class="n">void</span><span class="w"> </span><span class="n">swim</span><span class="p">(</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="k">while</span><span class="p">(</span><span class="n">k</span><span class="o">></span><span class="mi">1</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">k/2</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="p">)</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">k</span><span class="p">,</span><span class="n">k</span><span class="o">/</span><span class="mi">2</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">/=</span><span class="w"> </span><span class="mi">2</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>⇒ <strong>use siftup for inserting elements</strong> <br/>
when inserting an element, first append it to the end of the array, then siftup the element to its right position. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public void insert(Key k){ </span></span>
<span class="code-line"><span class="err">a[N++]=x; </span></span>
<span class="code-line"><span class="err">swim(N); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p><strong>siftdown (sink)</strong> <br/>
if a node is smaller than its children: <em>exchange it with the bigger child.</em> 用公司高层变动来类比很形象... </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">private</span><span class="w"> </span><span class="n">void</span><span class="w"> </span><span class="n">sink</span><span class="p">(</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="k">while</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="mi">2</span><span class="o"><=</span><span class="n">N</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">maxIndex</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">k</span><span class="o">]</span><span class="p">,</span><span class="n">a</span><span class="o">[</span><span class="n">k*2</span><span class="o">]</span><span class="p">)</span><span class="vm">?</span><span class="w"> </span><span class="n">k</span><span class="o">*</span><span class="mi">2</span><span class="w"> </span><span class="err">:</span><span class="w"> </span><span class="n">k</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">k</span><span class="o">*</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span><span class="o"><=</span><span class="n">N</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">maxIndex</span><span class="o">]</span><span class="p">,</span><span class="n">a</span><span class="o">[</span><span class="n">k*2+1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="n">maxIndex</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="o">*</span><span class="n">k</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">maxIndex</span><span class="o">==</span><span class="n">k</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="n">exch</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">k</span><span class="p">,</span><span class="n">maxIndex</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxIndex</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><strong>⇒ use siftdown for deleting max operation</strong> <br/>
First put a[N] to the position of root, then sink. (注意要防止loitering) </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span><span class="w"> </span><span class="k">Key</span><span class="w"> </span><span class="n">delMax</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">Key</span><span class="w"> </span><span class="n">ret</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">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">a</span><span class="o">[</span><span class="n">1</span><span class="o">]</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">N</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">a</span><span class="o">[</span><span class="n">N--</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="n">avoid</span><span class="w"> </span><span class="n">loitering</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">sink</span><span class="p">(</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">return</span><span class="w"> </span><span class="n">ret</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">Analysis</h2>
<p>insert: ~lgN <br/>
remove max: ~lgN </p>
<p>improvements: </p>
<ul>
<li>use a <strong>d-way heap</strong> instead of a 2-way heap (for ex. 3-way heap, the children of i are i<em>3, i</em>3+1, i*3+2, the parent of i is i/3) </li>
<li>Fibonacci heap: insert is <em>~1</em>, remove max still ~lgN, (but too complicated to use in practice). </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image004.png"/> </p>
<p>considerations: </p>
<ul>
<li>use immutable keys to avoid client from changing the keys (<code>final</code> keyword in java) </li>
<li>underflow/overflow: throw exceptions for delMax() when empty, use resizing array for implementation. </li>
<li>expand the API with other operations: remove an arbitrary item, change the prority of an item (like handling an array)... </li>
</ul>
<h1 id="3-heapsort_1">3. Heapsort</h1>
<p>Basic idea: </p>
<ul>
<li>create a max heap with all N entries <strong>("heap-construction")</strong> </li>
<li>then repeatedly remove all N items <strong>("sortdown")</strong> </li>
</ul>
<h2 id="implementation_1">Implementation</h2>
<p><strong>heap construction</strong> <br/>
Construct the heap using a <strong>bottom-up</strong> method: build the heaps with small size first. </p>
<ul>
<li>one-node heaps (jus leaf node of heap): need not consider </li>
<li>larger heaps: children(subtrees) are already heap-order → just perform sink operation for root </li>
</ul>
<p>⇒ <em>sink elements at indices from N/2 to 1</em> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">for(int k=N/2; k>=1; k++) </span></span>
<span class="code-line"><span class="err">sink(k);</span></span>
</pre></div>
<p>→ complexity is <strong>linear</strong>, see analysis below. </p>
<p><strong>heap destruction </strong> <br/>
now that we have a max-heap, to get the sorted array <em>in-place,</em> simply do: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">while(N>1){ </span></span>
<span class="code-line"><span class="err">exch(a,1, N--); </span></span>
<span class="code-line"><span class="err">sink(1); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h2 id="analysis_1">Analysis</h2>
<p><strong>proposition</strong> <br/>
Heap construction uses <strong><2N</strong> compares and exchanges. <br/>
<em>proof</em> <br/>
一个有h层的heap, 高度为k的subtree有 2^(h-k)个, 每个subtree最多可以交换k次(从root交换到最底层) <br/>
→ 每一层最多有 k*2^(h-k) 次交换 <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image007.png"/> <br/>
(第一个等式可以用数学归纳法证...) <br/>
<a href="">详细证明见http://algs4.cs.princeton.edu/24pq/</a> (Q20答案) </p>
<p><strong>proposition</strong> <br/>
Heapsort use < 2NlgN compares and exchanges. </p>
<p>Heapsort is the first <em>in-place</em> sorting algorithm with NlgN <em>worst-case</em> performance. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image005.png"/> </p>
<p>⇒ heapsort is <em>optimal for time and space</em>, but: </p>
<ul>
<li>inner loop is longer than quicksort </li>
<li>poor use of cache memory (too many references to cache — look far away from the array entry when going down through the tree) </li>
<li>not stable: because it does long-distance exchanges </li>
</ul>
<p>Summary: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image006.png"/> </p>
<h1 id="4-event-driven-simulation_1">4. Event-driven simulation</h1>
<p>Goal: simulating the motion of N particles in elastic collision (using a priority queue). <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image008.png"/> </p>
<h2 id="model">model</h2>
<p><img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image009.png"/> <br/>
bouncing balls (without collision) <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image010.png"/> </p>
<p>Ball class: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image011.png"/> </p>
<p>challenge: which objects to check? How to do it efficently ? </p>
<p><strong>Time driven simulation</strong> </p>
<ul>
<li>upate the position of balls every <code>dt</code> seconds </li>
<li>if overlap detected: figure how the velocity change and continue the simulation. </li>
</ul>
<p>problem with time-driven simulation: </p>
<ul>
<li>1/2*N^2 overlap check per time unit </li>
<li>if dt small: too much calculation </li>
<li>if dt too large: might miss collisions </li>
</ul>
<h2 id="event-driven-simulation">Event driven simulation</h2>
<p>change state only when something happens </p>
<ul>
<li>particles move in straight line before collision </li>
<li>→ collision time can be <em>predicted</em> </li>
<li>maintain PQ for collisions, key=collision time </li>
</ul>
<p><strong>collision prediction</strong> <br/>
input: each particle has radius <code>s</code>, position <code>(rx, ry)</code>, velocity <code>(vx, vy)</code>. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image013.png"/> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image014.png"/> <br/>
(美国高中物理这么凶残?...) </p>
<h2 id="implementation_2">Implementation</h2>
<p>anyway, 这个是封装好的类: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image015.png"/> <br/>
注意, 使用了count记录一个particle到目前为止的碰撞次数. </p>
<p>initialization: <strong>quadratic</strong> </p>
<ul>
<li>Fill PQ with all <em>potential</em> particle-particle collisions </li>
<li>Fill (the same) PQ with all <em>potential</em> particle-wall collisions </li>
</ul>
<p><strong>Main loop </strong> </p>
<ul>
<li>get next event from PQ, event time = t </li>
<li>ignore if the event is <em>invalidated</em> </li>
<li>advance all particles to time t </li>
<li>update velocities </li>
<li>predict future collisions and insert to PQ </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image018.png"/> <br/>
判断invalidate: <br/>
一个event构造(predict)时记录粒子构造时刻的碰撞次数(countA, countB), <br/>
然后在调用invalidate的时候, 如果粒子的碰撞次数发生了改变, 则返回false. <br/>
(详见他们的代码: <a href="http://algs4.cs.princeton.edu/61event/CollisionSystem.java.html">http://algs4.cs.princeton.edu/61event/CollisionSystem.java.html</a>) </p>
<p>Event class: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image016.png"/> <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_1/pasted_image017.png"/> </p>
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