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[Algorithms I] Week 4-2b Binary Search Trees
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-binary-search-trees" title="1. Binary Search Trees">1. Binary Search Trees</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></ul></li><li><a class="toc-href" href="#2-oredered-operations-in-bst_1" title="2. Oredered Operations in BST">2. Oredered Operations in BST</a><ul><li><a class="toc-href" href="#minmax" title="min/max">min/max</a></li><li><a class="toc-href" href="#floorceiling" title="floor/ceiling">floor/ceiling</a></li><li><a class="toc-href" href="#rankselect" title="rank/select">rank/select</a></li><li><a class="toc-href" href="#iteration" title="iteration">iteration</a></li></ul></li><li><a class="toc-href" href="#3-deletions-in-bst_1" title="3. Deletions in BST">3. Deletions in BST</a><ul><li><a class="toc-href" href="#lazy-approch" title="lazy approch">lazy approch</a></li><li><a class="toc-href" href="#deleteminmax" title="deleteMin/Max">deleteMin/Max</a></li><li><a class="toc-href" href="#hibbard-deletion" title="Hibbard deletion">Hibbard deletion</a></li><li><a class="toc-href" href="#analysis_1" title="Analysis">Analysis</a></li></ul></li></ul></div>
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<p>(BST是锻炼递归代码的好题目) </p>
<h1 id="1-binary-search-trees">1. Binary Search Trees</h1>
<p>def. <strong>BST</strong> <br/>
A binary tree where each node has a key: <br/>
for every node, the key is larger than <em>all nodes in left</em> subtree, smaller than <em>all nodes in right</em> subtree. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_2b/pasted_image.png"/> <br/>
Fields: key, val, left, right </p>
<h2 id="implementation">Implementation</h2>
<p>An inner class of BST nodes: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private class Node{ </span></span>
<span class="code-line"><span class="err"> private Key key; </span></span>
<span class="code-line"><span class="err"> private Value val; </span></span>
<span class="code-line"><span class="err"> private Node left, right; </span></span>
<span class="code-line"><span class="err"> public Node(Key k, Value v){...} </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>skeleton implementation of BST: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public class BST<Key implements Comparable<Key>, Value>{ </span></span>
<span class="code-line"><span class="err">private Node root; </span></span>
<span class="code-line"><span class="err">private class Node{...} </span></span>
<span class="code-line"><span class="err">public Value get(Key k){...} </span></span>
<span class="code-line"><span class="err">public void put(Key k, Value v){} </span></span>
<span class="code-line"><span class="err">public void delete(Key k){} </span></span>
<span class="code-line"><span class="err">public Iterable<Key> iterator(){} </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p><strong>search</strong> <br/>
recursive version: <br/>
(或者把这个函数写到Node类里面也可以. ) </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private Value get(Node nd, Key k){ </span></span>
<span class="code-line"><span class="err"> if(nd==null) return null; // search miss </span></span>
<span class="code-line"><span class="err"> int cmp = k.compareTo(nd.key); </span></span>
<span class="code-line"><span class="err"> if(cmp==0) return nd.val; // search hit </span></span>
<span class="code-line"><span class="err"> else if (cmp>0) return get(nd.right, k); </span></span>
<span class="code-line"><span class="err"> else return get(nd.left, k); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>non-recursive version: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public Value get(Key k){ </span></span>
<span class="code-line"><span class="err"> Node nd=root; </span></span>
<span class="code-line"><span class="err"> while(root!=null){ </span></span>
<span class="code-line"><span class="err"> int cmp = k.compareTo(nd.key); </span></span>
<span class="code-line"><span class="err"> if (cmp==0) return nd.val; </span></span>
<span class="code-line"><span class="err"> else if(cmp>0) nd = nd.right; </span></span>
<span class="code-line"><span class="err"> else nd = nd.left; </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> return null; </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p><strong>insert</strong> <br/>
recursive version: <br/>
(注意这个recursive函数的返回值不是void! 这里是一个trick: 返回的是在分叉以前的那个节点) </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private Node put(Node nd, Key k, Value v){ </span></span>
<span class="code-line"><span class="err"> if(nd==null) return new Node(k, v); </span></span>
<span class="code-line"><span class="err"> int cmp = k.compareTo(nd.key); </span></span>
<span class="code-line"><span class="err"> if(cmp==0) nd.val = v; </span></span>
<span class="code-line"><span class="err"> else if(cmp>0) nd.right = put(nd.right, k, v); </span></span>
<span class="code-line"><span class="err"> else nd.left = put(nd.left, k, v); </span></span>
<span class="code-line"><span class="err"> return nd; </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>non-recursive version: <br/>
不如递归版本优美... </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public void put(Key k, Value v){ </span></span>
<span class="code-line"><span class="err"> Node nd = root; </span></span>
<span class="code-line"><span class="err"> while(true){ </span></span>
<span class="code-line"><span class="err"> int cmp = k.compareTo(nd.key); </span></span>
<span class="code-line"><span class="err"> if(cmp==0) { </span></span>
<span class="code-line"><span class="err"> nd.val = v; break; </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> else if(cmp>0){ </span></span>
<span class="code-line"><span class="err"> if(nd.right!=null) nd = nd.right; </span></span>
<span class="code-line"><span class="err"> else {nd.right = new Node(k,v); break;} </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> else if (nd.left!=null) { </span></span>
<span class="code-line"><span class="err"> if(nd.left!=null) nd = nd.left; </span></span>
<span class="code-line"><span class="err"> else {nd.left = new Node(k,v); break;} </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h2 id="analysis">Analysis</h2>
<p>complexity: depth of the BST. </p>
<p>shape of BST: depends on how the keys come in (order of insertion). <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_2b/pasted_image001.png"/> <br/>
if keys come in <em>random</em> order: could be pretty well balanced. </p>
<p><strong>BST and quick-sort partitionning</strong> <br/>
<em>The root of BST is just the pivot in quick sort partitioning * <br/>
if all keys are distinct ⇒ one-to-one correspondence between quick sort and BST. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_2b/pasted_image002.png"/> <br/>
⇒ <strong>proposition</strong> <br/>
if all keys are distinct and come in randome order, the average number of compares for a search/insert is ~2lnN (or 1.39lgN). <br/>
</em>proof.* <br/>
证明见quicksort那里的数学推导... </p>
<p><strong>proposition </strong>(Reed, 2003) <br/>
N distinct keys come in random order, average tree height = 4.300lnN </p>
<p><strong>Worst-case:</strong> <br/>
The tree becomes just like a linked list: ~N for insertion and search </p>
<h1 id="2-oredered-operations-in-bst_1">2. Oredered Operations in BST</h1>
<p>task: ordered opeartions </p>
<ul>
<li><code>min()/max()</code>: min/max key </li>
<li><code>deleteMin()/deleteMax()</code> </li>
<li><code>floor(Key k)/ceiling(Key k)</code>: largest key <=k / smallest key >=k </li>
<li><code>rank(Key k)</code>: nb of keys < key </li>
<li><code>select(int i)</code>: key with rank=i </li>
<li><code>Iterator<Key> keys(lo, hi)</code>: iterates through [lo, hi] </li>
</ul>
<h2 id="minmax">min/max</h2>
<p>easy <br/>
min: left-most <br/>
max: right-most <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_2b/pasted_image003.png"/> </p>
<h2 id="floorceiling">floor/ceiling</h2>
<p>a little more complexed... </p>
<p><strong>floor (ceiling is similar)</strong> </p>
<ul>
<li>if k==nd.key </li>
</ul>
<p>return nd.val </p>
<ul>
<li>if k<nd.key </li>
</ul>
<p>the floor <em>must</em> be in the left subtree </p>
<ul>
<li>if k>nd.key <ul>
<li>如果min(nd.right) > k: 返回nd.val </li>
<li>如果min(nd.right) <= k: go to right <div class="highlight"><pre><span class="code-line"><span></span><span class="err">public Value floor(Node nd, Key k){// largest element with key <= k </span></span>
<span class="code-line"><span class="err">int cmp = k.compareTo(nd.key); </span></span>
<span class="code-line"><span class="err">if(cmp==0) return nd.val;//case 1 </span></span>
<span class="code-line"><span class="err">else if(cmp<0) return floor(nd.left, k);//case 2 </span></span>
<span class="code-line"><span class="err">if (nd.right==null || min(nd.right).compareTo(k)>0) //case 3 </span></span>
<span class="code-line"><span class="err"> return nd.val; </span></span>
<span class="code-line"><span class="err">else return floor(nd.right); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
</li>
</ul>
</li>
</ul>
<p>他提供的版本和我写的不一样: 递归函数floor返回的也是一个Node: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_2b/pasted_image004.png"/> </p>
<h2 id="rankselect">rank/select</h2>
<p>In each node, store the number of nodes in the subtree: <em>add an extra field</em>. </p>
<p><strong>size</strong> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private class Node{ </span></span>
<span class="code-line"><span class="err"> private int count; </span></span>
<span class="code-line"><span class="err"> //... </span></span>
<span class="code-line"><span class="err">} </span></span>
<span class="code-line"><span class="err">public int size(){ </span></span>
<span class="code-line"><span class="err"> return size(root); </span></span>
<span class="code-line"><span class="err">} </span></span>
<span class="code-line"><span class="err">public int size(Node nd){ </span></span>
<span class="code-line"><span class="err"> if(nd==null) return 0;// this is why we do not put size() inside the class Node! </span></span>
<span class="code-line"><span class="err"> return nd.count; </span></span>
<span class="code-line"><span class="err">} </span></span>
<span class="code-line"><span class="err">public void put(Node nd, Key k, Value v){ </span></span>
<span class="code-line"><span class="err"> //..... </span></span>
<span class="code-line"><span class="err"> nd.count = size(nd.left)+size(nd.right)+1;//maintain count for each node </span></span>
<span class="code-line"><span class="err"> return nd; </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p><strong>rank</strong> <br/>
(return nb of keys < k) </p>
<ul>
<li>if nd.key==k </li>
</ul>
<p>return size(nd.left) </p>
<ul>
<li>if nd.key>k </li>
</ul>
<p>return rank(nd.left, k) </p>
<ul>
<li>if nd.key<k </li>
</ul>
<p>return size(nd.left)+1+rank(nd,right, k) </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private int rank(Node nd, Key k){ </span></span>
<span class="code-line"><span class="err"> if(nd==null) return 0;//remember null case </span></span>
<span class="code-line"><span class="err"> int cmp = k.compareTo(nd.key); </span></span>
<span class="code-line"><span class="err"> if(cmp==0) return size(nd.left) </span></span>
<span class="code-line"><span class="err"> else if (cmp<0) return rank(nd.left, k); </span></span>
<span class="code-line"><span class="err"> else return size(nd.left)+1+rank(nd.right,k); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>select() similar... </p>
<h2 id="iteration">iteration</h2>
<p><strong>Inorder traversal 中序遍历</strong> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public Iterable<Key> keys(){ </span></span>
<span class="code-line"><span class="err"> Queue<Key> q = new Queue<Key>(); </span></span>
<span class="code-line"><span class="err"> inorder(root, q); </span></span>
<span class="code-line"><span class="err"> return q; </span></span>
<span class="code-line"><span class="err">} </span></span>
<span class="code-line"><span class="err">private void inorder(Node nd, Queue<Key> q){ </span></span>
<span class="code-line"><span class="err"> if(nd==null) return; </span></span>
<span class="code-line"><span class="err"> inorder(nd.left); </span></span>
<span class="code-line"><span class="err"> q.enqueue(nd.key); </span></span>
<span class="code-line"><span class="err"> inorder(nd.right); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p><img alt="" class="img-responsive" src="../images/algoI_week4_2b/pasted_image005.png"/> </p>
<p><strong>property</strong> <br/>
inorder-traversal gives the keys <em>in ascending order</em>. <br/>
(proof by induction) </p>
<h1 id="3-deletions-in-bst_1">3. Deletions in BST</h1>
<p>one final function to implement: delete(Key k), deleteMin(), deleteMax() <br/>
→ and remember to update the count field... </p>
<p>(感觉这篇文章其实就讲的很清楚了: <a href="http://www.algolist.net/Data_structures/Binary_search_tree/Removal">http://www.algolist.net/Data_structures/Binary_search_tree/Removal</a> 这个在递归函数里使用了parent这个参数) </p>
<h2 id="lazy-approch">lazy approch</h2>
<p><code>put(k, null)</code>, and leave the key in the tree (tombstone) <br/>
→ not good if have large number of tombstons... </p>
<h2 id="deleteminmax">deleteMin/Max</h2>
<p>go the the left-most node → replace it with its right node. <br/>
Recusive function with the <em>returning-node trick</em>: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private Node deleteMin(Node nd){ </span></span>
<span class="code-line"><span class="err"> if(nd==null) return null; // this might not happen </span></span>
<span class="code-line"><span class="err"> if(nd.left==null) return nd.right; </span></span>
<span class="code-line"><span class="err"> else nd.left = deleteMin(nd.left); </span></span>
<span class="code-line"><span class="err"> nd.count = size(nd.left)+1+size(right);//remember to maintain the count field </span></span>
<span class="code-line"><span class="err"> return nd; </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>这个递归的技巧又一次使用了. </p>
<h2 id="hibbard-deletion">Hibbard deletion</h2>
<p>first find node with the key to delete, 3 cases: </p>
<ul>
<li>0 children: </li>
</ul>
<p>simply set parent link to null <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_2b/pasted_image006.png"/> </p>
<ul>
<li>1 child: </li>
</ul>
<p>replace parent link with the child <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_2b/pasted_image007.png"/> </p>
<ul>
<li><strong>2 children (most subtle)</strong> <ul>
<li>first replace node key with smallest key in right subtree </li>
<li>remove the smallest key in right subtree </li>
</ul>
</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week4_2b/pasted_image008.png"/> </p>
<p><strong>code of Hibbard deletion</strong> <br/>
Again (for the 3rd time) use the <em>return-nd trick</em>... </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private Node delete(Node nd, Key k){ </span></span>
<span class="code-line"><span class="err"> if(nd==null) return null;// search miss </span></span>
<span class="code-line"><span class="err"> int cmp = k.compareTo(nd.key); </span></span>
<span class="code-line"><span class="err"> if(cmp>0) nd.right = delete(nd.right, k); </span></span>
<span class="code-line"><span class="err"> else if(cmp<0) nd.left = delete(nd.left,k); </span></span>
<span class="code-line"><span class="err"> else{ </span></span>
<span class="code-line"><span class="err"> //if nd is the node to delete </span></span>
<span class="code-line"><span class="err"> if(nd.left==null) return nd.right; </span></span>
<span class="code-line"><span class="err"> if(nd.right==null) return nd.left; </span></span>
<span class="code-line"><span class="err"> Key k2 = min(nd.right); </span></span>
<span class="code-line"><span class="err"> nd.key = k2; </span></span>
<span class="code-line"><span class="err"> nd.right = delete(nd.right, k2); </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err"> nd.count = size(nd.left)+1+size(nd.right); </span></span>
<span class="code-line"><span class="err"> return nd; </span></span>
<span class="code-line"><span class="err">} </span></span>
<span class="code-line"><span class="err">public void delete(Key k){ </span></span>
<span class="code-line"><span class="err"> root = delete(root, k); </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>感觉用了recursive return-nd 这个trick的实现很漂亮.... 比那篇博客里放一个参数进递归函数以及用auxroot的办法要好不少... </p>
<h2 id="analysis_1">Analysis</h2>
<p>problem: <em>not symmetric</em> <br/>
If random insert and delete for a while ⇒ tree become much less balanced ! Tree height tend to be sqrt(N). </p>
<p><strong>summery</strong> <br/>
BST is much better in average case, but not guaranteed for worst case. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week4_2b/pasted_image009.png"/> </p>
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<div id="toc"><ul><li><a class="toc-href" href="#1-binary-search-trees" title="1. Binary Search Trees">1. Binary Search Trees</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></ul></li><li><a class="toc-href" href="#2-oredered-operations-in-bst_1" title="2. Oredered Operations in BST">2. Oredered Operations in BST</a><ul><li><a class="toc-href" href="#minmax" title="min/max">min/max</a></li><li><a class="toc-href" href="#floorceiling" title="floor/ceiling">floor/ceiling</a></li><li><a class="toc-href" href="#rankselect" title="rank/select">rank/select</a></li><li><a class="toc-href" href="#iteration" title="iteration">iteration</a></li></ul></li><li><a class="toc-href" href="#3-deletions-in-bst_1" title="3. Deletions in BST">3. Deletions in BST</a><ul><li><a class="toc-href" href="#lazy-approch" title="lazy approch">lazy approch</a></li><li><a class="toc-href" href="#deleteminmax" title="deleteMin/Max">deleteMin/Max</a></li><li><a class="toc-href" href="#hibbard-deletion" title="Hibbard deletion">Hibbard deletion</a></li><li><a class="toc-href" href="#analysis_1" title="Analysis">Analysis</a></li></ul></li></ul></div>
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