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数组，结构体，二叉树
引入
有时候我们会遇到一些大规模的区间查找和区间修改问题，比如让你维护一个 \(10^5\) 长度的数列，要求操作有区间求和、区间加（区间每个数加上一个值），让你在一秒内完成 \(10^5\) 次操作。
暴力是肯定不行的，数据范围太大，操作太多，会超时。
所以我们就有一种专门解决大范围区间修改查询的数据结构：线段树。"><meta property="og:type" content="article"><meta property="og:title" content="线段树"><meta property="og:url" content="https://schtonn.github.io/blog/posts/segment-tree/index.html"><meta property="og:site_name" content="schtonn"><meta property="og:description" content="前置知识
数组，结构体，二叉树
引入
有时候我们会遇到一些大规模的区间查找和区间修改问题，比如让你维护一个 \(10^5\) 长度的数列，要求操作有区间求和、区间加（区间每个数加上一个值），让你在一秒内完成 \(10^5\) 次操作。
暴力是肯定不行的，数据范围太大，操作太多，会超时。
所以我们就有一种专门解决大范围区间修改查询的数据结构：线段树。"><meta property="og:locale" content="en_US"><meta property="og:image" content="https://schtonn.github.io/blog/images/segment-1.png"><meta property="article:published_time" content="2020-03-02T03:37:36.000Z"><meta property="article:modified_time" content="2022-10-19T15:02:06.675Z"><meta property="article:author" content="Alex"><meta property="article:tag" content="struct"><meta property="article:tag" content="tree"><meta name="twitter:card" content="summary"><meta name="twitter:image" content="https://schtonn.github.io/blog/images/segment-1.png"><link rel="canonical" href="https://schtonn.github.io/blog/posts/segment-tree/"><script id="page-configurations">CONFIG.page={sidebar:"",isHome:!1,isPost:!0,lang:"en"}</script><title>线段树 | schtonn</title><noscript><style>.sidebar-inner,.use-motion .brand,.use-motion .collection-header,.use-motion .comments,.use-motion .menu-item,.use-motion .pagination,.use-motion .post-block,.use-motion .post-body,.use-motion 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href="https://schtonn.github.io/blog/posts/segment-tree/"><span hidden itemprop="author" itemscope itemtype="http://schema.org/Person"><meta itemprop="image" content="/blog/images/avatar.gif"><meta itemprop="name" content="Alex"><meta itemprop="description" content="blog"></span><script type="text/javascript" src="/blog/js/md5.js"></script><script></script><script>document.oncopy=function(e){window.event&&(e=window.event);try{var t=e.srcElement;return"INPUT"==t.tagName&&"text"==t.type.toLowerCase()||"TEXTAREA"==t.tagName}catch(e){return!1}}</script><span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization"><meta itemprop="name" content="schtonn"></span><header class="post-header"><h1 class="post-title" itemprop="name headline"> 线段树</h1><div class="post-meta"><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-calendar-o"></i></span> <span class="post-meta-item-text">Posted on</span> <time title="Created: 2020-Mar-02 11:37:36" itemprop="dateCreated datePublished" datetime="2020-03-02T11:37:36+08:00">2020-Mar-02</time></span><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-calendar-check-o"></i></span> <span class="post-meta-item-text">Edited on</span> <time title="Modified: 2022-Oct-19 23:02:06" itemprop="dateModified" datetime="2022-10-19T23:02:06+08:00">2022-Oct-19</time></span><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-comment-o"></i></span> <span class="post-meta-item-text">Valine:</span><a title="valine" href="/blog/posts/segment-tree/#valine-comments" itemprop="discussionUrl"><span class="post-comments-count valine-comment-count" data-xid="/blog/posts/segment-tree/" itemprop="commentCount"></span></a></span></div></header><div class="post-body" itemprop="articleBody"><h2 id="前置知识">前置知识</h2><p>数组，结构体，二叉树</p><h2 id="引入">引入</h2><p>有时候我们会遇到一些大规模的区间查找和区间修改问题，比如让你维护一个 <span class="math inline">\(10^5\)</span> 长度的数列，要求操作有区间求和、区间加（区间每个数加上一个值），让你在一秒内完成 <span class="math inline">\(10^5\)</span> 次操作。</p><p>暴力是肯定不行的，数据范围太大，操作太多，会超时。</p><p>所以我们就有一种专门解决大范围区间修改查询的数据结构：线段树。</p><a id="more"></a><h2 id="线段树">线段树</h2><p>线段树本质上是把整个数列拆分了，用一个一个区间来表示。</p><p>比如有 <span class="math inline">\(n\)</span> 个节点，根节点代表整个数列的区间 <span class="math inline">\([1,n]\)</span>，根节点的两个子节点代表根区间二分成两部分，也就是 <span class="math inline">\([1,\operatorname{mid}]\)</span> 和 <span class="math inline">\([\operatorname{mid}+1,n]\)</span>。 而子节点也是一棵线段树。 一直往下延伸，叶子结点就代表着单一位置 <span class="math inline">\([a,a]\)</span>。 下图是一个 <span class="math inline">\(n=7\)</span> 的示例：</p><figure> <img data-src="/blog/images/segment-1.png" alt="segment-1"><figcaption>segment-1</figcaption></figure><p>每一个节点都存储着它所代表的区间的信息，比如这个区间的元素和。</p><h2 id="查找">查找</h2><h3 id="解释">解释</h3><p>如查找 <span class="math inline">\([l,r]\)</span>，则执行以下步骤： 1. 首先在根节点查找 <span class="math inline">\([l,r]\)</span> 2. 判断当前所在区间是否在当前查找的区间内部，若在内部则直接返回当前区间数据。 3. 若当前区间的<strong>右子节点区间的左边界</strong>在<strong>当前查找的区间的左边界</strong>的右侧，说明当前查找区间完全在右子节点一侧，则返回在右子节点查找 <span class="math inline">\([l,r]\)</span> 的结果。 4. 否则，若当前区间的<strong>左子节点区间的右边界</strong>在<strong>当前查找区间的右边界</strong>的左侧，说明当前查找区间完全在左子节点一侧，则返回在左子节点查找 <span class="math inline">\([l,r]\)</span> 的结果。 5. 否则，说明当前查找区间横跨左右子节点，那么返回：在左子节点查找<strong>当前查找区间的左边界-左子节点区间的右边界</strong>与在右子节点查找<strong>右子节点的左边界-当前查找区间的右边界</strong>结果的和。</p><p>简单来说就是不断向下递归寻找，直到当前的区间能包住全部或者一部分要求区间。</p><h3 id="举例">举例</h3><p>设若查找区间 <span class="math inline">\([3,6]\)</span>，首先从根节点开始： - 当前区间 <span class="math inline">\([1,7]\)</span> 不在 <span class="math inline">\([3,6]\)</span> 内部，经过判断 <span class="math inline">\([3,6]\)</span> 横跨左右子节点，分别在左右子节点查找查找：<span class="math inline">\([3,3]\)</span> 和 <span class="math inline">\([4,6]\)</span>。 - 于是在 <span class="math inline">\([1,3]\)</span> 内查找 <span class="math inline">\([3,3]\)</span>，经过两次下放到右子树最终返回。 - 接着在 <span class="math inline">\([4,7]\)</span> 内查找 <span class="math inline">\([4,6]\)</span>，<span class="math inline">\(4,6\)</span> 横跨左右子树，继续查找：<span class="math inline">\([4-5]\)</span> 和 <span class="math inline">\([6-6]\)</span>。 - 进入 <span class="math inline">\([4,5]\)</span>，在区间内部，直接返回。由 <span class="math inline">\([6,7]\)</span> 进入 <span class="math inline">\([6,6]\)</span> 后返回，查找完毕。</p><h2 id="修改">修改</h2><p>我们会发现，修改时如果像查找一样做，那么有一些细小的叶子修改就改不到，如刚才举例的 <span class="math inline">\([4,5]\)</span>，如果只修改 <span class="math inline">\([4,5]\)</span> 区间，那么单独查询 <span class="math inline">\([4,4]\)</span> 和 <span class="math inline">\([5,5]\)</span> 的时候就会出错。要是每一个修改都下放到叶子节点，那这个算法就和暴力一样了。所以我们需要一些巧妙地解决办法。</p><h3 id="lazy">lazy</h3><p>我们在树的节点上加一个标记：<span class="math inline">\(\texttt{lazy}\)</span>。</p><p>所谓 <span class="math inline">\(\texttt{lazy}\)</span>，就是要懒，就是要在事情迫不得已要做的时候把它做了，所以我们每次那些细小的叶子修改就不做他，直接记录在 <span class="math inline">\(\texttt{lazy}\)</span> 标记上，等到要查询的时候，再从标记上下放到子结点上，查询到哪里就下放到哪里。</p><p>在刚才的例子里，修改 <span class="math inline">\([4,5]\)</span> 时就把一个同样值的 <span class="math inline">\(\texttt{lazy}\)</span> 也放在了 <span class="math inline">\([4,5]\)</span> 点里，递归查找 <span class="math inline">\([4,4]\)</span> 时就会把 <span class="math inline">\(\texttt{lazy}\)</span> 下放，这时才修改了 <span class="math inline">\([4,4]\)</span>。</p><h3 id="注意">注意</h3><p>修改后的结点，需要同时把所有祖先结点全部更新。</p><h2 id="优势">优势</h2><p>暴力算法的复杂度是 <span class="math inline">\(O(nm)\)</span> 的，而线段树的复杂度是 <span class="math inline">\(O(n\log m)\)</span> 的，因为每次树深入一层，区间长度都会减半。</p><h2 id="代码luogu-p3372">代码(<a href="https://www.luogu.com.cn/problem/P3372" target="_blank" rel="noopener">luogu P3372</a>)</h2><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">"iostream"</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> N 1000010</span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> lson id<<1 <span class="comment">//此处偷了懒，因为完全二叉树的性质可以推出，左子树编号是根节点的一倍，右子树是根节点一倍加一，</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> rson id<<1|1 <span class="comment">//使用位运算提高速度。</span></span></span><br><span class="line"><span class="keyword">int</span> a[N],n,m,op,x,y,k;</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">node</span> {</span></span><br><span class="line"> <span class="keyword">int</span> l,r,sum,lazy;</span><br><span class="line">}t[<span class="number">2</span>*N];</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">update</span><span class="params">(<span class="keyword">int</span> id)</span></span>{<span class="comment">//更新函数（向上更新修改的）</span></span><br><span class="line"> t[id].sum=t[lson].sum+t[lson].lazy*(t[lson].r-t[lson].l+<span class="number">1</span>)+</span><br><span class="line"> t[rson].sum+t[rson].lazy*(t[rson].r-t[rson].l+<span class="number">1</span>);</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">pushdown</span><span class="params">(<span class="keyword">int</span> id)</span></span>{<span class="comment">//下推函数（下推lazy的）</span></span><br><span class="line"> <span class="keyword">if</span>(t[id].lazy){</span><br><span class="line"> t[lson].lazy+=t[id].lazy;</span><br><span class="line"> t[rson].lazy+=t[id].lazy;</span><br><span class="line"> t[id].sum+=t[id].lazy*(t[id].r-t[id].l+<span class="number">1</span>);</span><br><span class="line"> t[id].lazy=<span class="number">0</span>;</span><br><span class="line"> }</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">buildtree</span><span class="params">(<span class="keyword">int</span> id,<span class="keyword">int</span> l,<span class="keyword">int</span> r)</span></span>{<span class="comment">//建树</span></span><br><span class="line"> t[id].l=l;</span><br><span class="line"> t[id].r=r;</span><br><span class="line"> t[id].lazy=<span class="number">0</span>;</span><br><span class="line"> <span class="keyword">if</span>(l==r){</span><br><span class="line"> t[id].sum=a[l];</span><br><span class="line"> <span class="keyword">return</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">int</span> mid=(l+r)>><span class="number">1</span>;</span><br><span class="line"> buildtree(lson,l,mid);</span><br><span class="line"> buildtree(rson,mid+<span class="number">1</span>,r);</span><br><span class="line"> update(id);</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">change</span><span class="params">(<span class="keyword">int</span> id,<span class="keyword">int</span> l,<span class="keyword">int</span> r,<span class="keyword">int</span> c)</span></span>{<span class="comment">//修改</span></span><br><span class="line"> <span class="keyword">if</span>(t[id].l>=l&&t[id].r<=r){</span><br><span class="line"> t[id].lazy+=c;</span><br><span class="line"> <span class="keyword">return</span>;</span><br><span class="line"> }</span><br><span class="line"> pushdown(id);</span><br><span class="line"> <span class="keyword">if</span>(t[lson].r>=r)change(lson,l,r,c);</span><br><span class="line"> <span class="keyword">else</span> <span class="keyword">if</span>(t[rson].l<=l)change(rson,l,r,c);</span><br><span class="line"> <span class="keyword">else</span>{</span><br><span class="line"> change(lson,l,t[lson].r,c);</span><br><span class="line"> change(rson,t[rson].l,r,c);</span><br><span class="line"> }</span><br><span class="line"> update(id);</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">query</span><span class="params">(<span class="keyword">int</span> id,<span class="keyword">int</span> l,<span class="keyword">int</span> r)</span></span>{<span class="comment">//求和</span></span><br><span class="line"> <span class="keyword">if</span>(t[id].l>=l&&t[id].r<=r){</span><br><span class="line"> <span class="keyword">return</span> t[id].sum+t[id].lazy*(t[id].r-t[id].l+<span class="number">1</span>);</span><br><span class="line"> }</span><br><span class="line"> pushdown(id);</span><br><span class="line"> <span class="keyword">if</span>(t[lson].r>=r)<span class="keyword">return</span> query(lson,l,r);</span><br><span class="line"> <span class="keyword">else</span> <span class="keyword">if</span>(t[rson].l<=l)<span class="keyword">return</span> query(rson,l,r);</span><br><span class="line"> <span class="keyword">else</span>{</span><br><span class="line"> <span class="keyword">return</span> query(lson,l,t[lson].r)+query(rson,t[rson].l,r);</span><br><span class="line"> }</span><br><span class="line"> update(id);</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>{</span><br><span class="line"> <span class="built_in">cin</span>>>n>>m;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++){</span><br><span class="line"> <span class="built_in">cin</span>>>a[i];</span><br><span class="line"> }</span><br><span class="line"> buildtree(<span class="number">1</span>,<span class="number">1</span>,n);</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i<=m;i++){</span><br><span class="line"> <span class="built_in">cin</span>>>op;</span><br><span class="line"> <span class="keyword">if</span>(op==<span class="number">1</span>){</span><br><span class="line"> <span class="built_in">cin</span>>>x>>y>>k;</span><br><span class="line"> change(<span class="number">1</span>,x,y,k);</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">else</span>{</span><br><span class="line"> <span class="built_in">cin</span>>>x>>y;</span><br><span class="line"> <span class="built_in">cout</span><<query(<span class="number">1</span>,x,y)<<<span class="built_in">endl</span>;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure></div><div><ul class="post-copyright"><li class="post-copyright-author"> <strong>Post author:</strong> Alex</li><li class="post-copyright-link"> <strong>Post link:</strong> <a href="https://schtonn.github.io/blog/posts/segment-tree/" title="线段树">https://schtonn.github.io/blog/posts/segment-tree/</a></li><li class="post-copyright-license"> <strong>Copyright Notice:</strong> All articles in this blog are licensed under<a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" rel="noopener" target="_blank"><i 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class="nav-number">4.</span> <span class="nav-text">查找</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#解释"><span class="nav-number">4.1.</span> <span class="nav-text">解释</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#举例"><span class="nav-number">4.2.</span> <span class="nav-text">举例</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#修改"><span class="nav-number">5.</span> <span class="nav-text">修改</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#lazy"><span class="nav-number">5.1.</span> <span class="nav-text">lazy</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#注意"><span class="nav-number">5.2.</span> <span class="nav-text">注意</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#优势"><span class="nav-number">6.</span> <span class="nav-text">优势</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#代码luogu-p3372"><span class="nav-number">7.</span> <span class="nav-text">代码(luogu P3372)</span></a></li></ol></div></div><div class="site-overview-wrap sidebar-panel"><div class="site-author motion-element" itemprop="author" itemscope itemtype="http://schema.org/Person"><p class="site-author-name" itemprop="name">Alex</p><div class="site-description" itemprop="description">blog</div></div><div class="site-state-wrap motion-element"><nav class="site-state"><div class="site-state-item site-state-posts"> <a href="/blog/archives"><span class="site-state-item-count">35</span> <span class="site-state-item-name">posts</span></a></div><div class="site-state-item site-state-tags"> <a href="/blog/tags/"><span class="site-state-item-count">8</span> <span class="site-state-item-name">tags</span></a></div></nav></div><div class="links-of-author motion-element"><span class="links-of-author-item"><a href="https://github.com/schtonn" title="GitHub → 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