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button">RSS</a></div></div></header><cover></cover><main><div class="post-content"><div class="post-title"><h1 class="post-title__text">【题解】POJ2018 Best Cow Fences</h1><div class="post-title__meta"><a href="/archives/2020/05/" class="post-meta__date button">2020-05-05</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="#%E5%BC%80%E5%B7%A5"><span class="toc-text">开工！</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E8%BF%98%E6%B2%A1%E5%AE%8C%E5%91%A2"><span class="toc-text">还没完呢</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="#%E5%BC%80%E5%B7%A5"><span class="toc-text">开工！</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E8%BF%98%E6%B2%A1%E5%AE%8C%E5%91%A2"><span class="toc-text">还没完呢</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("2020-05-05"),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>最近终于开始学《提高篇》了…然后就刷到了这一题…</p><p>《提高篇》书上写的这道题的思路说实话我看了很久都不太能理解（当然现在明白了），所以就以我的角度来谈谈这道题该怎么下手做出正解～</p><span id="more"></span><h2 id="题目"><a class="markdownIt-Anchor" href="#题目"></a> 题目</h2><p>原题是 POJ2018 但是好像也是从 USACO 搬运过来的？</p><p><a target="_blank" rel="noopener" href="http://poj.org/problem?id=2018">http://poj.org/problem?id=2018</a></p><p>同时附上一个翻译过的版本：</p><p><a target="_blank" rel="noopener" href="https://oj.hans362.cn/problem/5">https://oj.hans362.cn/problem/5</a></p><p><strong>需要注意的是，本题在2018年第一次印刷的《提高篇》书上题干表述有误，“子序列”应更改为“子串”才对</strong></p><p><img src="https://hans362-img.oss.0vv0.top/2020/05/05/15886597191286.jpg?width=1920" class="lazy" data-srcset="https://hans362-img.oss.0vv0.top/2020/05/05/15886597191286.jpg?width=1920" srcset="/loading.gif" alt=""></p><h2 id="开工"><a class="markdownIt-Anchor" href="#开工"></a> 开工！</h2><p>首先呢，这道题是一道二分答案，因为它符合求最小值的最大值这种问题模式</p><p>那么二分答案要做的第一件事就是确定答案的范围，在本题中需要求解的是最大的平均数，按理来说是有一个范围的，但是其实没必要求得那么精确，咱就简单粗暴一点把答案范围定在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mo>−</mo><mn>1</mn><msup><mn>0</mn><mn>6</mn></msup><mo separator="true">,</mo><mn>1</mn><msup><mn>0</mn><mn>6</mn></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-10^6,10^6]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-.25em"></span><span class="mopen">[</span><span class="mord">−</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:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></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:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span>，反正精确范围肯定是这个范围的子集</p><p>紧接着由于答案是平均数，产生小数是很自然的，因此要选择实数域上的二分</p><p>考虑到最后题目让我们输出的是最大平均数的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1000</mn></mrow><annotation encoding="application/x-tex">1000</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 class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span></span> 倍，因此设置精度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mi>p</mi><mi>s</mi><mo>=</mo><mn>1</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup></mrow><annotation encoding="application/x-tex">eps = 1*10^{-5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.625em;vertical-align:-.19444em"></span><span class="mord mathnormal">e</span><span class="mord mathnormal">p</span><span class="mord mathnormal">s</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:.64444em;vertical-align:0"></span><span class="mord">1</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:.8141079999999999em;vertical-align:0"></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:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span></p><p>然后就开始设计二分判定，众所周知二分是把求解转化为判定的一种思想，本题中我们需要判定的是“是否存在一个长度不小于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">L</span></span></span></span> 的子串，且该子串平均数不小于二分的值”</p><p>为了便于操作按照惯用套路咱把数列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">A</span></span></span></span> 中每一项都减去二分的值，运用化归思想将问题转化为判定“是否存在一个长度不小于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">L</span></span></span></span> 的子串，且该子串的和非负”</p><p>由于是存在性问题（区别于恒成立问题），只需要保证数列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">A</span></span></span></span> 中所有长度不小于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">L</span></span></span></span> 的子串的子串和中的最大值非负即可，注意到求和是一个很浪费时间的过程，所以我们就把它用前缀和的方式预处理掉</p><p>学过数列的同学们应该不难理解，假设数列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">A</span></span></span></span> 某一子串为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>A</mi><mrow><mi>j</mi><mo>+</mo><mn>2</mn></mrow></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>A</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">A_{j+1},A_{j+2},...,A_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span> ，定义 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.83333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-.05764em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span> 为数列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">A</span></span></span></span> 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> 项的和，那么</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><mtext>该子串的子串和</mtext><mo>=</mo><msub><mi>A</mi><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow><mi>j</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><msub><mi>A</mi><mi>i</mi></msub><mo>=</mo><msub><mi>S</mi><mi>i</mi></msub><mo>−</mo><msub><mi>S</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">该子串的子串和 = A_{j+1}+A_{j+2}+...+A_i = S_i - S_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord cjk_fallback">该</span><span class="mord cjk_fallback">子</span><span class="mord cjk_fallback">串</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">子</span><span class="mord cjk_fallback">串</span><span class="mord cjk_fallback">和</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:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></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:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></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:.66666em;vertical-align:-.08333em"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</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:.83333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></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:.83333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-.05764em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></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:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:-.05764em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span></span></span></span></span></p><p>那么最后一个问题，怎么遍历数列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">A</span></span></span></span> 中所有长度不小于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">L</span></span></span></span> 的子串呢？这个容易呀～<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>:</mo><mi>L</mi><mo>→</mo><mi>n</mi><mo separator="true">,</mo><mi>j</mi><mo>:</mo><mn>0</mn><mo>→</mo><mi>i</mi><mo>−</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">i: L → n, j: 0 → i-L</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 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:.68333em;vertical-align:0"></span><span class="mord mathnormal">L</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:.85396em;vertical-align:-.19444em"></span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal" style="margin-right:.05724em">j</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:.64444em;vertical-align:0"></span><span class="mord">0</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:.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:.68333em;vertical-align:0"></span><span class="mord mathnormal">L</span></span></span></span> 循环嵌套一下就搞定</p><p>最后得到如下代码：（注释写得挺详细的应该看得懂吧w</p><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string"><bits/stdc++.h></span></span>
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<span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;
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<span class="hljs-type">const</span> <span class="hljs-type">int</span> MAXN = <span class="hljs-number">1e5</span> + <span class="hljs-number">1</span>;
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<span class="hljs-type">int</span> n;
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<span class="hljs-type">double</span> a[MAXN]; <span class="hljs-comment">//原始数列 A</span>
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<span class="hljs-type">double</span> b[MAXN]; <span class="hljs-comment">//数列 A 中每个数都减去二分平均数得到的新数列</span>
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<span class="hljs-type">double</span> s[MAXN]; <span class="hljs-comment">//前缀和</span>
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<span class="hljs-type">int</span> L;
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<span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{
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cin >> n;
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cin >> L;
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<span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i <= n; i++) cin >> a[i];
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<span class="hljs-type">double</span> eps = <span class="hljs-number">1e-5</span>; <span class="hljs-comment">//设置精度为10^{-5}</span>
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<span class="hljs-comment">//显然答案应该落在-1e6~1e6</span>
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<span class="hljs-type">double</span> l = <span class="hljs-number">-1e6</span>;
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<span class="hljs-type">double</span> r = <span class="hljs-number">1e6</span>;
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<span class="hljs-keyword">while</span> (r - l > eps) { <span class="hljs-comment">//控制精度</span>
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<span class="hljs-type">double</span> mid = (l + r) / <span class="hljs-number">2</span>;
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<span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i <= n; i++) {
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b[i] = a[i] - mid; <span class="hljs-comment">//把 A 数列中每个数都减去二分平均数</span>
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}
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<span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i <= n; i++) {
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s[i] = s[i - <span class="hljs-number">1</span>] + b[i]; <span class="hljs-comment">//递推求前缀和</span>
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}
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<span class="hljs-comment">//开始判定</span>
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<span class="hljs-type">double</span> ans = <span class="hljs-number">-1e10</span>; <span class="hljs-comment">//初始化数列 A 中所有长度不小于 L 的子串的子串和中的最大值为一个很小很小的数</span>
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<span class="hljs-comment">//遍历所有长度不小于 L 的子串</span>
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<span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = L; i <= n; i++) {
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<span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j <= i - L; j++) {
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ans = <span class="hljs-built_in">max</span>(ans, s[i] - s[j]); <span class="hljs-comment">//不断更新最大值</span>
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}
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}
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<span class="hljs-keyword">if</span> (ans >= <span class="hljs-number">0</span>) <span class="hljs-comment">//如果最大值非负，那么存在，于是更新左端点，在二分点的右侧继续寻找</span>
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l = mid;
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<span class="hljs-keyword">else</span> <span class="hljs-comment">//如果最大值都是负的，那么彻底没戏，不存在，于是更新右端点，在二分点的左侧继续寻找</span>
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r = mid;
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}
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cout << (<span class="hljs-type">int</span>)(r * <span class="hljs-number">1000</span>) << endl; <span class="hljs-comment">//输出结果，别忘了题目要求的乘以1000</span>
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<span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;
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}</code></pre><h2 id="还没完呢"><a class="markdownIt-Anchor" href="#还没完呢"></a> 还没完呢</h2><p>如果你真的按照上面我说的去做了，那么你大概会得到 57 分，并且发现过不了的点都是 TLE</p><p>（什么！你 AC 了？！要么数据太水要么你不是这样做的）</p><p>虽然总比爆零要好，但是咱要追求完美怎么能罢休～</p><p>分析一下，上面算法中可以优化的应该就是那个两重循环了，我们粗略地把判定时间复杂度看成 <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><msup><mi>N</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(N^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:.10903em">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> （但是实际上应该比 <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><msup><mi>N</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(N^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:.10903em">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 要好一点），那么有没有 <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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(N)</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="mclose">)</span></span></span></span> 的判定方法呢？</p><p>我们又回到了这个问题：怎么优雅地遍历数列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">A</span></span></span></span> 中所有长度不小于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">L</span></span></span></span> 的子串呢？难道真的需要两重循环吗？</p><p>前面提到过，假设数列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">A</span></span></span></span> 某一子串为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>A</mi><mrow><mi>j</mi><mo>+</mo><mn>2</mn></mrow></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>A</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">A_{j+1},A_{j+2},...,A_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span> ，定义 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.83333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-.05764em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span> 为数列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">A</span></span></span></span> 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> 项的和，那么</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><mtext>该子串的子串和</mtext><mo>=</mo><msub><mi>A</mi><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow><mi>j</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><msub><mi>A</mi><mi>i</mi></msub><mo>=</mo><msub><mi>S</mi><mi>i</mi></msub><mo>−</mo><msub><mi>S</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">该子串的子串和 = A_{j+1}+A_{j+2}+...+A_i = S_i - S_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord cjk_fallback">该</span><span class="mord cjk_fallback">子</span><span class="mord cjk_fallback">串</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">子</span><span class="mord cjk_fallback">串</span><span class="mord cjk_fallback">和</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:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></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:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></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:.66666em;vertical-align:-.08333em"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</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:.83333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></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:.83333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-.05764em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></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:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:-.05764em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span></span></span></span></span></p><p>当 <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> 取某个值时，观察上面的式子，发现 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.83333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-.05764em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span> 为定值，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">S_{j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:-.05764em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span></span></span></span> 随 <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> 可变，因为要求最大的子串和，所以减数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">S_{j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:-.05764em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span></span></span></span> 越小越好</p><p>当 <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> 每次增加 <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>，即 <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> 变成 <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> 那么 <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> 的取值范围会由 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi>i</mi><mo>−</mo><mi>L</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,i-L]</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="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></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 mathnormal">L</span><span class="mclose">]</span></span></span></span> 变到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>L</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,i+1-L]</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="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></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:.72777em;vertical-align:-.08333em"></span><span class="mord">1</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">L</span><span class="mclose">]</span></span></span></span> 仔细观察发现 <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> 可取的值只多了一个</p><p>所以我们完全没有必要对 <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> 做循环，我们只需要在 <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> 的每一次循环中记录下当前的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">S_{j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:-.05764em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span></span></span></span> 并和上一轮最小值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi>m</mi><mi>p</mi><mi mathvariant="normal">_</mi><mi>m</mi><mi>i</mi><mi>n</mi></mrow><annotation encoding="application/x-tex">tmp\_min</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.9695199999999999em;vertical-align:-.31em"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">m</span><span class="mord mathnormal">p</span><span class="mord" style="margin-right:.02778em">_</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord mathnormal">n</span></span></span></span> 取较小的那个存入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi>m</mi><mi>p</mi><mi mathvariant="normal">_</mi><mi>m</mi><mi>i</mi><mi>n</mi></mrow><annotation encoding="application/x-tex">tmp\_min</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.9695199999999999em;vertical-align:-.31em"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">m</span><span class="mord mathnormal">p</span><span class="mord" style="margin-right:.02778em">_</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord mathnormal">n</span></span></span></span></p><p>于是得到以下代码：</p><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string"><bits/stdc++.h></span></span>
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<span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;
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<span class="hljs-type">const</span> <span class="hljs-type">int</span> MAXN = <span class="hljs-number">1e5</span> + <span class="hljs-number">1</span>;
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<span class="hljs-type">int</span> n;
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<span class="hljs-type">double</span> a[MAXN]; <span class="hljs-comment">//原始数列 A</span>
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<span class="hljs-type">double</span> b[MAXN]; <span class="hljs-comment">//数列 A 中每个数都减去二分平均数得到的新数列</span>
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<span class="hljs-type">double</span> s[MAXN]; <span class="hljs-comment">//前缀和</span>
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<span class="hljs-type">int</span> L;
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<span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{
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cin >> n;
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cin >> L;
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<span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i <= n; i++) cin >> a[i];
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<span class="hljs-type">double</span> eps = <span class="hljs-number">1e-5</span>; <span class="hljs-comment">//设置精度为10^{-5}</span>
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<span class="hljs-comment">//显然答案应该落在-1e6~1e6</span>
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<span class="hljs-type">double</span> l = <span class="hljs-number">-1e6</span>;
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<span class="hljs-type">double</span> r = <span class="hljs-number">1e6</span>;
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<span class="hljs-keyword">while</span> (r - l > eps) { <span class="hljs-comment">//控制精度</span>
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<span class="hljs-type">double</span> mid = (l + r) / <span class="hljs-number">2</span>;
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<span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i <= n; i++) {
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b[i] = a[i] - mid; <span class="hljs-comment">//把 A 数列中每个数都减去二分平均数</span>
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}
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<span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i <= n; i++) {
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s[i] = s[i - <span class="hljs-number">1</span>] + b[i]; <span class="hljs-comment">//递推求前缀和</span>
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}
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<span class="hljs-comment">//开始判定</span>
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<span class="hljs-type">double</span> ans = <span class="hljs-number">-1e10</span>; <span class="hljs-comment">//初始化数列 A 中所有长度不小于 L 的子串的子串和中的最大值为一个很小很小的数</span>
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<span class="hljs-type">double</span> tmp_min = <span class="hljs-number">1e10</span>; <span class="hljs-comment">//初始化上一轮最小值为一个很大很大的数</span>
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<span class="hljs-comment">//遍历所有长度不小于 L 的子串</span>
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<span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = L; i <= n; i++)
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{
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tmp_min = <span class="hljs-built_in">min</span>(tmp_min, s[i - L]); <span class="hljs-comment">//刷新最小值</span>
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ans = <span class="hljs-built_in">max</span>(ans, s[i] - tmp_min); <span class="hljs-comment">//不断更新最大值</span>
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}
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<span class="hljs-keyword">if</span> (ans >= <span class="hljs-number">0</span>) <span class="hljs-comment">//如果最大值非负，那么存在，于是更新左端点，在二分点的右侧继续寻找</span>
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l = mid;
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<span class="hljs-keyword">else</span> <span class="hljs-comment">//如果最大值都是负的，那么彻底没戏，不存在，于是更新右端点，在二分点的左侧继续寻找</span>
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r = mid;
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}
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cout << (<span class="hljs-type">int</span>)(r * <span class="hljs-number">1000</span>) << endl; <span class="hljs-comment">//输出结果，别忘了题目要求的乘以1000</span>
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<span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;
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}</code></pre><p>然后就可以 AC 啦～</p></body></html></div><div class="license"><div class="license-title">【题解】POJ2018 Best Cow Fences</div><div class="license-link"><a href="https://blog.hans362.cn/post/poj2018-solution/">https://blog.hans362.cn/post/poj2018-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">2020-05-05</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/weekly-21/" 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">周记#21 - 网课时代完结撒花～</div></div></a></div><div class="nav__next"><a href="/post/thinkpad-x201s-hackintosh/" class="nav__link"><div><div class="nav__label">下一篇</div><div class="nav__title">ThinkPad X201s 黑苹果安装记录</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/poj2018-solution/",url:"https://blog.hans362.cn/post/poj2018-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> |