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  219.         <h3 id="前置知识"><a class="markdownIt-Anchor" href="#前置知识"></a> 前置知识</h3>
    220. 

  220. <p>数组，结构体，二叉树</p>
    221. 

  221. <h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3>
    222. 

  222. <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="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></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="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span> 次操作。<br />
    223. 

  223. 暴力是肯定不行的，数据范围太大，操作太多，会超时。<br />
    224. 

  224. 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>
    225. 

  225. <a id="more"></a>
    226. 

  226. <h3 id="线段树"><a class="markdownIt-Anchor" href="#线段树"></a> 线段树</h3>
    227. 

  227. <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>
    228. 

  228. <p>比如有 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span> 个节点，根节点代表整个数列的区间 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1,n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">]</span></span></span></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="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo separator="true">,</mo><mi mathvariant="normal">mid</mi><mo>⁡</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1,\operatorname{mid}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mord mathrm">m</span><span class="mord mathrm">i</span><span class="mord mathrm">d</span></span><span class="mclose">]</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">mid</mi><mo>⁡</mo><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\operatorname{mid},n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mop"><span class="mord mathrm">m</span><span class="mord mathrm">i</span><span class="mord mathrm">d</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">]</span></span></span></span>。<br />
    229. 

  229. 而子节点也是一棵线段树。<br />
    230. 

  230. 一直往下延伸，叶子结点就代表着单一位置 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a,a]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">]</span></span></span></span>。<br />
    231. 

  231. 下图是一个 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>=</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">n=7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">7</span></span></span></span> 的示例：</p>
    232. 

  232. <p><img data-src="/images/segment-1.png" alt="segment-1" /></p>
    233. 

  233. <p>每一个节点都存储着它所代表的区间的信息，比如这个区间的和。</p>
    234. 

  234. <h3 id="查找"><a class="markdownIt-Anchor" href="#查找"></a> 查找</h3>
    235. 

  235. <h4 id="解释"><a class="markdownIt-Anchor" href="#解释"></a> 解释</h4>
    236. 

  236. <p>如查找 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mi>l</mi><mo separator="true">,</mo><mi>r</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[l,r]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mclose">]</span></span></span></span>，则执行以下步骤：</p>
    237. 

  237. <ol>
    238. 

  238. <li>首先在根节点查找 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mi>l</mi><mo separator="true">,</mo><mi>r</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[l,r]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mclose">]</span></span></span></span></li>
    239. 

  239. <li>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。</li>
    240. 

  240. <li>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="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mi>l</mi><mo separator="true">,</mo><mi>r</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[l,r]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mclose">]</span></span></span></span> 的结果。</li>
    241. 

  241. <li>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="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mi>l</mi><mo separator="true">,</mo><mi>r</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[l,r]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mclose">]</span></span></span></span> 的结果。</li>
    242. 

  242. <li>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。</li>
    243. 

  243. </ol>
    244. 

  244. <h4 id="举例"><a class="markdownIt-Anchor" href="#举例"></a> 举例</h4>
    245. 

  245. <p>设若查找区间 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>3</mn><mo separator="true">,</mo><mn>6</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[3,6]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">6</span><span class="mclose">]</span></span></span></span>，首先从根节点开始：<br />
    246. 

  246. -当前区间 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo separator="true">,</mo><mn>7</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1,7]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">7</span><span class="mclose">]</span></span></span></span> 不在 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>3</mn><mo separator="true">,</mo><mn>6</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[3,6]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">6</span><span class="mclose">]</span></span></span></span> 内部，经过判断 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>3</mn><mo separator="true">,</mo><mn>6</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[3,6]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">6</span><span class="mclose">]</span></span></span></span> 横跨左右子节点，分别在左右子节点查找查找：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>3</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[3,3]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">]</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>4</mn><mo separator="true">,</mo><mn>6</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[4,6]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">6</span><span class="mclose">]</span></span></span></span>。<br />
    247. 

  247. -于是在 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1,3]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">]</span></span></span></span> 内查找 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>3</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[3,3]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">]</span></span></span></span>，经过两次下放到右子树最终返回。<br />
    248. 

  248. -接着在 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>4</mn><mo separator="true">,</mo><mn>7</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[4,7]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">7</span><span class="mclose">]</span></span></span></span> 内查找 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>4</mn><mo separator="true">,</mo><mn>6</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[4,6]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">6</span><span class="mclose">]</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>4</mn><mo separator="true">,</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">4,6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">6</span></span></span></span> 横跨左右子树，继续查找：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>4</mn><mo>−</mo><mn>5</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[4-5]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mclose">]</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>6</mn><mo>−</mo><mn>6</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[6-6]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">6</span><span class="mclose">]</span></span></span></span>。<br />
    249. 

  249. -进入 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>4</mn><mo separator="true">,</mo><mn>5</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[4,5]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">5</span><span class="mclose">]</span></span></span></span>，在区间内部，直接返回。由 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>6</mn><mo separator="true">,</mo><mn>7</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[6,7]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">7</span><span class="mclose">]</span></span></span></span> 进入 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>6</mn><mo separator="true">,</mo><mn>6</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[6,6]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">6</span><span class="mclose">]</span></span></span></span> 后返回，查找完毕。</p>
    250. 

  250. <h3 id="修改"><a class="markdownIt-Anchor" href="#修改"></a> 修改</h3>
    251. 

  251. <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="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>4</mn><mo separator="true">,</mo><mn>5</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[4,5]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">5</span><span class="mclose">]</span></span></span></span>，如果只修改 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>4</mn><mo separator="true">,</mo><mn>5</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[4,5]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">5</span><span class="mclose">]</span></span></span></span> 区间，那么单独查询 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>4</mn><mo separator="true">,</mo><mn>4</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[4,4]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">4</span><span class="mclose">]</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>5</mn><mo separator="true">,</mo><mn>5</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[5,5]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">5</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">5</span><span class="mclose">]</span></span></span></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>
    252. 

  252. <h4 id="lazy"><a class="markdownIt-Anchor" href="#lazy"></a> lazy</h4>
    253. 

  253. <p>我们在树的节点上加一个标记：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">lazy</mtext></mrow><annotation encoding="application/x-tex">\texttt{lazy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">lazy</span></span></span></span></span>。</p>
    254. 

  254. <p>所谓 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">lazy</mtext></mrow><annotation encoding="application/x-tex">\texttt{lazy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">lazy</span></span></span></span></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="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">lazy</mtext></mrow><annotation encoding="application/x-tex">\texttt{lazy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">lazy</span></span></span></span></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>
    255. 

  255. <h4 id="注意"><a class="markdownIt-Anchor" href="#注意"></a> 注意</h4>
    256. 

  256. <p>修改后的结点，需要同时把所有祖先结点全部更新。</p>
    257. 

  257. <h3 id="优势"><a class="markdownIt-Anchor" href="#优势"></a> 优势</h3>
    258. 

  258. <p>暴力算法的复杂度是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nm)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mord mathdefault">m</span><span class="mclose">)</span></span></span></span> 的，而线段树的复杂度是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">m</span><span class="mclose">)</span></span></span></span> 的，因为每次树深入一层，区间长度都会减半。</p>
    259. 

  259. <h3 id="代码luogu-p3372"><a class="markdownIt-Anchor" href="#代码luogu-p3372"></a> 代码(<a href="https://www.luogu.com.cn/problem/P3372" target="_blank" rel="noopener">luogu P3372</a>)</h3>
    260. 

  260. <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&lt;&lt;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&lt;&lt;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> &#123;</span></span><br><span class="line">	<span class="keyword">int</span> l,r,sum,lazy;</span><br><span class="line">&#125;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>&#123;<span class="comment">//更新函数</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">&#125;</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>&#123;<span class="comment">//下推函数</span></span><br><span class="line">	<span class="keyword">if</span>(t[id].lazy)&#123;</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">	&#125;</span><br><span class="line">&#125;</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>&#123;<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)&#123;</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">	&#125;</span><br><span class="line">	<span class="keyword">int</span> mid=(l+r)&gt;&gt;<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">&#125;</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>&#123;<span class="comment">//修改</span></span><br><span class="line">	<span class="keyword">if</span>(t[id].l&gt;=l&amp;&amp;t[id].r&lt;=r)&#123;</span><br><span class="line">		t[id].lazy+=c;</span><br><span class="line">		<span class="keyword">return</span>;</span><br><span class="line">	&#125;</span><br><span class="line">	pushdown(id);</span><br><span class="line">	<span class="keyword">if</span>(t[lson].r&gt;=r)change(lson,l,r,c);</span><br><span class="line">	<span class="keyword">else</span> <span class="keyword">if</span>(t[rson].l&lt;=l)change(rson,l,r,c);</span><br><span class="line">	<span class="keyword">else</span>&#123;</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">	&#125;</span><br><span class="line">	update(id);</span><br><span class="line">&#125;</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>&#123;<span class="comment">//求和</span></span><br><span class="line">	<span class="keyword">if</span>(t[id].l&gt;=l&amp;&amp;t[id].r&lt;=r)&#123;</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">	&#125;</span><br><span class="line">	pushdown(id);</span><br><span class="line">	<span class="keyword">if</span>(t[lson].r&gt;=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&lt;=l)<span class="keyword">return</span> query(rson,l,r);</span><br><span class="line">	<span class="keyword">else</span>&#123;</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">	&#125;</span><br><span class="line">	update(id);</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>&#123;</span><br><span class="line">	<span class="built_in">cin</span>&gt;&gt;n&gt;&gt;m;</span><br><span class="line">	<span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=n;i++)&#123;</span><br><span class="line">		<span class="built_in">cin</span>&gt;&gt;a[i];</span><br><span class="line">	&#125;</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&lt;=m;i++)&#123;</span><br><span class="line">		<span class="built_in">cin</span>&gt;&gt;op;</span><br><span class="line">		<span class="keyword">if</span>(op==<span class="number">1</span>)&#123;</span><br><span class="line">			<span class="built_in">cin</span>&gt;&gt;x&gt;&gt;y&gt;&gt;k;</span><br><span class="line">			change(<span class="number">1</span>,x,y,k);</span><br><span class="line">		&#125;</span><br><span class="line">		<span class="keyword">else</span>&#123;</span><br><span class="line">			<span class="built_in">cin</span>&gt;&gt;x&gt;&gt;y;</span><br><span class="line">			<span class="built_in">cout</span>&lt;&lt;query(<span class="number">1</span>,x,y)&lt;&lt;<span class="built_in">endl</span>;</span><br><span class="line">		&#125;</span><br><span class="line">	&#125;</span><br><span class="line">	<span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
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