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            <title><![CDATA[Bitovi]]></title>
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<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="题干-coci-20232024-5-bitovi">题干 [COCI 2023/2024 #5] Bitovi<a href="https://your-docusaurus-site.example.com/blog/blog/%E9%A2%98%E8%A7%A3/COCI/2023-2024-05/A-Bitovi.md#%E9%A2%98%E5%B9%B2-coci-20232024-5-bitovi" class="hash-link" aria-label="题干 [COCI 2023/2024 #5] Bitovi的直接链接" title="题干 [COCI 2023/2024 #5] Bitovi的直接链接" translate="no">​</a></h2>
<details class="details_lb9f alert alert--info details_b_Ee" data-collapsed="true"><summary>Details</summary><div><div class="collapsibleContent_i85q"><h2>题目背景</h2><p><strong>译自 <a href="https://hsin.hr/coci/archive/2023_2024" target="_blank" rel="noopener noreferrer" class="">COCI 2023/2024 Contest #5</a> T2「<a href="https://hsin.hr/coci/archive/2023_2024/contest5_tasks.pdf" target="_blank" rel="noopener noreferrer" class="">Bitovi</a>」</strong></p><h2>题目描述</h2><p>哪个先出现，鸡还是蛋？作为一个百万富翁活上一百年好，还是贫穷中度过七天好？如何成为国际象棋大师？如何拉起百叶窗？如何通过期末考试？如何训练龙？这些都是一些有趣的问题，我们可以在竞赛结束后再去思考，但现在我们提出一个不那么有趣的计算机科学问题。</p><p>给定两组数集 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.0502em">B</span></span></span></span>，大小均为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span></span></span></span>。每次操作，你可以从集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 中选择一个任意元素，并改变其二进制表示中的一个任意数位。结果数不能是改变前集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 中的元素。</p><p>例如，数字 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">5</span></span></span></span> 的二进制是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mn>0101</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">0101_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em"></span><span class="mord">010</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>。通过一次操作，它可以变成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>13</mn><mo>=</mo><msub><mn>1101</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">13=1101_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">13</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em"></span><span class="mord">110</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>=</mo><msub><mn>0001</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">1=0001_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em"></span><span class="mord">000</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mo>=</mo><msub><mn>0111</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">7=0111_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">7</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em"></span><span class="mord">011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mo>=</mo><msub><mn>0100</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">4=0100_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em"></span><span class="mord">010</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>。</p><p>确定一系列操作，通过这些操作集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 变得和集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.0502em">B</span></span></span></span> 相等。如果两个集合大小相同，并且集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 中没有不属于集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.0502em">B</span></span></span></span> 的元素，则认为两个集合是相等的。</p><p>注意：操作的数量不必是最小的，但必须满足任务的限制。</p><h2>输入格式</h2><p>第一行包含一个整数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span></span></span></span>（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>N</mi><mo>≤</mo><msup><mn>2</mn><mn>15</mn></msup></mrow><annotation encoding="application/x-tex">1 \le N \le 2^{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">15</span></span></span></span></span></span></span></span></span></span></span></span>），集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.0502em">B</span></span></span></span> 的大小。</p><p>第二行包含 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span></span></span></span> 个不同的整数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></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:0.5806em;vertical-align:-0.15em"></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:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span></span></span></span>（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><msub><mi>a</mi><mi>i</mi></msub><mo>&lt;</mo><msup><mn>2</mn><mn>15</mn></msup></mrow><annotation encoding="application/x-tex">0 \le a_i &lt; 2^{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em"></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:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">15</span></span></span></span></span></span></span></span></span></span></span></span>），表示集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 的元素。</p><p>第二行包含 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span></span></span></span> 个不同的整数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mi>i</mi></msub></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:0.8444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span></span></span></span>（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><msub><mi>b</mi><mi>i</mi></msub><mo>&lt;</mo><msup><mn>2</mn><mn>15</mn></msup></mrow><annotation encoding="application/x-tex">0 \le b_i &lt; 2^{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">15</span></span></span></span></span></span></span></span></span></span></span></span>），表示集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.0502em">B</span></span></span></span> 的元素。</p><h2>输出格式</h2><p>第一行输出操作的数目，不超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>19</mn></msup></mrow><annotation encoding="application/x-tex">2^{19}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">19</span></span></span></span></span></span></span></span></span></span></span></span>。</p><p>接下来若干行，每行输出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x,y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span></span></span></span>（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&lt;</mo><msup><mn>2</mn><mn>15</mn></msup></mrow><annotation encoding="application/x-tex">0 \le x, y &lt; 2^{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">15</span></span></span></span></span></span></span></span></span></span></span></span>），表示将集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 中的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span> 修改为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span></span></span></span>。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span></span></span></span> 在二进制位中只能有一位不同。并且，必须满足 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x\in A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>∉</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">y\not \in A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace nobreak"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span>。</p><h2>输入输出样例 #1</h2><h3>输入 #1</h3><div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">3</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">0 1 2</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">1 2 3</span><br></div></code></pre></div></div><h3>输出 #1</h3><div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">2</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">1 3</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">0 1</span><br></div></code></pre></div></div><h2>输入输出样例 #2</h2><h3>输入 #2</h3><div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">3</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">4 8 31</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">0 4 8</span><br></div></code></pre></div></div><h3>输出 #2</h3><div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">5</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">31 30</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">30 28</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">28 24</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">24 16</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">16 0</span><br></div></code></pre></div></div><h2>输入输出样例 #3</h2><h3>输入 #3</h3><div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">5</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">0 1 2 4 5</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">7 6 5 3 2</span><br></div></code></pre></div></div><h3>输出 #3</h3><div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">9</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">1 3</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">3 7</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">0 1</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">1 0</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">2 6</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">0 2</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">7 3</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">5 7</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">4 5</span><br></div></code></pre></div></div><h3>说明/提示</h3><h3>样例解释 1</h3><p>如果我们先操作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mtext>&nbsp;</mtext><mn>1</mn></mrow><annotation encoding="application/x-tex">0\ 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span><span class="mspace">&nbsp;</span><span class="mord">1</span></span></span></span>，再操作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mtext>&nbsp;</mtext><mn>3</mn></mrow><annotation encoding="application/x-tex">1\ 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span><span class="mspace">&nbsp;</span><span class="mord">3</span></span></span></span>，两次操作间，集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 将有两个相同的元素，这是不被允许的。另一种可能的方案是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mtext>&nbsp;</mtext><mn>3</mn></mrow><annotation encoding="application/x-tex">2\ 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span><span class="mspace">&nbsp;</span><span class="mord">3</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mtext>&nbsp;</mtext><mn>2</mn></mrow><annotation encoding="application/x-tex">0\ 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span><span class="mspace">&nbsp;</span><span class="mord">2</span></span></span></span>。</p><h3>样例解释 2</h3><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>31</mn><mo>=</mo><msub><mn>11111</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">31=11111_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">31</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em"></span><span class="mord">1111</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>。按照从最低有效位到最高有效位修改，依次可以得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>30</mn><mo separator="true">,</mo><mn>28</mn><mo separator="true">,</mo><mn>24</mn><mo separator="true">,</mo><mn>16</mn></mrow><annotation encoding="application/x-tex">30,28,24,16</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em"></span><span class="mord">30</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">28</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">24</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">16</span></span></span></span> 和 <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:0.6444em"></span><span class="mord">0</span></span></span></span>。在所有的操作后，<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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.0502em">B</span></span></span></span> 相同。</p><h3>子任务</h3><table><thead><tr><th style="text-align:center">Subtask</th><th style="text-align:center">Points</th><th style="text-align:center">Constraints</th></tr></thead><tbody><tr><td style="text-align:center">1</td><td style="text-align:center">10</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>b</mi><mi>i</mi></msub><mo>≤</mo><msup><mn>2</mn><mn>14</mn></msup></mrow><annotation encoding="application/x-tex">a_i,b_i \le 2^{14}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></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:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">14</span></span></span></span></span></span></span></span></span></span></span></span></td></tr><tr><td style="text-align:center">2</td><td style="text-align:center">15</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>≤</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">N \le 7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">7</span></span></span></span></td></tr><tr><td style="text-align:center">3</td><td style="text-align:center">30</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>≤</mo><msup><mn>2</mn><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">N \le 2^7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></td></tr><tr><td style="text-align:center">4</td><td style="text-align:center">15</td><td style="text-align:center">无额外限制</td></tr></tbody></table></div></div></details>
<hr>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="分析">分析<a href="https://your-docusaurus-site.example.com/blog/blog/%E9%A2%98%E8%A7%A3/COCI/2023-2024-05/A-Bitovi.md#%E5%88%86%E6%9E%90" class="hash-link" aria-label="分析的直接链接" title="分析的直接链接" translate="no">​</a></h2>
<h5><p>首先我们先对整个序列进行一次"去重"，将两个序列中相同的数先去掉，保证每个数需要操作至少一次</p><p>首先我们对每对去操作变化，每次就是一个<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\log</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mop">lo<span style="margin-right:0.0139em">g</span></span></span></span></span>的复杂度，但是如何处理重复的情况呢</p><p>假设现在存在一个变化情况<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span>-&gt;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span></span></span></span>-&gt;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal" style="margin-right:0.044em">z</span></span></span></span>但是<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span></span></span></span>已经存在，此时我们不妨先用那个已经存在的<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span></span></span></span>去操作</p><p>在最后，我们再执行<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span>-&gt;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span></span></span></span>就行了，相当于我们把这个操作暂缓处理</p><p>时间复杂度<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><mi>l</mi><mi>o</mi><mi>g</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nlogn)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.0278em">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.0197em">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></h5>
<hr>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="ac代码">AC代码<a href="https://your-docusaurus-site.example.com/blog/blog/%E9%A2%98%E8%A7%A3/COCI/2023-2024-05/A-Bitovi.md#ac%E4%BB%A3%E7%A0%81" class="hash-link" aria-label="AC代码的直接链接" title="AC代码的直接链接" translate="no">​</a></h2>
<details class="details_lb9f alert alert--info details_b_Ee" data-collapsed="true"><summary>Details</summary><div><div class="collapsibleContent_i85q"><div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">include</span><span class="token macro property string" style="color:#e3116c">&lt;bits/stdc++.h&gt;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">using</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">namespace</span><span class="token plain"> std</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">ll</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">long</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">long</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">pii</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression" style="color:#36acaa">pair</span><span class="token macro property expression operator" style="color:#393A34">&lt;</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">int</span><span class="token macro property expression punctuation" style="color:#393A34">,</span><span class="token macro property expression keyword" style="color:#00009f">int</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression operator" style="color:#393A34">&gt;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">QWQ</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">return</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression number" style="color:#36acaa">0</span><span class="token macro property expression punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">QAQ</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">return</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> N</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">20</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">set</span><span class="token operator" style="color:#393A34">&lt;</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">&gt;</span><span class="token plain"> a</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> b</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> c</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">stack</span><span class="token operator" style="color:#393A34">&lt;</span><span class="token plain"> pii </span><span class="token operator" style="color:#393A34">&gt;</span><span class="token plain"> rep</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">vector</span><span class="token operator" style="color:#393A34">&lt;</span><span class="token plain"> pii </span><span class="token operator" style="color:#393A34">&gt;</span><span class="token plain"> ans</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">inline</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">void</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">ope</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">,</span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> y </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">if</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> a</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">find</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> y </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">!=</span><span class="token plain"> a</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">end</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"> rep</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">push</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain">y </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> QAQ</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    a</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">erase</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> x </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> a</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">insert</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> y </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    ans</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">push_back</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain">y </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">inline</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">lowbit</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> x </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"> QAQ x</span><span class="token operator" style="color:#393A34">&amp;</span><span class="token operator" style="color:#393A34">-</span><span class="token plain">x</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">inline</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">void</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">turn</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">,</span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> y </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">while</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> x </span><span class="token operator" style="color:#393A34">!=</span><span class="token plain"> y </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> diff</span><span class="token operator" style="color:#393A34">=</span><span class="token plain">x</span><span class="token operator" style="color:#393A34">^</span><span class="token plain">y</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> diff</span><span class="token operator" style="color:#393A34">=</span><span class="token function" style="color:#d73a49">lowbit</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> diff </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token function" style="color:#d73a49">ope</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain">x</span><span class="token operator" style="color:#393A34">^</span><span class="token plain">diff </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        x</span><span class="token operator" style="color:#393A34">^=</span><span class="token plain">diff</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">while</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">!</span><span class="token plain">rep</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">empty</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token keyword" style="color:#00009f">auto</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">nx</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain">ny</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token plain">rep</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">top</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        rep</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">pop</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token function" style="color:#d73a49">turn</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> nx</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain">ny </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">main</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token comment" style="color:#999988;font-style:italic">// freopen( "1.in","r",stdin );</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token comment" style="color:#999988;font-style:italic">// freopen( "1.out","w",stdout );</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    ios</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">sync_with_stdio</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token boolean" style="color:#36acaa">false</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    cin</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">tie</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">nullptr</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> cout</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">tie</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">nullptr</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    cin</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> i</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token plain">n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">++</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        cin</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">x</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        a</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">insert</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> x </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> c</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">insert</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> x </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> i</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token plain">n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">++</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        cin</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">x</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        b</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">insert</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> x </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">while</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">!</span><span class="token plain">b</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">empty</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> y</span><span class="token operator" style="color:#393A34">=</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">b</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">begin</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token keyword" style="color:#00009f">if</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> a</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">find</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> y </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">!=</span><span class="token plain"> a</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">end</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"> b</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">erase</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> y </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> c</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">erase</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> y </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">          </span><span class="token keyword" style="color:#00009f">continue</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> x</span><span class="token operator" style="color:#393A34">=</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">c</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">begin</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token function" style="color:#d73a49">turn</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain">y </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        b</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">erase</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> y </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> c</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">erase</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> x </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    cout</span><span class="token operator" style="color:#393A34">&lt;&lt;</span><span class="token plain">ans</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">size</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token operator" style="color:#393A34">&lt;&lt;</span><span class="token string" style="color:#e3116c">"\n"</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">auto</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">&amp;</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">x</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain">y</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">:</span><span class="token plain">ans </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> cout</span><span class="token operator" style="color:#393A34">&lt;&lt;</span><span class="token plain">x</span><span class="token operator" style="color:#393A34">&lt;&lt;</span><span class="token string" style="color:#e3116c">" "</span><span class="token operator" style="color:#393A34">&lt;&lt;</span><span class="token plain">y</span><span class="token operator" style="color:#393A34">&lt;&lt;</span><span class="token string" style="color:#e3116c">"\n"</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    QWQ</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><br></div></code></pre></div></div></div></div></details>]]></content:encoded>
            <category>题解</category>
            <category>COCI</category>
            <category>二进制</category>
        </item>
        <item>
            <title><![CDATA[Piratski kod]]></title>
            <link>https://your-docusaurus-site.example.com/blog/blog/题解/COCI/2023-2024-05/B-Piratski kod.md</link>
            <guid>https://your-docusaurus-site.example.com/blog/blog/题解/COCI/2023-2024-05/B-Piratski kod.md</guid>
            <pubDate>Tue, 25 Aug 2026 00:00:00 GMT</pubDate>
            <description><![CDATA[DP，组合数学]]></description>
            <content:encoded><![CDATA[<h2>DP，组合数学</h2>
<!-- -->
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="题干-coci-20232024-5-piratski-kod">题干 [COCI 2023/2024 #5] Piratski kod<a href="https://your-docusaurus-site.example.com/blog/blog/%E9%A2%98%E8%A7%A3/COCI/2023-2024-05/B-Piratski%20kod.md#%E9%A2%98%E5%B9%B2-coci-20232024-5-piratski-kod" class="hash-link" aria-label="题干 [COCI 2023/2024 #5] Piratski kod的直接链接" title="题干 [COCI 2023/2024 #5] Piratski kod的直接链接" translate="no">​</a></h2>
<details class="details_lb9f alert alert--info details_b_Ee" data-collapsed="true"><summary>Details</summary><div><div class="collapsibleContent_i85q"><h2>题目背景</h2><p><strong>译自 <a href="https://hsin.hr/coci/archive/2023_2024" target="_blank" rel="noopener noreferrer" class="">COCI 2023/2024 Contest #5</a> T2「<a href="https://hsin.hr/coci/archive/2023_2024/contest5_tasks.pdf" target="_blank" rel="noopener noreferrer" class="">Bitovi</a>」</strong></p><h2>题目描述</h2><p>哪个先出现，鸡还是蛋？作为一个百万富翁活上一百年好，还是贫穷中度过七天好？如何成为国际象棋大师？如何拉起百叶窗？如何通过期末考试？如何训练龙？这些都是一些有趣的问题，我们可以在竞赛结束后再去思考，但现在我们提出一个不那么有趣的计算机科学问题。</p><p>给定两组数集 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.0502em">B</span></span></span></span>，大小均为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span></span></span></span>。每次操作，你可以从集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 中选择一个任意元素，并改变其二进制表示中的一个任意数位。结果数不能是改变前集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 中的元素。</p><p>例如，数字 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">5</span></span></span></span> 的二进制是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mn>0101</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">0101_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em"></span><span class="mord">010</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>。通过一次操作，它可以变成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>13</mn><mo>=</mo><msub><mn>1101</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">13=1101_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">13</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em"></span><span class="mord">110</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>=</mo><msub><mn>0001</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">1=0001_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em"></span><span class="mord">000</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mo>=</mo><msub><mn>0111</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">7=0111_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">7</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em"></span><span class="mord">011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mo>=</mo><msub><mn>0100</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">4=0100_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em"></span><span class="mord">010</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>。</p><p>确定一系列操作，通过这些操作集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 变得和集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.0502em">B</span></span></span></span> 相等。如果两个集合大小相同，并且集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 中没有不属于集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.0502em">B</span></span></span></span> 的元素，则认为两个集合是相等的。</p><p>注意：操作的数量不必是最小的，但必须满足任务的限制。</p><h2>输入格式</h2><p>第一行包含一个整数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span></span></span></span>（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>N</mi><mo>≤</mo><msup><mn>2</mn><mn>15</mn></msup></mrow><annotation encoding="application/x-tex">1 \le N \le 2^{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">15</span></span></span></span></span></span></span></span></span></span></span></span>），集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.0502em">B</span></span></span></span> 的大小。</p><p>第二行包含 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span></span></span></span> 个不同的整数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></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:0.5806em;vertical-align:-0.15em"></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:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span></span></span></span>（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><msub><mi>a</mi><mi>i</mi></msub><mo>&lt;</mo><msup><mn>2</mn><mn>15</mn></msup></mrow><annotation encoding="application/x-tex">0 \le a_i &lt; 2^{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em"></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:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">15</span></span></span></span></span></span></span></span></span></span></span></span>），表示集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 的元素。</p><p>第二行包含 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span></span></span></span> 个不同的整数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mi>i</mi></msub></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:0.8444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span></span></span></span>（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><msub><mi>b</mi><mi>i</mi></msub><mo>&lt;</mo><msup><mn>2</mn><mn>15</mn></msup></mrow><annotation encoding="application/x-tex">0 \le b_i &lt; 2^{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">15</span></span></span></span></span></span></span></span></span></span></span></span>），表示集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.0502em">B</span></span></span></span> 的元素。</p><h2>输出格式</h2><p>第一行输出操作的数目，不超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>19</mn></msup></mrow><annotation encoding="application/x-tex">2^{19}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">19</span></span></span></span></span></span></span></span></span></span></span></span>。</p><p>接下来若干行，每行输出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x,y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span></span></span></span>（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>&lt;</mo><msup><mn>2</mn><mn>15</mn></msup></mrow><annotation encoding="application/x-tex">0 \le x, y &lt; 2^{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">15</span></span></span></span></span></span></span></span></span></span></span></span>），表示将集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 中的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span> 修改为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span></span></span></span>。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span></span></span></span> 在二进制位中只能有一位不同。并且，必须满足 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x\in A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>∉</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">y\not \in A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">y</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace nobreak"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span>。</p><h2>输入输出样例 #1</h2><h3>输入 #1</h3><div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">3</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">0 1 2</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">1 2 3</span><br></div></code></pre></div></div><h3>输出 #1</h3><div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">2</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">1 3</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">0 1</span><br></div></code></pre></div></div><h2>输入输出样例 #2</h2><h3>输入 #2</h3><div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">3</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">4 8 31</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">0 4 8</span><br></div></code></pre></div></div><h3>输出 #2</h3><div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">5</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">31 30</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">30 28</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">28 24</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">24 16</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">16 0</span><br></div></code></pre></div></div><h2>输入输出样例 #3</h2><h3>输入 #3</h3><div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">5</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">0 1 2 4 5</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">7 6 5 3 2</span><br></div></code></pre></div></div><h3>输出 #3</h3><div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">9</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">1 3</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">3 7</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">0 1</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">1 0</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">2 6</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">0 2</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">7 3</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">5 7</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">4 5</span><br></div></code></pre></div></div><h2>说明/提示</h2><h3>样例解释 1</h3><p>如果我们先操作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mtext>&nbsp;</mtext><mn>1</mn></mrow><annotation encoding="application/x-tex">0\ 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span><span class="mspace">&nbsp;</span><span class="mord">1</span></span></span></span>，再操作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mtext>&nbsp;</mtext><mn>3</mn></mrow><annotation encoding="application/x-tex">1\ 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span><span class="mspace">&nbsp;</span><span class="mord">3</span></span></span></span>，两次操作间，集合 <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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 将有两个相同的元素，这是不被允许的。另一种可能的方案是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mtext>&nbsp;</mtext><mn>3</mn></mrow><annotation encoding="application/x-tex">2\ 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span><span class="mspace">&nbsp;</span><span class="mord">3</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mtext>&nbsp;</mtext><mn>2</mn></mrow><annotation encoding="application/x-tex">0\ 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span><span class="mspace">&nbsp;</span><span class="mord">2</span></span></span></span>。</p><h3>样例解释 2</h3><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>31</mn><mo>=</mo><msub><mn>11111</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">31=11111_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">31</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em"></span><span class="mord">1111</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>。按照从最低有效位到最高有效位修改，依次可以得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>30</mn><mo separator="true">,</mo><mn>28</mn><mo separator="true">,</mo><mn>24</mn><mo separator="true">,</mo><mn>16</mn></mrow><annotation encoding="application/x-tex">30,28,24,16</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em"></span><span class="mord">30</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">28</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">24</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">16</span></span></span></span> 和 <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:0.6444em"></span><span class="mord">0</span></span></span></span>。在所有的操作后，<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:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.0502em">B</span></span></span></span> 相同。</p><h3>子任务</h3><table><thead><tr><th style="text-align:center">Subtask</th><th style="text-align:center">Points</th><th style="text-align:center">Constraints</th></tr></thead><tbody><tr><td style="text-align:center">1</td><td style="text-align:center">10</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>b</mi><mi>i</mi></msub><mo>≤</mo><msup><mn>2</mn><mn>14</mn></msup></mrow><annotation encoding="application/x-tex">a_i,b_i \le 2^{14}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></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:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">14</span></span></span></span></span></span></span></span></span></span></span></span></td></tr><tr><td style="text-align:center">2</td><td style="text-align:center">15</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>≤</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">N \le 7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">7</span></span></span></span></td></tr><tr><td style="text-align:center">3</td><td style="text-align:center">30</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>≤</mo><msup><mn>2</mn><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">N \le 2^7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em"></span><span class="mord mathnormal" style="margin-right:0.109em">N</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></td></tr><tr><td style="text-align:center">4</td><td style="text-align:center">15</td><td style="text-align:center">无额外限制</td></tr></tbody></table></div></div></details>
<hr>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="分析">分析<a href="https://your-docusaurus-site.example.com/blog/blog/%E9%A2%98%E8%A7%A3/COCI/2023-2024-05/B-Piratski%20kod.md#%E5%88%86%E6%9E%90" class="hash-link" aria-label="分析的直接链接" title="分析的直接链接" translate="no">​</a></h2>
<h5><p>便于描述，以下将「没有两个连续的<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:0.6444em"></span><span class="mord">1</span></span></span></span>的序列」称为「散块」，将「末尾的两个字符都是<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:0.6444em"></span><span class="mord">1</span></span></span></span>的序列」称为「整块」</p><p>不难发现一个价值不为<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:0.6444em"></span><span class="mord">0</span></span></span></span>的二进制序列其实就是一个整块加上一个散块，所以问题转化为如何求散块和整块的价值</p><p>我们设计<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>i</mi><mo separator="true">,</mo><mn>0</mn><mi mathvariant="normal">/</mi><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">f_{i,0/1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0496em;vertical-align:-0.3552em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.5198em;margin-left:-0.1076em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">0/1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em"><span></span></span></span></span></span></span></span></span></span>的含义为：长度为<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:0.6595em"></span><span class="mord mathnormal">i</span></span></span></span>的散块，结尾为<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi mathvariant="normal">/</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">0/1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">0/1</span></span></span></span>的价值总和</p><p>考虑怎么转移</p><p>显然转移应该从<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mi mathvariant="normal">/</mi><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">f_{i−1,0/1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0496em;vertical-align:-0.3552em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.5198em;margin-left:-0.1076em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">0/1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em"><span></span></span></span></span></span></span></span></span></span>增加<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi mathvariant="normal">/</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">0/1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">0/1</span></span></span></span>得到<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>i</mi><mo separator="true">,</mo><mn>0</mn><mi mathvariant="normal">/</mi><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">f_{i,0/1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0496em;vertical-align:-0.3552em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.5198em;margin-left:-0.1076em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">0/1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em"><span></span></span></span></span></span></span></span></span></span>​</p><p>增加一个<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:0.6444em"></span><span class="mord">0</span></span></span></span>好办，因为不会增加价值，所以得到转移方程：</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>i</mi><mo separator="true">,</mo><mn>0</mn></mrow></msub><mtext>​</mtext><mo>=</mo><msub><mi>f</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn></mrow></msub><mtext>​</mtext><mo>+</mo><msub><mi>f</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub><mtext>​</mtext></mrow><annotation encoding="application/x-tex">f_{i,0}​=f_{i−1,0}​+f_{i−1,1}​</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct 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:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span></span></span></span>，</p><p>增加一个<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:0.6444em"></span><span class="mord">1</span></span></span></span>会对每一条序列都增加<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mi>i</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">Fib</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal" style="margin-right:0.1389em">F</span><span class="mord mathnormal">ib</span></span></span></span>的价值</p><p>注意到是每一条，所以还要再设计一个数组<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><msub><mi>m</mi><mrow><mi>i</mi><mo separator="true">,</mo><mn>0</mn><mi mathvariant="normal">/</mi><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">num_{i,0/1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7858em;vertical-align:-0.3552em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">0/1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em"><span></span></span></span></span></span></span></span></span></span>​，意思是长度为<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:0.6595em"></span><span class="mord mathnormal">i</span></span></span></span>，结尾为<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi mathvariant="normal">/</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">0/1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">0/1</span></span></span></span>的散块的个数</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span>数组的转移是简单的，注意不能连续放两个<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:0.6444em"></span><span class="mord">1</span></span></span></span>就行：</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><msub><mi>m</mi><mrow><mi>i</mi><mo separator="true">,</mo><mn>0</mn></mrow></msub><mtext>​</mtext><mo>=</mo><mi>n</mi><mi>u</mi><msub><mi>m</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn></mrow></msub><mtext>​</mtext><mo>+</mo><mi>n</mi><mi>u</mi><msub><mi>m</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub><mtext>​，</mtext><mi>n</mi><mi>u</mi><msub><mi>m</mi><mrow><mi>i</mi><mo separator="true">,</mo><mn>1</mn></mrow></msub><mtext>​</mtext><mo>=</mo><mi>n</mi><mi>u</mi><msub><mi>m</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">num_{i,0}​=num_{i−1,0}​+num_{i−1,1}​，num_{i,1}​=num_{i−1,0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8694em;vertical-align:-0.2861em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct 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:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mord cjk_fallback">，</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct 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:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span></span></span></span>​</p><p>回到刚才<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span></span></span></span>数组的转移，现在已经求出了序列的个数，转移方程就出来了：</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>i</mi><mo separator="true">,</mo><mn>1</mn></mrow></msub><mtext>​</mtext><mo>=</mo><msub><mi>f</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn></mrow></msub><mtext>​</mtext><mo>+</mo><mi>n</mi><mi>u</mi><msub><mi>m</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn></mrow></msub><mtext>​</mtext><mo>×</mo><mi>F</mi><mi>i</mi><msub><mi>b</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">f_{i,1}​=f_{i−1,0}​+num_{i−1,0}​\times Fib_{i+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct 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:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.8694em;vertical-align:-0.2861em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em"></span><span class="mord mathnormal" style="margin-right:0.1389em">F</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</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:0.2083em"><span></span></span></span></span></span></span></span></span></span></p><p>现在散块算完了，该算整块了,整块和散块的计算是类似的</p><p>定义<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">g_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span></span></span></span>为长度为<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:0.6595em"></span><span class="mord mathnormal">i</span></span></span></span>的整块的价值总和<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mi>u</mi><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">sum_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span></span></span></span>​为长度为<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:0.6595em"></span><span class="mord mathnormal">i</span></span></span></span>的整块个数</p><p>根据整块的定义，可以发现一个以<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:0.6444em"></span><span class="mord">1</span></span></span></span>结尾的散块在末尾加上不计算价值的<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:0.6444em"></span><span class="mord">1</span></span></span></span>就成为了整块，同时两个整块拼接在一起仍然是整块</p><p>所以为了避免重复计算或者漏算，计算<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">g_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span></span></span></span>时将这个整块拆分成一个前面计算过的整块和一个用散块加<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:0.6444em"></span><span class="mord">1</span></span></span></span>形成的整块。</p><p>状态转移方程可以推理得到：</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mi>i</mi></msub><mtext>​</mtext><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mtext>​</mtext><msub><mi>f</mi><mi>j</mi></msub><mtext>​​</mtext><mo>×</mo><mi>s</mi><mi>u</mi><msub><mi>m</mi><mrow><mi>i</mi><mo>−</mo><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mtext>​</mtext><mo>+</mo><msub><mi>g</mi><mrow><mi>i</mi><mo>−</mo><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mtext>​​</mtext><mo>×</mo><mi>n</mi><mi>u</mi><msub><mi>m</mi><mrow><mi>j</mi><mo separator="true">,</mo><mn>1</mn></mrow></msub><mtext>​</mtext></mrow><annotation encoding="application/x-tex">g_i​=\sum_{j=1}^{i-1}​f_j​​\times sum_{i−j−1}​+g_{i−j−1}​​\times num_{j,1}​</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1.4004em;vertical-align:-0.4358em"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9646em"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em"><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:0.0572em">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</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:0.4358em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">​</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="mord">​​</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.8694em;vertical-align:-0.2861em"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0572em">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:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.8694em;vertical-align:-0.2861em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0572em">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:0.2861em"><span></span></span></span></span></span></span><span class="mord">​​</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.0572em">j</span><span class="mpunct 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:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span></span></span></span></p><p>其中<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:0.854em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0572em">j</span></span></span></span>枚举的是散块的长度，任意一个整块和任意一个散块合并都会产生贡献，所以用到乘法原理</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">sum</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span>数组的转移是类似的：</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mi>u</mi><msub><mi>m</mi><mi>i</mi></msub><mtext>​</mtext><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mtext>​</mtext><mi>s</mi><mi>u</mi><mi>m</mi><mrow><mi>i</mi><mo>−</mo><mi>j</mi><mo>−</mo><mn>1</mn></mrow><mtext>​​</mtext><mo>×</mo><mi>n</mi><mi>u</mi><mi>m</mi><mrow><mi>j</mi><mo separator="true">,</mo><mn>1</mn></mrow><mtext>​</mtext></mrow><annotation encoding="application/x-tex">sum_i​=\sum_{j=1}^{i-1}​sum{i−j−1}​​\times num{j,1}​</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1.4004em;vertical-align:-0.4358em"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9646em"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em"><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:0.0572em">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</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:0.4358em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">​</span><span class="mord mathnormal">s</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal" style="margin-right:0.0572em">j</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">1</span></span><span class="mord">​​</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0572em">j</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">1</span></span><span class="mord">​</span></span></span></span></p><p>好了，现在散块和整块的价值都计算完毕，最后的价值不为<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:0.6444em"></span><span class="mord">0</span></span></span></span>的序列是由一个整块和散块合并而成的，枚举整块的长度即可：</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mi>n</mi><msub><mi>s</mi><mi>i</mi></msub><mtext>​</mtext><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>2</mn></mrow><mi>i</mi></msubsup><mtext>​</mtext><msub><mi>g</mi><mi>j</mi></msub><mtext>​</mtext><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><msub><mi>m</mi><mrow><mi>i</mi><mo>−</mo><mi>j</mi><mo separator="true">,</mo><mn>0</mn></mrow></msub><mtext>​</mtext><mo>+</mo><mi>n</mi><mi>u</mi><msub><mi>m</mi><mrow><mi>i</mi><mo>−</mo><mi>j</mi><mo separator="true">,</mo><mn>1</mn></mrow></msub><mtext>​</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ans_i​=\sum_{j=2}^{i}​g_j​\times (num_{i−j,0}​+num_{i−j,1}​)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em"></span><span class="mord mathnormal">an</span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><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:0.15em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1.4004em;vertical-align:-0.4358em"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9646em"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em"><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:0.0572em">j</span><span class="mrel mtight">=</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord">​</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em"></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0572em">j</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0572em">j</span><span class="mpunct 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:0.2861em"><span></span></span></span></span></span></span><span class="mord">​</span><span class="mclose">)</span></span></span></span></p><p>时间复杂度<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:1.0641em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.0278em">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></h5>
<hr>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="ac代码">AC代码<a href="https://your-docusaurus-site.example.com/blog/blog/%E9%A2%98%E8%A7%A3/COCI/2023-2024-05/B-Piratski%20kod.md#ac%E4%BB%A3%E7%A0%81" class="hash-link" aria-label="AC代码的直接链接" title="AC代码的直接链接" translate="no">​</a></h2>
<details class="details_lb9f alert alert--info details_b_Ee" data-collapsed="true"><summary>Details</summary><div><div class="collapsibleContent_i85q"><div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">include</span><span class="token macro property string" style="color:#e3116c">&lt;bits/stdc++.h&gt;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">using</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">namespace</span><span class="token plain"> std</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">ll</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">long</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">long</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">pii</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression" style="color:#36acaa">pair</span><span class="token macro property expression operator" style="color:#393A34">&lt;</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">int</span><span class="token macro property expression punctuation" style="color:#393A34">,</span><span class="token macro property expression keyword" style="color:#00009f">int</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression operator" style="color:#393A34">&gt;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">QWQ</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">return</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression number" style="color:#36acaa">0</span><span class="token macro property expression punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">QAQ</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">return</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> N</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">5e3</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">10</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> mod</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1e9</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">7</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> fib</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">N</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> g</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">N</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> sum</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">N</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">N</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">2</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> num</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">N</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">2</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">main</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token comment" style="color:#999988;font-style:italic">// freopen( "1.in","r",stdin );</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token comment" style="color:#999988;font-style:italic">// freopen( "1.out","w",stdout );</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    ios</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">sync_with_stdio</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token boolean" style="color:#36acaa">false</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    cin</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">tie</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">nullptr</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> cout</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">tie</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">nullptr</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    cin</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    fib</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> fib</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">2</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> i</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">3</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token plain">n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">++</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> fib</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> fib</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">+</span><span class="token plain">fib</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">2</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token operator" style="color:#393A34">%</span><span class="token plain">mod</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    num</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> i</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token plain">n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">++</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">+</span><span class="token plain">f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token operator" style="color:#393A34">%</span><span class="token plain">mod</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">1ll</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">fib</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">num</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token operator" style="color:#393A34">%</span><span class="token plain">mod</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        num</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> num</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">+</span><span class="token plain">num</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token operator" style="color:#393A34">%</span><span class="token plain">mod</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">		num</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token plain">num</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    sum</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> i</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token plain">n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">++</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> j</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">j</span><span class="token operator" style="color:#393A34">&lt;</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">j</span><span class="token operator" style="color:#393A34">++</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> g</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">+=</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">1ll</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">j</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">sum</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token plain">j</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">%</span><span class="token plain">mod</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">1ll</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">g</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token plain">j</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">num</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">j</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">%</span><span class="token plain">mod </span><span class="token punctuation" style="color:#393A34">)</span><span class="token operator" style="color:#393A34">%</span><span class="token plain">mod </span><span class="token punctuation" style="color:#393A34">)</span><span class="token operator" style="color:#393A34">%=</span><span class="token plain">mod</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> sum</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">+=</span><span class="token number" style="color:#36acaa">1ll</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">sum</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token plain">j</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">num</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">j</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">%</span><span class="token plain">mod </span><span class="token punctuation" style="color:#393A34">)</span><span class="token operator" style="color:#393A34">%=</span><span class="token plain">mod</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> </span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> i</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token plain">n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">++</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">	</span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">		</span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> ans</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">		</span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> j</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">2</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">j</span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">j</span><span class="token operator" style="color:#393A34">++</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> ans</span><span class="token operator" style="color:#393A34">+=</span><span class="token number" style="color:#36acaa">1ll</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">g</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">j</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">*</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> num</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token plain">j</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">+</span><span class="token plain">num</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token plain">j</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token operator" style="color:#393A34">%</span><span class="token plain">mod </span><span class="token punctuation" style="color:#393A34">)</span><span class="token operator" style="color:#393A34">%=</span><span class="token plain">mod</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">		cout</span><span class="token operator" style="color:#393A34">&lt;&lt;</span><span class="token plain">ans</span><span class="token operator" style="color:#393A34">&lt;&lt;</span><span class="token string" style="color:#e3116c">" "</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">	</span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    QWQ</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><br></div></code></pre></div></div></div></div></details>]]></content:encoded>
            <category>题解</category>
            <category>COCI</category>
            <category>DP</category>
            <category>组合数学</category>
        </item>
        <item>
            <title><![CDATA[Rolete]]></title>
            <link>https://your-docusaurus-site.example.com/blog/blog/题解/COCI/2023-2024-05/C-Rolete.md</link>
            <guid>https://your-docusaurus-site.example.com/blog/blog/题解/COCI/2023-2024-05/C-Rolete.md</guid>
            <pubDate>Tue, 25 Aug 2026 00:00:00 GMT</pubDate>
            <description><![CDATA[贪心，二分，三分]]></description>
            <content:encoded><![CDATA[<h2>贪心，二分，三分</h2>
<!-- -->
<hr>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="分析">分析<a href="https://your-docusaurus-site.example.com/blog/blog/%E9%A2%98%E8%A7%A3/COCI/2023-2024-05/C-Rolete.md#%E5%88%86%E6%9E%90" class="hash-link" aria-label="分析的直接链接" title="分析的直接链接" translate="no">​</a></h2>
<h5><p>考虑贪心</p><p>首先，当高度极高的时候它的代价一定是<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:0.6444em"></span><span class="mord">0</span></span></span></span>(不用拉就都满足)，我们考虑从高到低计算</p><p>当我们从计算高度<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal">h</span></span></span></span>的答案时，我们直接考虑从<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">h+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span></span></span></span>的最优答案去转移</p><p>因为要少花费用，所以我们每次在上一个基础上将所以更高的<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><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:0.7278em;vertical-align:-0.0833em"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>就可以了</p><p>此时我们发现要么将更高的全部单个降低，要么用一次全局降低，选择费用较低的并记录即可</p><p>以下是一段比较感性的证明</p><p>如果你<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span></span></span></span>的情况用这种方法得到了最优解，并且<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span>的时候用这种方法得不到最优解</p><p>那么你一定是因为调整上面的某一步操作会让之前的代价更大但是此时的代价最小</p><p>显然如果会这样要么是之前用<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:0.6444em"></span><span class="mord">1</span></span></span></span>代价更大但是如果用 1 可以减小本次使用 2 操作的代价</p><p>但是如果你之前用 2 现在用 1 的操作代价是一样的，所以你之前用<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span></span></span></span>是不劣的</p><p>或者是之前用<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span></span></span></span>代价更大但是用<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span></span></span></span>操作可以拉下去一些本来不需要拉的窗帘，让你<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:0.6444em"></span><span class="mord">1</span></span></span></span>操作的代价更小</p><p>但是你之前用<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:0.6444em"></span><span class="mord">1</span></span></span></span>这下用<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span></span></span></span>肯定也是不劣的</p><p>而对于这道题，我们还存在一个二分套三分的代码，它的思路也可以数学严谨的证明这个贪心的正确性</p><p>时间复杂度<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>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.0278em">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em">q</span><span class="mclose">)</span></span></span></span></p></h5>
<hr>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="ac代码">AC代码<a href="https://your-docusaurus-site.example.com/blog/blog/%E9%A2%98%E8%A7%A3/COCI/2023-2024-05/C-Rolete.md#ac%E4%BB%A3%E7%A0%81" class="hash-link" aria-label="AC代码的直接链接" title="AC代码的直接链接" translate="no">​</a></h2>
<details class="details_lb9f alert alert--info details_b_Ee" data-collapsed="true"><summary>Details</summary><div><div class="collapsibleContent_i85q"><div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">include</span><span class="token macro property string" style="color:#e3116c">&lt;bits/stdc++.h&gt;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">using</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">namespace</span><span class="token plain"> std</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">ll</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">long</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">long</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">pii</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression" style="color:#36acaa">pair</span><span class="token macro property expression operator" style="color:#393A34">&lt;</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">int</span><span class="token macro property expression punctuation" style="color:#393A34">,</span><span class="token macro property expression keyword" style="color:#00009f">int</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression operator" style="color:#393A34">&gt;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">QWQ</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">return</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression number" style="color:#36acaa">0</span><span class="token macro property expression punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">QAQ</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">return</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">intt</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression" style="color:#36acaa">__int128_t</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> N</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1e5</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">10</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> m</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1e5</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> pre</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">N</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> bck</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">N</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> all</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">N</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> </span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">intt ans</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">N</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">inline</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">void</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">print</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> intt x </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">	</span><span class="token keyword" style="color:#00009f">if</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> x </span><span class="token operator" style="color:#393A34">&gt;</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">9</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">print</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> x</span><span class="token operator" style="color:#393A34">/</span><span class="token number" style="color:#36acaa">10</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">	</span><span class="token function" style="color:#d73a49">putchar</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> x</span><span class="token operator" style="color:#393A34">%</span><span class="token number" style="color:#36acaa">10</span><span class="token operator" style="color:#393A34">+</span><span class="token char">'0'</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">main</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token comment" style="color:#999988;font-style:italic">// freopen( "1.in","r",stdin );</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token comment" style="color:#999988;font-style:italic">// freopen( "1.out","w",stdout );</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    ios</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">sync_with_stdio</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token boolean" style="color:#36acaa">false</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    cin</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">tie</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">nullptr</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> cout</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">tie</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">nullptr</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> n</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> t</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> s</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> k</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    cin</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">n</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">t</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">s</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">k</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> i</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> h</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token plain">n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">++</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        cin</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">h</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        pre</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">h</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">++</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> bck</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">h</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">++</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> i</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token plain">m</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">++</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain">   pre</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">+=</span><span class="token plain">pre</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> i</span><span class="token operator" style="color:#393A34">=</span><span class="token plain">m</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">&gt;=</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">--</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> bck</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">+=</span><span class="token plain">bck</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> i</span><span class="token operator" style="color:#393A34">=</span><span class="token plain">m</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">&gt;=</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">--</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        intt fee1</span><span class="token operator" style="color:#393A34">=</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> intt </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain">bck</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">1</span><span class="token operator" style="color:#393A34">+</span><span class="token plain">all</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">t</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        intt fee2</span><span class="token operator" style="color:#393A34">=</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> intt </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain">s</span><span class="token operator" style="color:#393A34">+</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> intt </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain">k</span><span class="token operator" style="color:#393A34">*</span><span class="token plain">pre</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">all</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token keyword" style="color:#00009f">if</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> fee1 </span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token plain"> fee2 </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> ans</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token plain">ans</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">+</span><span class="token plain">fee1</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> all</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token plain">all</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token keyword" style="color:#00009f">else</span><span class="token plain">               ans</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token plain">ans</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">+</span><span class="token plain">fee2</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> all</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token plain">all</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> q</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> cin</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">q</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">q</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">q</span><span class="token operator" style="color:#393A34">--</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> cin</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">x</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">print</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> ans</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">x</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">,</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">putchar</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token char">' '</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    QWQ</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><br></div></code></pre></div></div></div></div></details>]]></content:encoded>
            <category>题解</category>
            <category>COCI</category>
            <category>贪心</category>
        </item>
        <item>
            <title><![CDATA[Trokut]]></title>
            <link>https://your-docusaurus-site.example.com/blog/blog/题解/COCI/2023-2024-05/D-Trokut.md</link>
            <guid>https://your-docusaurus-site.example.com/blog/blog/题解/COCI/2023-2024-05/D-Trokut.md</guid>
            <pubDate>Tue, 25 Aug 2026 00:00:00 GMT</pubDate>
            <description><![CDATA[博弈论，Ad-hoc]]></description>
            <content:encoded><![CDATA[<p>博弈论，Ad-hoc</p>
<!-- -->
<p>博弈论经典操作之先找小的特殊情况，再DP推广</p>
<p>首先我们容易想到每次连接两个点其实会把整个问题分成两个更小的子问题</p>
<p>于是我们考虑分讨这两个子问题的状态</p>
<div class="language-text codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token plain">&lt;1&gt; 0 and 0</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">后手无论怎么选都会输，先手必胜</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">&lt;2&gt; 0 and 1</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">后手进入唯一必胜态，先手必败</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">&lt;3&gt; 1 and 0</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">同&lt;2&gt;，先手必败</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">&lt;4&gt;1 and 1</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">后手选择必胜态进入，含义是在该集合中一定存在使先手最后要连接已涂色点</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">但是先手可以选择另一个必胜态，相当与将一个必胜态转移为 0 后选择另一个 1</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">同&lt;2&gt;&lt;3&gt;，先手必胜</span><br></div></code></pre></div></div>
<p>即异或和决定先后手必胜</p>
<p>然后我们思考一下边界，当划分后没有点了或者只有一个点了此时先手只能连接两个选择过的点，必败</p>
<p>即f[0]=f[1]=0</p>
<p>然后我们写出这个<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:1.0641em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.0278em">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>代码就可以过前面<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>1000</mn></mrow><annotation encoding="application/x-tex">n\leq 1000</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1000</span></span></span></span>的点了</p>
<p>然后，最阴间的人类智慧观察环节来了</p>
<p>打表后“我们瞪眼可以发现”从第69位开始后面每34个为一个循环，打表即可</p>
<p>时间复杂度<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>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.0278em">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.1389em">T</span><span class="mclose">)</span></span></span></span></p>
<div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><div class="token-line" style="color:#393A34"><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">include</span><span class="token macro property string" style="color:#e3116c">&lt;bits/stdc++.h&gt;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">using</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">namespace</span><span class="token plain"> std</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">ll</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">long</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">long</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">pii</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression" style="color:#36acaa">pair</span><span class="token macro property expression operator" style="color:#393A34">&lt;</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">int</span><span class="token macro property expression punctuation" style="color:#393A34">,</span><span class="token macro property expression keyword" style="color:#00009f">int</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression operator" style="color:#393A34">&gt;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">QWQ</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">return</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token macro property expression number" style="color:#36acaa">0</span><span class="token macro property expression punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token macro property directive-hash" style="color:#36acaa">#</span><span class="token macro property directive keyword" style="color:#00009f">define</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property macro-name" style="color:#36acaa">QAQ</span><span class="token macro property" style="color:#36acaa"> </span><span class="token macro property expression keyword" style="color:#00009f">return</span><span class="token macro property expression" style="color:#36acaa"> </span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> N</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">5e3</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">10</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">N</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">inline</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">void</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">init</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> i</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">2</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token number" style="color:#36acaa">200</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">++</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        map</span><span class="token operator" style="color:#393A34">&lt;</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token punctuation" style="color:#393A34">,</span><span class="token keyword" style="color:#00009f">bool</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">&gt;</span><span class="token plain"> vis</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> j</span><span class="token operator" style="color:#393A34">=</span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">;</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> j</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">2</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">j</span><span class="token operator" style="color:#393A34">++</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> vis</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">j</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">^</span><span class="token plain">f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token operator" style="color:#393A34">-</span><span class="token plain">j</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">2</span><span class="token punctuation" style="color:#393A34">]</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">=</span><span class="token boolean" style="color:#36acaa">true</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">        </span><span class="token keyword" style="color:#00009f">while</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> vis</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">find</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">!=</span><span class="token plain"> vis</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">end</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">i</span><span class="token punctuation" style="color:#393A34">]</span><span class="token operator" style="color:#393A34">++</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">inline</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">void</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">sovel</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    cin</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">n</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">if</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> n </span><span class="token operator" style="color:#393A34">&lt;=</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">69</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> cout</span><span class="token operator" style="color:#393A34">&lt;&lt;</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token plain">n</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">?</span><span class="token plain"> </span><span class="token string" style="color:#e3116c">"Lucija\n"</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">:</span><span class="token plain"> </span><span class="token string" style="color:#e3116c">"Ivan\n"</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">else</span><span class="token plain">          cout</span><span class="token operator" style="color:#393A34">&lt;&lt;</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> f</span><span class="token punctuation" style="color:#393A34">[</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> n</span><span class="token operator" style="color:#393A34">-</span><span class="token number" style="color:#36acaa">70</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token operator" style="color:#393A34">%</span><span class="token number" style="color:#36acaa">34</span><span class="token operator" style="color:#393A34">+</span><span class="token number" style="color:#36acaa">70</span><span class="token punctuation" style="color:#393A34">]</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">?</span><span class="token plain"> </span><span class="token string" style="color:#e3116c">"Lucija\n"</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">:</span><span class="token plain"> </span><span class="token string" style="color:#e3116c">"Ivan\n"</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">main</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token comment" style="color:#999988;font-style:italic">// freopen( "1.in","r",stdin );</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token comment" style="color:#999988;font-style:italic">// freopen( "1.out","w",stdout );</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    ios</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">sync_with_stdio</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token boolean" style="color:#36acaa">false</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    cin</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">tie</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">nullptr</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"> cout</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">tie</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">nullptr</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">	</span><span class="token function" style="color:#d73a49">init</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> t</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    </span><span class="token keyword" style="color:#00009f">for</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain"> cin</span><span class="token operator" style="color:#393A34">&gt;&gt;</span><span class="token plain">t</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">t</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain">t</span><span class="token operator" style="color:#393A34">--</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">sovel</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:#393A34"><span class="token plain">    QWQ</span><br></div><div class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><br></div></code></pre></div></div>]]></content:encoded>
            <category>题解</category>
            <category>COCI</category>
            <category>博弈论</category>
        </item>
    </channel>
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