Home Explore Help
Register Sign In
schtonn
/
public
1
1
Fork 0
Files Issues 0 Pull Requests 0 Wiki
Tree: 251bb196a5
Branches Tags
public
/
posts
/
segment-tree
/
index.html

index.html 56 KB
History Raw

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509510511512513514515516517518519520521522523524525526527528529530531532533534535536537538539540541542543544545546547548549550551 
  1. <!DOCTYPE html>
    2. 

  2. <html lang="en">
    3. 

  3. <head>
    4. 

  4.   <meta charset="UTF-8">
    5. 

  5. <meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=2">
    6. 

  6. <meta name="theme-color" content="#222">
    7. 

  7. <meta name="generator" content="Hexo 4.2.0">
    8. 

  8.   <link rel="apple-touch-icon" sizes="180x180" href="/images/apple-touch-icon-next.png">
    9. 

  9.   <link rel="icon" type="image/png" sizes="32x32" href="/images/favicon-32x32-next.png">
    10. 

  10.   <link rel="icon" type="image/png" sizes="16x16" href="/images/favicon-16x16-next.png">
    11. 

  11.   <link rel="mask-icon" href="/images/logo.svg" color="#222">
    12. 

  12. 

  13. <link rel="stylesheet" href="/css/main.css">
    14. 

  14. 

  15. 

  16. <link rel="stylesheet" href="/lib/font-awesome/css/font-awesome.min.css">
    17. 

  17.   <link rel="stylesheet" href="/lib/pace/pace-theme-minimal.min.css">
    18. 

  18.   <script src="/lib/pace/pace.min.js"></script>
    19. 

  19. 

  20. <script id="hexo-configurations">
    21. 

  21.     var NexT = window.NexT || {};
    22. 

  22.     var CONFIG = {"hostname":"schtonn.github.io","root":"/","scheme":"Muse","version":"7.8.0","exturl":false,"sidebar":{"position":"left","display":"post","padding":18,"offset":12,"onmobile":false},"copycode":{"enable":true,"show_result":true,"style":"flat"},"back2top":{"enable":true,"sidebar":false,"scrollpercent":true},"bookmark":{"enable":false,"color":"#222","save":"auto"},"fancybox":false,"mediumzoom":false,"lazyload":true,"pangu":false,"comments":{"style":"tabs","active":"valine","storage":true,"lazyload":false,"nav":null,"activeClass":"valine"},"algolia":{"hits":{"per_page":10},"labels":{"input_placeholder":"Search for Posts","hits_empty":"We didn't find any results for the search: ${query}","hits_stats":"${hits} results found in ${time} ms"}},"localsearch":{"enable":false,"trigger":"auto","top_n_per_article":1,"unescape":false,"preload":false},"motion":{"enable":true,"async":false,"transition":{"post_block":"fadeIn","post_header":"slideDownIn","post_body":"slideDownIn","coll_header":"slideLeftIn","sidebar":"slideUpIn"}}};
    23. 

  23.   </script>
    24. 

  24. 

  25.   <meta name="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 10510^5105 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 10510^5105 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。">
    26. 

  26. <meta property="og:type" content="article">
    27. 

  27. <meta property="og:title" content="线段树">
    28. 

  28. <meta property="og:url" content="https://schtonn.github.io/posts/segment-tree/index.html">
    29. 

  29. <meta property="og:site_name" content="Alex&#39;s Blog">
    30. 

  30. <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 10510^5105 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 10510^5105 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。">
    31. 

  31. <meta property="og:locale" content="en_US">
    32. 

  32. <meta property="og:image" content="https://schtonn.github.io/images/segment-1.png">
    33. 

  33. <meta property="article:published_time" content="2020-03-02T03:37:36.000Z">
    34. 

  34. <meta property="article:modified_time" content="2020-03-29T07:59:45.949Z">
    35. 

  35. <meta property="article:author" content="Alex">
    36. 

  36. <meta property="article:tag" content="graph">
    37. 

  37. <meta name="twitter:card" content="summary">
    38. 

  38. <meta name="twitter:image" content="https://schtonn.github.io/images/segment-1.png">
    39. 

  39. 

  40. <link rel="canonical" href="https://schtonn.github.io/posts/segment-tree/">
    41. 

  41. 

  42. 

  43. <script id="page-configurations">
    44. 

  44.   // https://hexo.io/docs/variables.html
    45. 

  45.   CONFIG.page = {
    46. 

  46.     sidebar: "",
    47. 

  47.     isHome : false,
    48. 

  48.     isPost : true,
    49. 

  49.     lang   : 'en'
    50. 

  50.   };
    51. 

  51. </script>
    52. 

  52. 

  53.   <title>线段树 | Alex's Blog</title>
    54. 

  54.   
    55. 

  55. 

  56. 

  57. 

  58. 

  59. 

  60. 

  61.   <noscript>
    62. 

  62.   <style>
    63. 

  63.   .use-motion .brand,
    64. 

  64.   .use-motion .menu-item,
    65. 

  65.   .sidebar-inner,
    66. 

  66.   .use-motion .post-block,
    67. 

  67.   .use-motion .pagination,
    68. 

  68.   .use-motion .comments,
    69. 

  69.   .use-motion .post-header,
    70. 

  70.   .use-motion .post-body,
    71. 

  71.   .use-motion .collection-header { opacity: initial; }
    72. 

  72. 

  73.   .use-motion .site-title,
    74. 

  74.   .use-motion .site-subtitle {
    75. 

  75.     opacity: initial;
    76. 

  76.     top: initial;
    77. 

  77.   }
    78. 

  78. 

  79.   .use-motion .logo-line-before i { left: initial; }
    80. 

  80.   .use-motion .logo-line-after i { right: initial; }
    81. 

  81.   </style>
    82. 

  82. </noscript>
    83. 

  83. 

  84. </head>
    85. 

  85. 

  86. <body itemscope itemtype="http://schema.org/WebPage">
    87. 

  87.   <div class="container use-motion">
    88. 

  88.     <div class="headband"></div>
    89. 

  89. 

  90.     <header class="header" itemscope itemtype="http://schema.org/WPHeader">
    91. 

  91.       <div class="header-inner"><div class="site-brand-container">
    92. 

  92.   <div class="site-nav-toggle">
    93. 

  93.     <div class="toggle" aria-label="Toggle navigation bar">
    94. 

  94.       <span class="toggle-line toggle-line-first"></span>
    95. 

  95.       <span class="toggle-line toggle-line-middle"></span>
    96. 

  96.       <span class="toggle-line toggle-line-last"></span>
    97. 

  97.     </div>
    98. 

  98.   </div>
    99. 

  99. 

  100.   <div class="site-meta">
    101. 

  101. 

  102.     <a href="/" class="brand" rel="start">
    103. 

  103.       <span class="logo-line-before"><i></i></span>
    104. 

  104.       <h1 class="site-title">Alex's Blog</h1>
    105. 

  105.       <span class="logo-line-after"><i></i></span>
    106. 

  106.     </a>
    107. 

  107.       <p class="site-subtitle" itemprop="description">Middle school OIer</p>
    108. 

  108.   </div>
    109. 

  109. 

  110.   <div class="site-nav-right">
    111. 

  111.     <div class="toggle popup-trigger">
    112. 

  112.     </div>
    113. 

  113.   </div>
    114. 

  114. </div>
    115. 

  115. 

  116. 

  117. 

  118. 

  119. <nav class="site-nav">
    120. 

  120.   <ul id="menu" class="menu">
    121. 

  121.         <li class="menu-item menu-item-home">
    122. 

  122. 

  123.     <a href="/" rel="section"><i class="fa fa-fw fa-home"></i>Home</a>
    124. 

  124. 

  125.   </li>
    126. 

  126.         <li class="menu-item menu-item-tags">
    127. 

  127. 

  128.     <a href="/tags/" rel="section"><i class="fa fa-fw fa-tags"></i>Tags</a>
    129. 

  129. 

  130.   </li>
    131. 

  131.         <li class="menu-item menu-item-archives">
    132. 

  132. 

  133.     <a href="/archives/" rel="section"><i class="fa fa-fw fa-archive"></i>Archives</a>
    134. 

  134. 

  135.   </li>
    136. 

  136.         <li class="menu-item menu-item-games">
    137. 

  137. 

  138.     <a href="/games/" rel="section"><i class="fa fa-fw fa-gamepad"></i>Games</a>
    139. 

  139. 

  140.   </li>
    141. 

  141.         <li class="menu-item menu-item-sans">
    142. 

  142. 

  143.     <a href="/sans/" rel="section"><i class="fa fa-fw fa-heart"></i>sans</a>
    144. 

  144. 

  145.   </li>
    146. 

  146.   </ul>
    147. 

  147. </nav>
    148. 

  148. 

  149. 

  150. 

  151. 

  152. </div>
    153. 

  153.     </header>
    154. 

  154. 

  155.     
    156. 

  156.   <div class="back-to-top">
    157. 

  157.     <i class="fa fa-arrow-up"></i>
    158. 

  158.     <span>0%</span>
    159. 

  159.   </div>
    160. 

  160.   <div class="reading-progress-bar"></div>
    161. 

  161. 

  162. 

  163.     <main class="main">
    164. 

  164.       <div class="main-inner">
    165. 

  165.         <div class="content-wrap">
    166. 

  166.           
    167. 

  167. 

  168.           <div class="content post posts-expand">
    169. 

  169.             
    170. 

  170. 

  171.     
    172. 

  172.   
    173. 

  173.   
    174. 

  174.   <article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en">
    175. 

  175.     <link itemprop="mainEntityOfPage" href="https://schtonn.github.io/posts/segment-tree/">
    176. 

  176. 

  177.     <span hidden itemprop="author" itemscope itemtype="http://schema.org/Person">
    178. 

  178.       <meta itemprop="image" content="/images/avatar.gif">
    179. 

  179.       <meta itemprop="name" content="Alex">
    180. 

  180.       <meta itemprop="description" content="The blog of a middle school OIer">
    181. 

  181.     </span>
    182. 

  182. 

  183.     <span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization">
    184. 

  184.       <meta itemprop="name" content="Alex's Blog">
    185. 

  185.     </span>
    186. 

  186.       <header class="post-header">
    187. 

  187.         <h1 class="post-title" itemprop="name headline">
    188. 

  188.           线段树
    189. 

  189.         </h1>
    190. 

  190. 

  191.         <div class="post-meta">
    192. 

  192.             <span class="post-meta-item">
    193. 

  193.               <span class="post-meta-item-icon">
    194. 

  194.                 <i class="fa fa-calendar-o"></i>
    195. 

  195.               </span>
    196. 

  196.               <span class="post-meta-item-text">Posted on</span>
    197. 

  197. 

  198.               <time title="Created: 2020-Mar-02 11:37:36" itemprop="dateCreated datePublished" datetime="2020-03-02T11:37:36+08:00">2020-Mar-02</time>
    199. 

  199.             </span>
    200. 

  200.               <span class="post-meta-item">
    201. 

  201.                 <span class="post-meta-item-icon">
    202. 

  202.                   <i class="fa fa-calendar-check-o"></i>
    203. 

  203.                 </span>
    204. 

  204.                 <span class="post-meta-item-text">Edited on</span>
    205. 

  205.                 <time title="Modified: 2020-Mar-29 15:59:45" itemprop="dateModified" datetime="2020-03-29T15:59:45+08:00">2020-Mar-29</time>
    206. 

  206.               </span>
    207. 

  207. 

  208.           
    209. 

  209.             <span id="/posts/segment-tree/" class="post-meta-item leancloud_visitors" data-flag-title="线段树" title="Views">
    210. 

  210.               <span class="post-meta-item-icon">
    211. 

  211.                 <i class="fa fa-eye"></i>
    212. 

  212.               </span>
    213. 

  213.               <span class="post-meta-item-text">Views: </span>
    214. 

  214.               <span class="leancloud-visitors-count"></span>
    215. 

  215.             </span>
    216. 

  216.   
    217. 

  217.   <span class="post-meta-item">
    218. 

  218.     
    219. 

  219.       <span class="post-meta-item-icon">
    220. 

  220.         <i class="fa fa-comment-o"></i>
    221. 

  221.       </span>
    222. 

  222.       <span class="post-meta-item-text">Valine: </span>
    223. 

  223.     
    224. 

  224.     <a title="valine" href="/posts/segment-tree/#valine-comments" itemprop="discussionUrl">
    225. 

  225.       <span class="post-comments-count valine-comment-count" data-xid="/posts/segment-tree/" itemprop="commentCount"></span>
    226. 

  226.     </a>
    227. 

  227.   </span>
    228. 

  228.   
    229. 

  229.   
    230. 

  230. 

  231.         </div>
    232. 

  232.       </header>
    233. 

  233. 

  234.     
    235. 

  235.     
    236. 

  236.     
    237. 

  237.     <div class="post-body" itemprop="articleBody">
    238. 

  238. 

  239.       
    240. 

  240.         <h3 id="前置知识"><a class="markdownIt-Anchor" href="#前置知识"></a> 前置知识</h3>
    241. 

  241. <p>数组，结构体，二叉树</p>
    242. 

  242. <h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3>
    243. 

  243. <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 />
    244. 

  244. 暴力是肯定不行的，数据范围太大，操作太多，会超时。<br />
    245. 

  245. 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>
    246. 

  246. <a id="more"></a>
    247. 

  247. <h3 id="线段树"><a class="markdownIt-Anchor" href="#线段树"></a> 线段树</h3>
    248. 

  248. <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>
    249. 

  249. <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 />
    250. 

  250. 而子节点也是一棵线段树。<br />
    251. 

  251. 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 />
    252. 

  252. 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>
    253. 

  253. <p><img data-src="/images/segment-1.png" alt="segment-1" /></p>
    254. 

  254. <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>
    255. 

  255. <h3 id="查找"><a class="markdownIt-Anchor" href="#查找"></a> 查找</h3>
    256. 

  256. <h4 id="解释"><a class="markdownIt-Anchor" href="#解释"></a> 解释</h4>
    257. 

  257. <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>
    258. 

  258. <ol>
    259. 

  259. <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>
    260. 

  260. <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>
    261. 

  261. <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>
    262. 

  262. <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>
    263. 

  263. <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>
    264. 

  264. </ol>
    265. 

  265. <h4 id="举例"><a class="markdownIt-Anchor" href="#举例"></a> 举例</h4>
    266. 

  266. <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 />
    267. 

  267. -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 />
    268. 

  268. -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 />
    269. 

  269. -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 />
    270. 

  270. -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>
    271. 

  271. <h3 id="修改"><a class="markdownIt-Anchor" href="#修改"></a> 修改</h3>
    272. 

  272. <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>
    273. 

  273. <h4 id="lazy"><a class="markdownIt-Anchor" href="#lazy"></a> lazy</h4>
    274. 

  274. <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>
    275. 

  275. <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>
    276. 

  276. <h4 id="注意"><a class="markdownIt-Anchor" href="#注意"></a> 注意</h4>
    277. 

  277. <p>修改后的结点，需要同时把所有祖先结点全部更新。</p>
    278. 

  278. <h3 id="优势"><a class="markdownIt-Anchor" href="#优势"></a> 优势</h3>
    279. 

  279. <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>
    280. 

  280. <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>
    281. 

  281. <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>
    282. 

  282.     </div>
    283. 

  283. 

  284.     
    285. 

  285.     
    286. 

  286.     
    287. 

  287.         
    288. 

  288. 

  289. <div>
    290. 

  290. <ul class="post-copyright">
    291. 

  291.   <li class="post-copyright-author">
    292. 

  292.     <strong>Post author:  </strong>Alex
    293. 

  293.   </li>
    294. 

  294.   <li class="post-copyright-link">
    295. 

  295.     <strong>Post link: </strong>
    296. 

  296.     <a href="https://schtonn.github.io/posts/segment-tree/" title="线段树">https://schtonn.github.io/posts/segment-tree/</a>
    297. 

  297.   </li>
    298. 

  298.   <li class="post-copyright-license">
    299. 

  299.     <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 class="fa fa-fw fa-creative-commons"></i>BY-NC-SA</a> unless stating additionally.
    300. 

  300.   </li>
    301. 

  301. </ul>
    302. 

  302. </div>
    303. 

  303. 

  304. 

  305.       <footer class="post-footer">
    306. 

  306.           
    307. 

  307.           <div class="post-tags">
    308. 

  308.               <a href="/tags/graph/" rel="tag"><i class="fa fa-tag"></i> graph</a>
    309. 

  309.           </div>
    310. 

  310. 

  311.         
    312. 

  312. 

  313. 

  314.         
    315. 

  315.     <div class="post-nav">
    316. 

  316.       <div class="post-nav-item">
    317. 

  317.     <a href="/posts/fibonacci/" rel="prev" title="斐波那契数列-O(1)">
    318. 

  318.       <i class="fa fa-chevron-left"></i> 斐波那契数列-O(1)
    319. 

  319.     </a></div>
    320. 

  320.       <div class="post-nav-item">
    321. 

  321.     <a href="/posts/computer/" rel="next" title="How do computers work?">
    322. 

  322.       How do computers work? <i class="fa fa-chevron-right"></i>
    323. 

  323.     </a></div>
    324. 

  324.     </div>
    325. 

  325.       </footer>
    326. 

  326.     
    327. 

  327.   </article>
    328. 

  328.   
    329. 

  329.   
    330. 

  330.   
    331. 

  331. 

  332. 

  333. 

  334.           </div>
    335. 

  335.           
    336. 

  336.     <div class="comments" id="valine-comments"></div>
    337. 

  337. 

  338. <script>
    339. 

  339.   window.addEventListener('tabs:register', () => {
    340. 

  340.     let { activeClass } = CONFIG.comments;
    341. 

  341.     if (CONFIG.comments.storage) {
    342. 

  342.       activeClass = localStorage.getItem('comments_active') || activeClass;
    343. 

  343.     }
    344. 

  344.     if (activeClass) {
    345. 

  345.       let activeTab = document.querySelector(`a[href="#comment-${activeClass}"]`);
    346. 

  346.       if (activeTab) {
    347. 

  347.         activeTab.click();
    348. 

  348.       }
    349. 

  349.     }
    350. 

  350.   });
    351. 

  351.   if (CONFIG.comments.storage) {
    352. 

  352.     window.addEventListener('tabs:click', event => {
    353. 

  353.       if (!event.target.matches('.tabs-comment .tab-content .tab-pane')) return;
    354. 

  354.       let commentClass = event.target.classList[1];
    355. 

  355.       localStorage.setItem('comments_active', commentClass);
    356. 

  356.     });
    357. 

  357.   }
    358. 

  358. </script>
    359. 

  359. 

  360.         </div>
    361. 

  361.           
    362. 

  362.   
    363. 

  363.   <div class="toggle sidebar-toggle">
    364. 

  364.     <span class="toggle-line toggle-line-first"></span>
    365. 

  365.     <span class="toggle-line toggle-line-middle"></span>
    366. 

  366.     <span class="toggle-line toggle-line-last"></span>
    367. 

  367.   </div>
    368. 

  368. 

  369.   <aside class="sidebar">
    370. 

  370.     <div class="sidebar-inner">
    371. 

  371. 

  372.       <ul class="sidebar-nav motion-element">
    373. 

  373.         <li class="sidebar-nav-toc">
    374. 

  374.           Table of Contents
    375. 

  375.         </li>
    376. 

  376.         <li class="sidebar-nav-overview">
    377. 

  377.           Overview
    378. 

  378.         </li>
    379. 

  379.       </ul>
    380. 

  380. 

  381.       <!--noindex-->
    382. 

  382.       <div class="post-toc-wrap sidebar-panel">
    383. 

  383.           <div class="post-toc motion-element"><ol class="nav"><li class="nav-item nav-level-3"><a class="nav-link" href="#前置知识"><span class="nav-number">1.</span> <span class="nav-text"> 前置知识</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#引入"><span class="nav-number">2.</span> <span class="nav-text"> 引入</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#线段树"><span class="nav-number">3.</span> <span class="nav-text"> 线段树</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#查找"><span class="nav-number">4.</span> <span class="nav-text"> 查找</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><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-4"><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-3"><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-4"><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-4"><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-3"><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-3"><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>
    384. 

  384.       </div>
    385. 

  385.       <!--/noindex-->
    386. 

  386. 

  387.       <div class="site-overview-wrap sidebar-panel">
    388. 

  388.         <div class="site-author motion-element" itemprop="author" itemscope itemtype="http://schema.org/Person">
    389. 

  389.   <p class="site-author-name" itemprop="name">Alex</p>
    390. 

  390.   <div class="site-description" itemprop="description">The blog of a middle school OIer</div>
    391. 

  391. </div>
    392. 

  392. <div class="site-state-wrap motion-element">
    393. 

  393.   <nav class="site-state">
    394. 

  394.       <div class="site-state-item site-state-posts">
    395. 

  395.           <a href="/archives">
    396. 

  396.           <span class="site-state-item-count">10</span>
    397. 

  397.           <span class="site-state-item-name">posts</span>
    398. 

  398.         </a>
    399. 

  399.       </div>
    400. 

  400.       <div class="site-state-item site-state-tags">
    401. 

  401.             <a href="/tags/">
    402. 

  402.         <span class="site-state-item-count">4</span>
    403. 

  403.         <span class="site-state-item-name">tags</span></a>
    404. 

  404.       </div>
    405. 

  405.   </nav>
    406. 

  406. </div>
    407. 

  407.   <div class="links-of-author motion-element">
    408. 

  408.       <span class="links-of-author-item">
    409. 

  409.         <a href="https://github.com/schtonn" title="GitHub → https:&#x2F;&#x2F;github.com&#x2F;schtonn" rel="noopener" target="_blank"><i class="fa fa-fw fa-github"></i>GitHub</a>
    410. 

  410.       </span>
    411. 

  411.       <span class="links-of-author-item">
    412. 

  412.         <a href="mailto:m18519511495@163.com" title="E-Mail → mailto:m18519511495@163.com" rel="noopener" target="_blank"><i class="fa fa-fw fa-envelope"></i>E-Mail</a>
    413. 

  413.       </span>
    414. 

  414.   </div>
    415. 

  415. 

  416. 

  417.   <div class="links-of-blogroll motion-element">
    418. 

  418.     <div class="links-of-blogroll-title">
    419. 

  419.       <i class="fa fa-fw fa-link"></i>
    420. 

  420.       Links
    421. 

  421.     </div>
    422. 

  422.     <ul class="links-of-blogroll-list">
    423. 

  423.         <li class="links-of-blogroll-item">
    424. 

  424.           <a href="https://yonghong.github.io/" title="https:&#x2F;&#x2F;yonghong.github.io" rel="noopener" target="_blank">Yonghong</a>
    425. 

  425.         </li>
    426. 

  426.         <li class="links-of-blogroll-item">
    427. 

  427.           <a href="https://source.unsplash.com/random/1600x900" title="https:&#x2F;&#x2F;source.unsplash.com&#x2F;random&#x2F;1600x900" rel="noopener" target="_blank">Background</a>
    428. 

  428.         </li>
    429. 

  429.     </ul>
    430. 

  430.   </div>
    431. 

  431. 

  432.       </div>
    433. 

  433. 

  434.       
    435. 

  435.       <div id="treefrog" style="text-align: center;margin-top: 18px;">
    436. 

  436.         <object type="application/x-shockwave-flash" style="outline:none;" data="/flash/treefrog.swf?up_bodyColor=444444&up_pattern=0&up_flyColor=777777&up_tongueColor=555555&up_patternColor=000000&up_releaseFly=0&up_frogName=Froggie&up_backgroundImage=http://&up_bellySize=.5&up_footColor=444444&up_eyeColor=444444&up_backgroundColor=222222&" width="300" height="600"><param name="movie" value="http://cdn.abowman.com/widgets/treefrog/treefrog.swf?up_bodyColor=444444&up_pattern=0&up_flyColor=777777&up_tongueColor=555555&up_patternColor=000000&up_releaseFly=0&up_frogName=Froggie&up_backgroundImage=http://&up_bellySize=.5&up_footColor=444444&up_eyeColor=444444&up_backgroundColor=222222&"></param><param name="AllowScriptAccess" value="always"></param><param name="wmode" value="opaque"></param><param name="scale" value="noscale"/><param name="salign" value="tl"/></object>
    437. 

  437.       </div>
    438. 

  438.       
    439. 

  439. 

  440.     </div>
    441. 

  441.   </aside>
    442. 

  442.   <div id="sidebar-dimmer"></div>
    443. 

  443. 

  444. 

  445.       </div>
    446. 

  446.     </main>
    447. 

  447. 

  448.     <footer class="footer">
    449. 

  449.       <div class="footer-inner">
    450. 

  450.         
    451. 

  451. 

  452.         
    453. 

  453. 

  454. <div class="copyright">
    455. 

  455.   
    456. 

  456.   &copy; 
    457. 

  457.   <span itemprop="copyrightYear">2020</span>
    458. 

  458.   <span class="with-love">
    459. 

  459.     <i class="fa fa-user"></i>
    460. 

  460.   </span>
    461. 

  461.   <span class="author" itemprop="copyrightHolder">Alex Xiang</span>
    462. 

  462. </div>
    463. 

  463. 

  464.         
    465. 

  465. 

  466. 

  467. 

  468. 

  469. 

  470. 

  471. 

  472. 

  473.       </div>
    474. 

  474.     </footer>
    475. 

  475.   </div>
    476. 

  476. 

  477.   
    478. 

  478.   <script src="/lib/anime.min.js"></script>
    479. 

  479.   <script src="//cdn.jsdelivr.net/npm/lozad@1/dist/lozad.min.js"></script>
    480. 

  480.   <script src="/lib/velocity/velocity.min.js"></script>
    481. 

  481.   <script src="/lib/velocity/velocity.ui.min.js"></script>
    482. 

  482. 

  483. <script src="/js/utils.js"></script>
    484. 

  484. 

  485. <script src="/js/motion.js"></script>
    486. 

  486. 

  487. 

  488. <script src="/js/schemes/muse.js"></script>
    489. 

  489. 

  490. 

  491. <script src="/js/next-boot.js"></script>
    492. 

  492. 

  493. 

  494. 

  495. 

  496.   
    497. 

  497. 

  498. 

  499. 

  500. 

  501. 

  502. 

  503. 

  504. 

  505. 

  506. 

  507. 

  508. 

  509. 

  510. 

  511. 

  512.   
    513. 

  513. 

  514.   
    515. 

  515.       
    516. 

  516. <link rel="stylesheet" href="//cdn.jsdelivr.net/npm/katex@0/dist/katex.min.css">
    517. 

  517. 

  518. 

  519.   
    520. 

  520. 

  521. 

  522. <script>
    523. 

  523. NexT.utils.loadComments(document.querySelector('#valine-comments'), () => {
    524. 

  524.   NexT.utils.getScript('//unpkg.com/valine/dist/Valine.min.js', () => {
    525. 

  525.     var GUEST = ['nick', 'mail', 'link'];
    526. 

  526.     var guest = 'nick,mail,link';
    527. 

  527.     guest = guest.split(',').filter(item => {
    528. 

  528.       return GUEST.includes(item);
    529. 

  529.     });
    530. 

  530.     new Valine({
    531. 

  531.       el         : '#valine-comments',
    532. 

  532.       verify     : true,
    533. 

  533.       notify     : true,
    534. 

  534.       appId      : 'BmologYYnRqCv0SLHDeDdA17-gzGzoHsz',
    535. 

  535.       appKey     : 'w9mVebFMdCmY6Nh9vfcBGaGt',
    536. 

  536.       placeholder: "Say something.",
    537. 

  537.       avatar     : 'mm',
    538. 

  538.       meta       : guest,
    539. 

  539.       pageSize   : '10' || 10,
    540. 

  540.       visitor    : true,
    541. 

  541.       lang       : 'en' || 'zh-cn',
    542. 

  542.       path       : location.pathname,
    543. 

  543.       recordIP   : true,
    544. 

  544.       serverURLs : ''
    545. 

  545.     });
    546. 

  546.   }, window.Valine);
    547. 

  547. });
    548. 

  548. </script>
    549. 

  549. 

  550. </body>
    551. 

  551. </html>
    552.