|
|
<!DOCTYPE html><html lang="zh-CN"><head><meta charset="utf-8"><meta http-equiv="x-dns-prefetch-control" content="on"><meta name="viewport" content="width=device-width,initial-scale=1,minimum-scale=1,maximum-scale=1,user-scalable=no"><meta name="renderer" content="webkit"><meta name="force-rendering" content="webkit"><meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"><meta name="HandheldFriendly" content="True"><meta name="mobile-web-app-capable" content="yes"><link rel="shortcut icon" href="https://hans362-img.oss.0vv0.top/favicon.ico"><link rel="icon" type="image/png" sizes="16x16" href="https://hans362-img.oss.0vv0.top/favicon-16x16.png"><link rel="icon" type="image/png" sizes="32x32" href="https://hans362-img.oss.0vv0.top/favicon-32x32.png"><link rel="apple-touch-icon" sizes="180x180" href="https://hans362-img.oss.0vv0.top/apple-touch-icon.png"><link rel="mask-icon" href="https://hans362-img.oss.0vv0.top/safari-pinned-tab.svg"><title>【题解】P1855 榨取kkksc03 | Hans362 's Blog</title><meta name="keywords" content="OI, 题解, Hans362"><meta name="description" content="某天刷洛谷的时候看到这道题，被标题骗进来了（标题好爽（雾 毋庸置疑这是一道大水题，大概是看在 kkk 的面子上才标了「普及/提高-」难度 那么决定了，第一篇题解就给它吧！（其实只是想练习一下 MathJax 的使用（"><meta property="og:type" content="article"><meta property="og:title" content="【题解】P1855 榨取kkksc03"><meta property="og:url" content="https://blog.hans362.cn/post/p1855-solution/"><meta property="og:site_name" content="Hans362 's Blog"><meta property="og:description" content="某天刷洛谷的时候看到这道题，被标题骗进来了（标题好爽（雾 毋庸置疑这是一道大水题，大概是看在 kkk 的面子上才标了「普及/提高-」难度 那么决定了，第一篇题解就给它吧！（其实只是想练习一下 MathJax 的使用（"><meta property="og:locale" content="zh_CN"><meta property="article:published_time" content="2020-01-25T07:02:19.000Z"><meta property="article:modified_time" content="2025-04-11T10:35:15.357Z"><meta property="article:author" content="Hans362"><meta property="article:tag" content="OI"><meta property="article:tag" content="题解"><meta name="twitter:card" content="summary_large_image"><link rel="stylesheet" href="/css/style/main.css"><link rel="stylesheet" id="hl-default-theme" href="https://blog.hans362.cn/npm/highlight.js@10.1.2/styles/atom-one-light.css" media="none"><link rel="stylesheet" id="hl-dark-theme" href="https://blog.hans362.cn/npm/highlight.js@10.1.2/styles/atom-one-dark.css" media="none"><script src="/js/darkmode.js"></script><link rel="dns-prefetch" href="https://analytics.0vv0.top"><link rel="preconnect" href="https://hans362-img.oss.0vv0.top"><meta name="generator" content="Hexo 7.1.1"><link rel="alternate" href="/atom.xml" title="Hans362 's Blog" type="application/atom+xml"></head><body><div class="app-shell-loader">加载中...</div><div class="container" tabindex="-1"><header><div class="header__left"><a href="/" class="button"><span class="logo__text">Hans362 's Blog</span></a></div><div class="header__right"><div class="navbar__menus"><a href="/" class="button"><div class="navbar-menu">首页</div></a><a href="/archives/" class="button"><div class="navbar-menu">归档</div></a><a href="/tags/" class="button"><div class="navbar-menu">标签</div></a><a href="/bangumi/" class="button"><div class="navbar-menu">追番</div></a><a href="/links/" class="button"><div class="navbar-menu">友链</div></a><a href="/about/" class="button"><div class="navbar-menu">关于</div></a><a href="/atom.xml" class="button"><div class="navbar-menu">RSS</div></a></div><a href="/search/" class="button"><div id="btn-search"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1024 1024" width="24" height="24" fill="currentColor" stroke="currentColor" stroke-width="32"><path d="M192 448c0-141.152 114.848-256 256-256s256 114.848 256 256-114.848 256-256 256-256-114.848-256-256z m710.624 409.376l-206.88-206.88A318.784 318.784 0 0 0 768 448c0-176.736-143.264-320-320-320S128 271.264 128 448s143.264 320 320 320a318.784 318.784 0 0 0 202.496-72.256l206.88 206.88 45.248-45.248z"></path></svg></div></a><a href="javaScript:void(0);" rel="external nofollow noreferrer" class="button"><div id="btn-toggle-dark"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M21 12.79A9 9 0 1 1 11.21 3 7 7 0 0 0 21 12.79z"></path></svg></div></a><a href="#" class="button" id="b2t" aria-label="回到顶部" title="回到顶部"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1024 1024" width="32" height="32"><path d="M233.376 722.752L278.624 768 512 534.624 745.376 768l45.248-45.248L512 444.128zM192 352h640V288H192z" fill="currentColor"></path></svg> </a><a class="dropdown-icon button" tabindex="0"><div id="btn-dropdown"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 20 20" width="24" height="24" fill="none" stroke="currentColor" stroke-width="0.7" stroke-linecap="round" stroke-linejoin="round"><path fill="currentColor" d="M3.314,4.8h13.372c0.41,0,0.743-0.333,0.743-0.743c0-0.41-0.333-0.743-0.743-0.743H3.314c-0.41,0-0.743,0.333-0.743,0.743C2.571,4.467,2.904,4.8,3.314,4.8z M16.686,15.2H3.314c-0.41,0-0.743,0.333-0.743,0.743s0.333,0.743,0.743,0.743h13.372c0.41,0,0.743-0.333,0.743-0.743S17.096,15.2,16.686,15.2z M16.686,9.257H3.314c-0.41,0-0.743,0.333-0.743,0.743s0.333,0.743,0.743,0.743h13.372c0.41,0,0.743-0.333,0.743-0.743S17.096,9.257,16.686,9.257z"></path></svg></div></a><div class="dropdown-menus" id="dropdown-menus"><a href="/" class="dropdown-menu button">首页</a> <a href="/archives/" class="dropdown-menu button">归档</a> <a href="/tags/" class="dropdown-menu button">标签</a> <a href="/bangumi/" class="dropdown-menu button">追番</a> <a href="/links/" class="dropdown-menu button">友链</a> <a href="/about/" class="dropdown-menu button">关于</a> <a href="/atom.xml" class="dropdown-menu button">RSS</a></div></div></header><cover></cover><main><div class="post-content"><div class="post-title"><h1 class="post-title__text">【题解】P1855 榨取kkksc03</h1><div class="post-title__meta"><a href="/archives/2020/01/" class="post-meta__date button">2020-01-25</a> <span class="separate-dot"></span> <a href="/categories/OI-%E7%AC%94%E8%AE%B0/" class="button"><span class="post-meta__cats">OI 笔记</span></a><style>.post-meta__pv{color:var(--t-l);visibility:hidden;opacity:0;transition:.2s}</style><span class="separate-dot"></span> <span class="post-meta__pv"></span></div></div><aside class="post-side"><div class="post-side__toc"><div class="toc-title">文章目录</div><ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E9%A2%98%E7%9B%AE"><span class="toc-text">题目</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%80%9D%E8%B7%AF"><span class="toc-text">思路</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E4%BB%A3%E7%A0%81"><span class="toc-text">代码</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%8F%8D%E6%80%9D"><span class="toc-text">反思</span></a></li></ol></div></aside><a class="btn-toc button" id="btn-toc" tabindex="0"><svg viewBox="0 0 1024 1024" width="32" height="32" xmlns="http://www.w3.org/2000/svg"><path d="M128 256h64V192H128zM320 256h576V192H320zM128 544h64v-64H128zM320 544h576v-64H320zM128 832h64v-64H128zM320 832h576v-64H320z" fill="currentColor"></path></svg></a><div class="toc-menus" id="toc-menus"><div class="toc-title">文章目录</div><ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E9%A2%98%E7%9B%AE"><span class="toc-text">题目</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%80%9D%E8%B7%AF"><span class="toc-text">思路</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E4%BB%A3%E7%A0%81"><span class="toc-text">代码</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%8F%8D%E6%80%9D"><span class="toc-text">反思</span></a></li></ol></div><article class="post post__with-toc card"><div class="post__header"><div class="post__expire" id="post-expired-notify"><p><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16" style="fill:#f5a623;stroke:#f5a623"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg> 本文最后更新于 <span id="expire-date"></span> 天前，文中部分描述可能已经过时。</p></div><script>(()=>{var e=Date.parse("2020-01-25"),t=(new Date).getTime(),t=Math.floor((t-e)/864e5);120<=t&&(document.querySelectorAll("#expire-date")[0].innerHTML=t,document.querySelectorAll("#post-expired-notify")[0].style.display="block")})()</script></div><div class="post__content"><html><head><script>var meting_api="https://api-v2.hans362.cn/vip/?server=:server&type=:type&id=:id&r=:r"</script><script class="meting-secondary-script-marker" src="/js/Meting.min.js"></script></head><body><p>某天刷洛谷的时候看到这道题，被标题骗进来了（标题好爽（雾</p><p>毋庸置疑这是一道大水题，大概是看在 kkk 的面子上才标了「普及/提高-」难度</p><p>那么决定了，第一篇题解就给它吧！（其实只是想练习一下 MathJax 的使用（</p><span id="more"></span><h2 id="题目"><a class="markdownIt-Anchor" href="#题目"></a> 题目</h2><p><a target="_blank" rel="noopener" href="https://www.luogu.com.cn/problem/P1855">P1855 榨取kkksc03</a></p><h2 id="思路"><a class="markdownIt-Anchor" href="#思路"></a> 思路</h2><p>题目的废话很多（其实是洛谷的广告吧），抛开那些废话你会发现这是一道背包问题</p><p>我们只需将每个愿望消耗的时间和金钱看作费用，而每个愿望的价值都看作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">1</span></span></span></span>，那么这就是一道二维费用的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>01</mn></mrow><annotation encoding="application/x-tex">01</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">0</span><span class="mord">1</span></span></span></span> 背包问题</p><p>于是乎咱开一个三维数组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f[i][j][k]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.10764em">f</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mclose">]</span></span></span></span>，其数值表示前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 个愿望恰好消耗 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.85396em;vertical-align:-.19444em"></span><span class="mord mathnormal" style="margin-right:.05724em">j</span></span></span></span> 的时间和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span></span></span></span> 的金钱所产生的最大价值，由于我们已经将每个愿望的价值看作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">1</span></span></span></span>，产生的最大价值也就等价于最大的愿望数</p><p>我们再用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 表示第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 个愿望所消耗的时间，用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">b[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">b</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 表示第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 个愿望所消耗的金钱</p><p>根据题意，得出状态转移方程：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>=</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>j</mi><mo>−</mo><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo>−</mo><mi>b</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f[i][j][k]=max(f[i-1][j][k],f[i-1][j-a[i]][k-b[i]]+1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.10764em">f</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mclose">]</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">m</span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:.10764em">f</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">1</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal" style="margin-right:.10764em">f</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">1</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">b</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mclose">]</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><p>稍微解释一下，对于第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 个愿望，有“实现”还是“不实现”两种策略</p><p>如果我们实现第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 个愿望，那么前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 个愿望恰好消耗 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.85396em;vertical-align:-.19444em"></span><span class="mord mathnormal" style="margin-right:.05724em">j</span></span></span></span> 的时间和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span></span></span></span> 的金钱所产生的最大价值就等于前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.74285em;vertical-align:-.08333em"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">1</span></span></span></span> 个愿望恰好消耗 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>−</mo><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">j-a[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.85396em;vertical-align:-.19444em"></span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 的时间和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><mi>b</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k-b[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.77777em;vertical-align:-.08333em"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">b</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 的金钱所产生的最大价值再加上第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 个愿望的价值（也就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">1</span></span></span></span>）</p><p>如果我们不实现第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 个愿望，那么前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 个愿望恰好消耗 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.85396em;vertical-align:-.19444em"></span><span class="mord mathnormal" style="margin-right:.05724em">j</span></span></span></span> 的时间和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span></span></span></span> 的金钱所产生的最大价值就等于前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.74285em;vertical-align:-.08333em"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">1</span></span></span></span> 个愿望恰好消耗 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.85396em;vertical-align:-.19444em"></span><span class="mord mathnormal" style="margin-right:.05724em">j</span></span></span></span> 的时间和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span></span></span></span> 的金钱所产生的最大价值</p><p>对于两种策略再取二者中的最大值，就可以得到当前阶段的最优策略</p><p>再来看看边界条件，首先初始化三维数组中的每一项为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">0</span></span></span></span>，不多解释</p><p>其次状态转移方程里有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>−</mo><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">j-a[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.85396em;vertical-align:-.19444em"></span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><mi>b</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k-b[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.77777em;vertical-align:-.08333em"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">b</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span>，有没有想过这两项有可能减出来是负数呢？</p><p>所以代码中除了基本的三层循环外，还需要加入判断，一旦两者中存在一者为负，那么我们直接让它继承 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f[i-1][j][k]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.10764em">f</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">1</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mclose">]</span></span></span></span> 的数值，也就是：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>=</mo><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f[i][j][k]=f[i-1][j][k]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.10764em">f</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mclose">]</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.10764em">f</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">1</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mclose">]</span></span></span></span></span></p><p>（我才不会告诉你我这题 WA 了两次就是因为这个←_←）</p><h2 id="代码"><a class="markdownIt-Anchor" href="#代码"></a> 代码</h2><p>dp 的代码实现还是相对比较容易的，直接上代码：</p><pre><code class="hljs inform7">#include <bits/stdc++.h>
|
|
|
|
|
|
using namespace std;
|
|
|
|
|
|
int f<span class="hljs-comment">[101]</span><span class="hljs-comment">[201]</span><span class="hljs-comment">[201]</span>;
|
|
|
int a<span class="hljs-comment">[101]</span>;
|
|
|
int b<span class="hljs-comment">[101]</span>;
|
|
|
|
|
|
int main() {
|
|
|
int n, m, t;
|
|
|
int ans = 0;
|
|
|
cin >> n >> m >> t;
|
|
|
for (int i = 1; i <= n; i++) {
|
|
|
cin >> a<span class="hljs-comment">[i]</span> >> b<span class="hljs-comment">[i]</span>;
|
|
|
}
|
|
|
for (int i = 1; i <= n; i++) {
|
|
|
for (int j = 1; j <= m; j++) {
|
|
|
for (int k = 1; k <= t; k++) {
|
|
|
if (j >= a<span class="hljs-comment">[i]</span> && k >= b<span class="hljs-comment">[i]</span>) {
|
|
|
f<span class="hljs-comment">[i]</span><span class="hljs-comment">[j]</span><span class="hljs-comment">[k]</span> = max(f<span class="hljs-comment">[i - 1]</span><span class="hljs-comment">[j]</span><span class="hljs-comment">[k]</span>, f<span class="hljs-comment">[i - 1]</span><span class="hljs-comment">[j - a<span class="hljs-comment">[i]</span>]</span><span class="hljs-comment">[k - b<span class="hljs-comment">[i]</span>]</span> + 1);
|
|
|
} else {
|
|
|
f<span class="hljs-comment">[i]</span><span class="hljs-comment">[j]</span><span class="hljs-comment">[k]</span> = f<span class="hljs-comment">[i - 1]</span><span class="hljs-comment">[j]</span><span class="hljs-comment">[k]</span>;
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
cout << f<span class="hljs-comment">[n]</span><span class="hljs-comment">[m]</span><span class="hljs-comment">[t]</span> << endl;
|
|
|
return 0;
|
|
|
}</code></pre><p>原谅我代码写得这么丑ㄟ(▔ ,▔)ㄏ</p><h2 id="反思"><a class="markdownIt-Anchor" href="#反思"></a> 反思</h2><p>这个算法的时间复杂度是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo>∗</mo><mi>M</mi><mo>∗</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(N*M*T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:.10903em">N</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.13889em">T</span><span class="mclose">)</span></span></span></span>，空间复杂度也是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo>∗</mo><mi>M</mi><mo>∗</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(N*M*T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:.10903em">N</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.13889em">T</span><span class="mclose">)</span></span></span></span>，对于这道题的数据规模足矣，但是显然还有更加优化的做法</p><p>我们可以考虑降维成二维数组进一步优化空间复杂度，事实上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 的这一维是可以省去的，只需略微改动一下状态转移方程和循环结构，不断覆盖上一次的无用数据：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>=</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">[</mo><mi>j</mi><mo>−</mo><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo>−</mo><mi>b</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f[j][k]=max(f[j][k],f[j-a[i]][k-b[i]]+1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.10764em">f</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mclose">]</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">m</span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:.10764em">f</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal" style="margin-right:.10764em">f</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">b</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mclose">]</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><p>这样我们就得到了一个空间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>M</mi><mo>∗</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(M*T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.13889em">T</span><span class="mclose">)</span></span></span></span> 的做法～</p><p>当然还有记忆化搜索等做法，各位可以自己去研究哈～</p><p>那么本篇题解就是这样，kkksc03 已经被我们榨取得<s>一滴不剩</s>（大雾）</p></body></html></div><div class="license"><div class="license-title">【题解】P1855 榨取kkksc03</div><div class="license-link"><a href="https://blog.hans362.cn/post/p1855-solution/">https://blog.hans362.cn/post/p1855-solution/</a></div><div class="license-meta"><div class="license-meta-item"><div class="license-meta-title">本文作者</div><div class="license-meta-text">Hans362</div></div><div class="license-meta-item"><div class="license-meta-title">最后更新</div><div class="license-meta-text">2020-01-25</div></div><div class="license-meta-item"><div class="license-meta-title">许可协议</div><div class="license-meta-text"><a href="https://creativecommons.org/licenses/by-nc-sa/4.0/deed.zh" rel="nofollow noopener noreferrer" target="_blank">CC BY-NC-SA 4.0</a></div></div></div><div>转载或引用本文时请遵守许可协议，注明出处、不得用于商业用途！</div></div><div class="post-footer__cats"><a href="/categories/OI-%E7%AC%94%E8%AE%B0/" class="post-cats__link button">OI 笔记</a><a href="/tags/OI/" class="post-tags__link button"># OI</a><a href="/tags/%E9%A2%98%E8%A7%A3/" class="post-tags__link button"># 题解</a></div></article><div class="nav"><div class="nav__prev"><a href="/post/2019-ncov-information/" class="nav__link"><div><svg viewBox="0 0 1024 1024" xmlns="http://www.w3.org/2000/svg" width="24" height="24"><path d="M589.088 790.624L310.464 512l278.624-278.624 45.248 45.248L400.96 512l233.376 233.376z" fill="#808080"></path></svg></div><div><div class="nav__label">上一篇</div><div class="nav__title">【项目发布】2019-nCov 中国实时疫情信息聚合</div></div></a></div><div class="nav__next"><a href="/post/weekly-20/" class="nav__link"><div><div class="nav__label">下一篇</div><div class="nav__title">周记#20 - 终极高一生活感想</div></div><div><svg viewBox="0 0 1024 1024" xmlns="http://www.w3.org/2000/svg" width="24" height="24"><path d="M434.944 790.624l-45.248-45.248L623.04 512l-233.376-233.376 45.248-45.248L713.568 512z" fill="#808080"></path></svg></div></a></div></div><div class="post__sponsers card"><div class="sponser-label">喜欢这篇文章吗？考虑支持一下作者吧～</div><a class="sponser-button button" href="https://afdian.net/@hans362" rel="external nofollow noreferrer" target="_blank" data-type="afdian">爱发电</a> <a class="sponser-button button" data-type="alipay">支付宝<img class="sponser-qrcode" src="https://hans362-img.oss.0vv0.top/2021/08/05/68281340.jpg"></a></div><div class="post__comments post__with-toc card" id="comment"><h4>评论</h4><div id="disqus_thread">您所在的地区可能无法访问 Disqus 评论系统，请切换网络环境再尝试。</div></div></div></main><footer><p class="footer-copyright">Copyright © 2017 - 2025 <a href="/">Hans362 's Blog</a></p><p>Powered by <a href="https://hexo.io" target="_blank">Hexo</a> | Theme - <a href="https://github.com/ChrAlpha/hexo-theme-cards" target="_blank">Cards</a></p><script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script><ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-8746554831230893" data-ad-slot="6356225601" data-ad-format="auto" data-full-width-responsive="true"></ins><script>(adsbygoogle=window.adsbygoogle||[]).push({})</script></footer></div><script defer src="https://blog.hans362.cn/npm/vanilla-lazyload@17.8.3/dist/lazyload.min.js"></script><script>window.lazyLoadOptions={elements_selector:".lazy"}</script><script async defer data-website-id="5d181692-8a81-4c20-a282-cee87a6b90ef" src="https://analytics.0vv0.top/vue.js"></script><script src="/js/pageviews.js"></script><link rel="stylesheet" href="https://blog.hans362.cn/npm/katex@0.16.0/dist/katex.min.css" crossorigin="anonymous"><script>function loadComment(){let n,e;(n=document.createElement("script")).src="https://blog.hans362.cn/js/disqus.js",document.body.appendChild(n),n.onload=()=>{new DisqusJS({shortname:"hans362-s-blog",siteName:"Hans362 's Blog",api:"https://api-v3.hans362.cn/",apikey:"8Z1UVT4UOk22yNyk9MhpqQ0FLb27Hb1bpV066b4v9zOFie0GQ6VCoJ9TJwoGlCVF",admin:"hans362",identifier:"post/p1855-solution/",url:"https://blog.hans362.cn/post/p1855-solution/",nesting:"4"})},(e=document.createElement("link")).rel="stylesheet",e.href="https://blog.hans362.cn/css/disqusjs.css",document.head.appendChild(e)}var runningOnBrowser="undefined"!=typeof window,isBot=runningOnBrowser&&!("onscroll"in window)||"undefined"!=typeof navigator&&/(gle|ing|ro|msn)bot|crawl|spider|yand|duckgo/i.test(navigator.userAgent),supportsIntersectionObserver=runningOnBrowser&&"IntersectionObserver"in window;setTimeout(function(){var e;!isBot&&supportsIntersectionObserver?(e=new IntersectionObserver(function(n){n[0].isIntersecting&&(loadComment(),e.disconnect())},{threshold:[0]})).observe(document.getElementById("comment")):loadComment()},1)</script></body></html> |