线段树

分手后的思念是犯贱 2022-08-10 10:53 250阅读 0赞

1、概述

线段树，也叫区间树，是一个完全二叉树，它在各个节点保存一条线段（即“子数组”），因而常用于解决数列维护问题，它基本能保证每个操作的复杂度为O(lgN)。

2、线段树基本操作

线段树的基本操作主要包括构造线段树，区间查询和区间修改。

（1）    线段树构造

首先介绍构造线段树的方法：让根节点表示区间\[0,N-1\]，即所有N个数所组成的一个区间，然后，把区间分成两半，分别由左右子树表示。不难证明，这样的线段树的节点数只有2N-1个，是O(N)级别的，如图：

![segment\_tree][segment_tree]

显然，构造线段树是一个递归的过程，伪代码如下：

线段树除了最后一层外，前面每一层的结点都是满的，因此线段树的深度

h =ceil(log(2n -1))=O(log n)。

（2）    区间查询

区间查询指用户输入一个区间，获取该区间的有关信息，如区间中最大值，最小值，第N大的值等。

比如前面一个图中所示的树，如果询问区间是\[0,2\]，或者询问的区间是\[3,3\]，不难直接找到对应的节点回答这一问题。但并不是所有的提问都这么容易回答，比如\[0,3\]，就没有哪一个节点记录了这个区间的最小值。当然，解决方法也不难找到：把\[0,2\]和\[3,3\]两个区间（它们在整数意义上是相连的两个区间）的最小值“合并”起来，也就是求这两个最小值的最小值，就能求出\[0,3\]范围的最小值。同理，对于其他询问的区间，也都可以找到若干个相连的区间，合并后可以得到询问的区间。

区间查询的伪代码如下：

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      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// node 为线段树的结点类型，其中Left 和Right 分别表示区间左右端点</code> 
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       &nbsp; 
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      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// Lch 和Rch 分别表示指向左右孩子的指针</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,102,153)!important; background:none!important">void</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> Query(node *p, </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:gray!important; background:none!important">int</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> a, </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:gray!important; background:none!important">int</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> b) </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 当前考察结点为p，查询区间为(a,b]</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">{ </code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,102,153)!important; background:none!important">if</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> (a &lt;= p-&gt;Left &amp;&amp; p-&gt;Right &lt;= b)</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 如果当前结点的区间包含在查询区间内</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">{ </code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">...... </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 更新结果</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,102,153)!important; background:none!important">return</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">;</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">}</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">Push_Down(p); </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 等到下面的修改操作再解释这句</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:gray!important; background:none!important">int</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> mid = (p-&gt;Left + p-&gt;Right) / 2; </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 计算左右子结点的分隔点</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,102,153)!important; background:none!important">if</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> (a &lt; mid) Query(p-&gt;Lch, a, b); </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 和左孩子有交集，考察左子结点</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,102,153)!important; background:none!important">if</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> (b &gt; mid) Query(p-&gt;Rch, a, b); </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 和右孩子有交集，考察右子结点</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">}</code> 
     </div> 
    </div> </td> 
  </tr> 
 </tbody> 
</table>

可见，这样的过程一定选出了尽量少的区间，它们相连后正好涵盖了整个\[l,r\]，没有重复也没有遗漏。同时，考虑到线段树上每层的节点最多会被选取2个，一共选取的节点数也是O(log n)的，因此查询的时间复杂度也是O(log n)。

线段树并不适合所有区间查询情况，它的使用条件是“相邻的区间的信息可以被合并成两个区间的并区间的信息”。即问题是可以被分解解决的。

（3）    区间修改

当用户修改一个区间的值时，如果连同其子孙全部修改，则改动的节点数必定会远远超过O(log n)个。因而，如果要想把区间修改操作也控制在O(log n)的时间内，只修改O(log n)个节点的信息就成为必要。

借鉴前一节区间查询用到的思路：区间修改时如果修改了一个节点所表示的区间，也不用去修改它的儿子节点。然而，对于被修改节点的祖先节点，也必须更新它所记录的值，否则查询操作就肯定会出问题（正如修改单个节点的情况一样）。

这些选出的节点的祖先节点直接更新值即可，而选出的节点的子孙却显然不能这么简单地处理：每个节点的值必须能由两个儿子节点的值得到，如这幅图中的例子：

![update][]

这里，节点\[0,1\]的值应该是4，但是两个儿子的值又分别是3和5。如果查询\[0,0\]区间的RMQ，算出来的结果会是3，而正确答案显然是4。

问题显然在于，尽管修改了一个节点以后，不用修改它的儿子节点，但是它的儿子节点的信息事实上已经被改变了。这就需要我们在节点里增设一个域：标记。把对节点的修改情况储存在标记里面，这样，当我们自上而下地访问某节点时，就能把一路上所遇到的所有标记都考虑进去。

但是，在一个节点带上标记时，会给更新这个节点的值带来一些麻烦。继续上面的例子，如果我把位置0的数字从4改成了3，区间\[0,0\]的值应该变回3，但实际上，由于区间\[0,1\]有一个“添加了1”的标记，如果直接把值修改为3，则查询区间\[0,0\]的时候我们会得到3+1=4这个错误结果。但是，把这个3改成2，虽然正确，却并不直观，更不利于推广（参见下面的一个例子）。

为此我们引入延迟标记的一些概念。每个结点新增加一个标记，记录这个结点是否被进行了某种修改操作(这种修改操作会影响其子结点)。还是像上面的一样，对于任意区间的修改，我们先按照查询的方式将其划分成线段树中的结点，然后修改这些结点的信息，并给这些结点标上代表这种修改操作的标记。在修改和查询的时候，如果我们到了一个结点p ，并且决定考虑其子结点，那么我们就要看看结点p 有没有标记，如果有，就要按照标记修改其子结点的信息，并且给子结点都标上相同的标记，同时消掉p 的标记。代码框架为：

<table style="width:660px; margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; background:none!important"> 
 <tbody style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; background:none!important"> 
  <tr style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; background:none!important"> 
   <td style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; color:rgb(175,175,175)!important; background:none!important"> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      1 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      2 
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    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      3 
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    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      4 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      5 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      6 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      7 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      8 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      9 
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    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      10 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      11 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      12 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      13 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      14 
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    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      15 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      16 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      17 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      18 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      19 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      20 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      21 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      22 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      23 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      24 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      25 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      26 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      27 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      28 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      29 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      30 
    </div> 
    <div style="margin:0px!important; padding:0px 0.5em 0px 1em!important; border-width:0px 3px 0px 0px!important; border-right-style:solid!important; border-right-color:rgb(108,226,108)!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; text-align:right!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
      31 
    </div> </td> 
   <td style="width:625px; margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; background:none!important"> 
    <div style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:relative!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; background:none!important"> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// node 为线段树的结点类型，其中Left 和Right 分别表示区间左右端点</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// Lch 和Rch 分别表示指向左右孩子的指针</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,102,153)!important; background:none!important">void</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> Change(node *p, </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:gray!important; background:none!important">int</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> a, </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:gray!important; background:none!important">int</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> b) </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 当前考察结点为p，修改区间为(a,b]</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">{ </code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,102,153)!important; background:none!important">if</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> (a &lt;= p-&gt;Left &amp;&amp; p-&gt;Right &lt;= b)</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 如果当前结点的区间包含在修改区间内</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">{ </code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">...... </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 修改当前结点的信息，并标上标记</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,102,153)!important; background:none!important">return</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">;</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">}</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">Push_Down(p); </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 把当前结点的标记向下传递</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:gray!important; background:none!important">int</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> mid = (p-&gt;Left + p-&gt;Right) / 2; </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 计算左右子结点的分隔点</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,102,153)!important; background:none!important">if</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> (a &lt; mid) Change(p-&gt;Lch, a, b); </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 和左孩子有交集，考察左子结点</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-weight:bold!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,102,153)!important; background:none!important">if</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important"> (b &gt; mid) Change(p-&gt;Rch, a, b); </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 和右孩子有交集，考察右子结点</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">&nbsp;&nbsp;</code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">Update(p); </code> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; color:rgb(0,130,0)!important; background:none!important">// 维护当前结点的信息（因为其子结点的信息可能有更改）</code> 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important">
       &nbsp; 
     </div> 
     <div style="margin:0px!important; padding:0px 1em!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-size:1em!important; direction:ltr!important; white-space:pre!important"> 
      <code style="margin:0px!important; padding:0px!important; border:0px!important; bottom:auto!important; float:none!important; height:auto!important; left:auto!important; line-height:1.1em!important; outline:0px!important; overflow:visible!important; position:static!important; right:auto!important; top:auto!important; vertical-align:baseline!important; width:auto!important; font-family:Consolas,'Bitstream Vera Sans Mono','Courier New',Courier,monospace!important; font-size:1em!important; direction:ltr!important; display:inline!important; background:none!important">}</code> 
     </div> 
    </div> </td> 
  </tr> 
 </tbody> 
</table>

3、应用

下面给出线段树的几个应用：

（1）有一列数，初始值全部为0。每次可以进行以下三种操作中的一种：

a. 给指定区间的每个数加上一个特定值；

b.将指定区间的所有数置成一个统一的值；

c.询问一个区间上的最小值、最大值、所有数的和。

给出一系列a.b.操作后，输出c的结果。

\[问题分析\]

这个是典型的线段树的应用。在每个节点上维护一下几个变量：delta（区间增加值），same（区间被置为某个值），min（区间最小值），max（区间最大值），sum（区间和），其中delta和same属于“延迟标记”。

（2）在所有不大于30000的自然数范围内讨论一个问题：已知n条线段，把端点依次输入给你，然后有m（≤30000）个询问，每个询问输入一个点，要求这个点在多少条线段上出现过。

\[问题分析\]

在这个问题中，我们可以直接对问题处理的区间建立线段树，在线段树上维护区间被覆盖的次数。将n条线段插入线段树，然后对于询问的每个点，直接查询被覆盖的次数即可。

但是我们在这里用这道题目，更希望能够说明一个问题，那就是这道题目完全可以不用线段树。我们将每个线段拆成(L,+1)，(R+1,-1)的两个事件点，每个询问点也在对应坐标处加上一个询问的事件点，排序之后扫描就可以完成题目的询问。我们这里讨论的问题是一个离线的问题，因此我们也设计出了一个很简单的离线算法。线段树在处理在线问题的时候会更加有效，因为它维护了一个实时的信息。

![example2][]

这个题目也告诉我们，有的题目尽管可以使用线段树处理，但是如果我们能够抓住题目的特点，就可能获得更加优秀的算法。

（3）某次列车途经C个城市，城市编号依次为1到C，列车上共有S个座位，铁路局规定售出的车票只能是坐票，即车上所有的旅客都有座，售票系统是由计算机执行的，每一个售票申请包含三个参数，分别用O、D、N表示，O为起始站，D为目的地站，N为车票张数，售票系统对该售票申请作出受理或不受理的决定，只有在从O到D的区段内列车上都有N个或N个以上的空座位时该售票申请才被受理，请你写一个程序，实现这个自动售票系统。

\[问题分析\]

这里我们可以把所有的车站顺次放在一个数轴上，在数轴上建立线段树，在线段树上维护区间的delta与max。每次判断一个售票申请是否可行就是查询区间上的最大值；每个插入一个售票请求，就是给一个区间上所有的元素加上购票数。

这道题目在线段树上维护的信息既包括自下至上的递推，也包括了自上至下的传递，能够比较全面地对线段树的基本操作进行训练。

（4）给一个n\*n的方格棋盘，初始时每个格子都是白色。现在要刷M次黑色或白色的油漆。每次刷漆的区域都是一个平行棋盘边缘的矩形区域。

输入n,M,以及每次刷漆的区域和颜色，输出刷了M次之后棋盘上还有多少个棋格是白色。

\[问题分析\]

首先我们从简单入手，考虑一维的问题。即对于一个长度为n的白色线段，对它进行M次修改（每次更新某一子区域的颜色）。问最后还剩下的白色区域有多长。

对于这个问题，很容易想到建立一棵线段树的模型。复杂度为O(Mlgn)。

扩展到二维，需要把线段树进行调整，即首先在横坐标上建立线段树，它的每个节点是一棵建立在纵坐标上的线段树（即树中有树。称为二维线段树）。复杂度为O(M(logn)^2)。

![example4][]

4、总结

利用线段树，我们可以高效地询问和修改一个数列中某个区间的信息，并且代码也不算特别复杂。

但是线段树也是有一定的局限性的，其中最明显的就是数列中数的个数必须固定，即不能添加或删除数列中的数。

5、参考资料

（1）    杨弋文章：《线段树》：

[http://download.csdn.net/source/2255479][http_download.csdn.net_source_2255479]

（2）    林涛文章《线段树的应用》：

[http://wenku.baidu.com/view/d65cf31fb7360b4c2e3f64ac.html][http_wenku.baidu.com_view_d65cf31fb7360b4c2e3f64ac.html]

（3）    朱全民文章《线段树及其应用》：

[http://wenku.baidu.com/view/437ad3bec77da26925c5b0ba.html][http_wenku.baidu.com_view_437ad3bec77da26925c5b0ba.html]

（4）    线段树：

[http://wenku.baidu.com/view/32652a2d7375a417866f8f51.html][http_wenku.baidu.com_view_32652a2d7375a417866f8f51.html]

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