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[Algorithms I] Week 5-1 Balanced Search Trees
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-2-3-search-trees" title="1. 2-3 Search Trees">1. 2-3 Search Trees</a><ul><li><a class="toc-href" href="#analysis" title="Analysis">Analysis</a></li><li><a class="toc-href" href="#implementation" title="Implementation">Implementation</a></li></ul></li><li><a class="toc-href" href="#2-red-black-bst_1" title="2. Red-Black BST">2. Red-Black BST</a><ul><li><a class="toc-href" href="#properties" title="properties">properties</a></li><li><a class="toc-href" href="#representation" title="representation">representation</a></li><li><a class="toc-href" href="#elementary-operations" title="elementary operations">elementary operations</a></li><li><a class="toc-href" href="#implementation_1" title="Implementation">Implementation</a></li><li><a class="toc-href" href="#code" title="Code">Code</a></li><li><a class="toc-href" href="#analysis_1" title="Analysis">Analysis</a></li></ul></li><li><a class="toc-href" href="#3-b-trees_1" title="3. B-trees">3. B-trees</a><ul><li><a class="toc-href" href="#b-tree" title="B-tree">B-tree</a></li><li><a class="toc-href" href="#searching" title="Searching">Searching</a></li><li><a class="toc-href" href="#insertion" title="Insertion">Insertion</a></li><li><a class="toc-href" href="#analysis_2" title="Analysis">Analysis</a></li><li><a class="toc-href" href="#system-implementations" title="System implementations">System implementations</a></li></ul></li></ul></div>
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<p>goal: lgN for insert/search/delete operations (not necessarily <em>binary</em> trees..) <br/>
3 algo: 2-3 tree, (left leaning) red-black tree, B-tree </p>
<h1 id="1-2-3-search-trees">1. 2-3 Search Trees</h1>
<p>def. <strong>2-3 tree</strong> </p>
<ul>
<li>allow 1 or 2 keys per node, & 2 or 3 children per node: <ul>
<li>2-node: one key, 2 children (ordinary BST node) </li>
<li>3-node: 2 keys, 3 children (3 children: <em>less, between, more</em>) </li>
</ul>
</li>
<li><strong>perfect balance</strong>: every path from root to null link has the same length (2-3 tree的一个超好的性质, <em>类似于一个满二叉树</em>!) </li>
<li>symmetric order: inorder traversal gives ascending order (和BST类似) </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image.png"/> <br/>
<strong>search</strong> <br/>
Just follow the correct link... Natural generalization of search in BST... </p>
<p><strong>insert</strong> </p>
<ul>
<li>case 1: insert into a 2-node at bottom </li>
</ul>
<p>just convert a 2-node into a 3-node </p>
<ul>
<li>case 2: insert into a 3-node at bottom <ul>
<li>create a temporary 4-node (three keys) </li>
<li>move middle key in 4-node into parent, split the rest two keys into two 2-nodes </li>
</ul>
</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image001.png"/> <img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image002.png"/> <img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image003.png"/> </p>
<ul>
<li>if parent becom a 3-node → continue the process </li>
<li>if arrived at the root (root is a 4-node with three keys): split it into three 2-nodes </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image004.png"/> <img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image005.png"/> </p>
<p><strong>splitting a 4-node</strong>: can be done in constant time (<em>local transformation</em>). <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image006.png"/> </p>
<h2 id="analysis">Analysis</h2>
<p>Invariant: maintains symmetric order and perfect balance. <br/>
<em>proof.</em> <br/>
each transformation maintains the order and the balance, all possible transformations: <br/>
这个图很好, 3-node的插入一共有三种情况: 自身是root/父亲是2-node/父亲是3-node <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image007.png"/> </p>
<p><strong>performance</strong> <br/>
every path from root to null link has the same length. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image008.png"/> </p>
<h2 id="implementation">Implementation</h2>
<ul>
<li>direct implementation is complicated: </li>
<li>bottom line: <em>Could do it, but there's a better way.</em> </li>
</ul>
<h1 id="2-red-black-bst_1">2. Red-Black BST</h1>
<p>LLRB tree: left-leaning red-black tree. </p>
<blockquote>
<p><em>BST representation of the 2-3 trees</em> <br/>
<strong>use internal left-leaning links for 3 nodes</strong> </p>
</blockquote>
<p>红色link即为internal left leaning link (红黑树就是这么来的), 用红色link连接起来的组成一个(虚拟的)3-node 或4-node. </p>
<ul>
<li>3-node用一个red link表示: </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image009.png"/> </p>
<ul>
<li>4-node用两个red link表示: </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image030.png"/> ⇒ <img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image029.png"/> or <img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image027.png"/> or <img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image028.png"/> </p>
<p>example: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image010.png"/> </p>
<h2 id="properties">properties</h2>
<ul>
<li>no node has two red links (不可以一个节点连两个red link) </li>
<li>every path from path to null link has the same number of <em>black</em> links (想象所有red link都变为horizontal) </li>
<li>all red links lean left </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image011.png"/> </p>
<h2 id="representation">representation</h2>
<p>Each node has only one link from parent <br/>
⇒ add a boolean to encode color of links (the color of the link <em>from parent</em>). <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image012.png"/> </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"> Node left, right; </span></span>
<span class="code-line"><span class="err"> boolean color;//true means red </span></span>
<span class="code-line"><span class="err">} </span></span>
<span class="code-line"><span class="err">private boolean isRed(Node nd){ </span></span>
<span class="code-line"><span class="err"> if (nd==null) return false; </span></span>
<span class="code-line"><span class="err"> return nd.color; </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>insert to parent 操作: 只需把color变为RED即表示该节点 被变成了和父节点一起的一个(虚拟)节点. </p>
<h2 id="elementary-operations">elementary operations</h2>
<p><strong>left-rotation</strong> <br/>
(def: <em>convert a right-learning red link to left.</em> ) <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image013.png"/> <img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image014.png"/> <br/>
(symmetric ordering and perfect black balance are maintained) </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private Node rotateLeft(Node h){ </span></span>
<span class="code-line"><span class="err"> Node s = h.right; </span></span>
<span class="code-line"><span class="err"> h.right = s.left; </span></span>
<span class="code-line"><span class="err"> s.left = h; </span></span>
<span class="code-line"><span class="err"> s.color = h.color; // not = BLACK </span></span>
<span class="code-line"><span class="err"> h.color = RED; </span></span>
<span class="code-line"><span class="err"> return s; </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p><strong>right-rotation</strong> <br/>
(temporarily turn a left-leaning red link to right) <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image015.png"/> <img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image016.png"/> </p>
<p><code>private Node rotateRight(Node h){...}</code> </p>
<p>right rotation 是为了应对这种情况: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image024.png"/> rotateRight(c) ⇒ <img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image023.png"/> </p>
<p><strong>color-flip</strong> <br/>
(split a 4-node, with three kyes — two red links) <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image017.png"/> <img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image018.png"/> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">private void filpColor(Node h){ </span></span>
<span class="code-line"><span class="err"> h.color = RED; </span></span>
<span class="code-line"><span class="err"> h.left.color = BLACK; </span></span>
<span class="code-line"><span class="err"> h.right.color = BLACK; </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<h2 id="implementation_1">Implementation</h2>
<p><strong>Basic strategy</strong> <br/>
Maintain <em>one-to-one correspondence</em> with 2-3 tree by applying elementary operations. </p>
<ul>
<li><strong>search</strong> </li>
</ul>
<p><em>Exactly</em> the same as elementary BST. ( ⇒ The same code for floor and ceiling) </p>
<ul>
<li><strong>insert</strong> </li>
</ul>
<p><strong>Each insert will generate a red link</strong> (then should rotate to make it legal) </p>
<p>插入的时候有两种可能: </p>
<ol>
<li>insert into a 2-node at the bottom </li>
</ol>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image025.png"/> </p>
<ul>
<li>standart BST insert </li>
<li>if have red right link: rotateLeft </li>
</ul>
<p>ex: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image021.png"/> </p>
<ol>
<li>insert into a 3-node </li>
</ol>
<p>有三种可能: insert into left/middle/right, right最简单, left捎复杂, middle最复杂, 见下图: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image026.png"/> </p>
<ul>
<li>standard BST insert and color nodes </li>
<li>if necessary, rotate to balance 4-node, 比如: </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image031.png"/> </p>
<ul>
<li>flip colors to pass red link to upper level </li>
<li>if necessary, rotate to make all links left-leaning </li>
</ul>
<p>ex: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image032.png"/> <br/>
ex2: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image033.png"/> </p>
<p>视频最后一段的demo太帅了! 叹为观止!! </p>
<h2 id="code">Code</h2>
<p>原来<strong>只有4种(其实是3种)情况要调整</strong>: </p>
<ol>
<li><em>left = black, right = red</em> </li>
</ol>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image034.png"/> ⇒ rotateLeft(a) </p>
<ol>
<li><em>left =red, left.right = red [这个不会出现, 因为这对于下一层来说是case 1..]</em> </li>
</ol>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image039.png"/> ⇒ rotateLeft(e) ⇒ 变为case 3 </p>
<ol>
<li><em>left = red, left.left = red</em> </li>
</ol>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image035.png"/> ⇒ rotateRight(s) ⇒ 变为case 4 </p>
<ol>
<li><em>left = red. right = red</em> </li>
</ol>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image036.png"/> ⇒ flipColor(r) </p>
<p>几个状态之间的转化: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image038.png"/> </p>
<p><strong>只要三行代码即可处理LLRB tree !! 老爷子牛逼...... </strong> <br/>
(这个也是在2007年algo第四版的时候才刚刚弄出来的, 以前的代码要复杂) </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,RED); </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; // 这里不急着返回 -- same trick as for BSTs.. </span></span>
<span class="code-line"><span class="err"> else if(cmp<0) nd.left = put(nd.left, k, v); </span></span>
<span class="code-line"><span class="err"> else nd.right = put(nd.right, k, v); </span></span>
<span class="code-line"><span class="err"> // modifications to maintain LLRB tree property: </span></span>
<span class="code-line"><span class="err"> if( isRed(nd.right) && !isRed(nd.left) ) nd = rotateLeft(nd);//case 1 </span></span>
<span class="code-line"><span class="err"> //if( isRed(nd.left) && isRed(nd.left.right) ) nd.left = rotateLeft(nd.left);// case 2 -- never happen... </span></span>
<span class="code-line"><span class="err"> if( isRed(nd.left) && isRed(nd.left.right) ) nd = rotateRight(nd);// case 3 </span></span>
<span class="code-line"><span class="err"> if( isRed(nd.left) && isRed(nd.right) ) flipColor(nd);//case 4 </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>这三行代码越看越精妙...... </p>
<h2 id="analysis_1">Analysis</h2>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image041.png"/> </p>
<p><strong>worst case</strong>: the left path is <em>alternating red and black</em>. <br/>
⇒ longest path <= 2 * shortest path (height<= 2lgN) </p>
<p>practical applications: height ~ 1.0 lgN </p>
<p>summery: <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image042.png"/> </p>
<h1 id="3-b-trees_1">3. B-trees</h1>
<p>setting: data access in file system. <br/>
<em>Probe is much expensive than accessing data within a page.</em> </p>
<p>Goal: access data using a minimum number of probes. </p>
<h2 id="b-tree">B-tree</h2>
<p><strong>def.</strong> <br/>
external nodes: contain just keys, not links <br/>
internal nodes: contain key-link pairs </p>
<p><strong>def. B-tree</strong> <br/>
Generalize 2-3 trees by allowing up to M-1 keys per node: </p>
<ul>
<li>
<blockquote>
<p>= 2 keys in root </p>
</blockquote>
</li>
<li>
<blockquote>
<p>= M/2 keys in other nodes </p>
</blockquote>
</li>
<li>external nodes contain client keys </li>
<li>internal nodes contain copies of keys to guide search </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image043.png"/> </p>
<h2 id="searching">Searching</h2>
<p>similar to BST/2-3tree <br/>
ex. <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image044.png"/> </p>
<p>(Choose M as large as possible so that M links fit into a page) </p>
<h2 id="insertion">Insertion</h2>
<p>similar to 2-3 tree <br/>
<img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image045.png"/> </p>
<h2 id="analysis_2">Analysis</h2>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image046.png"/> </p>
<h2 id="system-implementations">System implementations</h2>
<p>system implementations of RBtree. <br/>
java: <br/>
<code>java.util.TreeMap</code>, <code>java.util.TreeSet</code>. </p>
<p>八卦1: </p>
<p><img alt="" class="img-responsive" src="../images/algoI_week5_1/pasted_image047.png"/> </p>
<p>八卦2:
Sedgewick 的朋友, <a href="https://fr.wikipedia.org/wiki/Philippe_Flajolet">Philippe Flajolet</a>, 是一个X! </p>
</div>
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<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoI_week4_2a.html' style="text-align:left">[Algorithms I] Week 4-2a Elementry Symbol Tables</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoI_week4_2b.html' style="text-align:left">[Algorithms I] Week 4-2b Binary Search Trees</a></li>
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<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoI_week5_2.html' style="text-align:left">[Algorithms I] Week 5-2 Geometric Applications of BSTs</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoI_week6.html' style="text-align:left">[Algorithms I] Week 6 Hash Tables</a></li>
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<div id="toc"><ul><li><a class="toc-href" href="#1-2-3-search-trees" title="1. 2-3 Search Trees">1. 2-3 Search Trees</a><ul><li><a class="toc-href" href="#analysis" title="Analysis">Analysis</a></li><li><a class="toc-href" href="#implementation" title="Implementation">Implementation</a></li></ul></li><li><a class="toc-href" href="#2-red-black-bst_1" title="2. Red-Black BST">2. Red-Black BST</a><ul><li><a class="toc-href" href="#properties" title="properties">properties</a></li><li><a class="toc-href" href="#representation" title="representation">representation</a></li><li><a class="toc-href" href="#elementary-operations" title="elementary operations">elementary operations</a></li><li><a class="toc-href" href="#implementation_1" title="Implementation">Implementation</a></li><li><a class="toc-href" href="#code" title="Code">Code</a></li><li><a class="toc-href" href="#analysis_1" title="Analysis">Analysis</a></li></ul></li><li><a class="toc-href" href="#3-b-trees_1" title="3. B-trees">3. B-trees</a><ul><li><a class="toc-href" href="#b-tree" title="B-tree">B-tree</a></li><li><a class="toc-href" href="#searching" title="Searching">Searching</a></li><li><a class="toc-href" href="#insertion" title="Insertion">Insertion</a></li><li><a class="toc-href" href="#analysis_2" title="Analysis">Analysis</a></li><li><a class="toc-href" href="#system-implementations" title="System implementations">System implementations</a></li></ul></li></ul></div>
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