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<span>1 分钟</span>
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<p><strong>待施工ing</strong></p>
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<time title="创建时间：2024-03-19 00:00:00 / 修改时间：22:05:37" itemprop="dateCreated datePublished" datetime="2024-03-19T00:00:00+08:00">2024-03-19</time>
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<h2 id="i.-起因">I. 起因</h2>
<p>之前打比赛多次担心写单哈希被卡，但 N
神之前说他高中打过的所有比赛，但凡是字符串的题写 unsigned long long
自然溢出从来没被卡过。我信以为真，认为现在已经没有人素质差到卡别人单模哈希。后来我打了一些
AtCoder 的比赛也写了很多 ull 自然溢出也没事。直到 <a
target="_blank" rel="noopener" href="https://codeforces.com/contest/1943">这场CF</a> N 神 B 题 ull
自然溢出被卡到裂开，以为思路错了，直到赛后才发现是单哈希被卡了。</p>
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<h2 id="brief">Brief</h2>
<p>NEUQ 寒假集训自主刷题记录，比较 trivival
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<h2 id="解题思路">解题思路</h2>
<p>先考虑几个<strong>必要条件</strong>：</p>
<ul>
<li><span class="math inline">\(x\)</span> 结点是 <code>(</code> 且
<span class="math inline">\(y\)</span> 结点是 <code>)</code>。</li>
<li><span class="math inline">\(x\)</span> 到 <span
class="math inline">\(y\)</span> 至少有一条路径经过偶数个结点。</li>
</ul>
<p>在此之外，我们注意到如果路径的开头存在连续两个
<code>(</code>，那么可以在上面来回走来刷左括号；如果路径的结尾存在两个
<code>)</code>，那么可以在上面来回走来刷右括号。</p>
<p>那么一个初步的想法就是：</p>
<ul>
<li>如果 <span class="math inline">\(x\)</span>
可以直接沿着左右括号交替的路径走到 <span
class="math inline">\(y\)</span>，那就做完了。</li>
<li>否则，从 <span class="math inline">\(x\)</span>
沿着交替的左右括号一直走，走到某一个
<code>((</code>，然后开始刷足够多的左括号，然后原路返回 <span
class="math inline">\(x\)</span>，再从 <span
class="math inline">\(x\)</span> 沿着某条经过偶数个结点的路径走到 <span
class="math inline">\(y\)</span>，从 <span
class="math inline">\(y\)</span> 沿着交替的左右括号一直走，走到某一个
<code>))</code>，然后开始刷右括号，刷到与剩下的左括号平衡，然后返回
<span class="math inline">\(y\)</span> 即可。</li>
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<h2 id="题目链接">题目链接</h2>
<p><a
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<h2 id="题目大意">题目大意</h2>
<p>总共有 <span class="math inline">\(N\)</span> 种不同颜色的袜子，第
<span class="math inline">\(i\)</span> 种颜色的袜子有 <span
class="math inline">\(A_i\)</span>
只。现在高桥要按照以下规则决定早上穿哪一双袜子：</p>
<ul>
<li><p>从当前剩余的袜子中等概率随机且不放回地抽取一只袜子。</p></li>
<li><p>若抽出的袜子颜色与之前某次抽出的袜子颜色相同，则停止操作并穿这种颜色的袜子；</p>
<p>若抽出的袜子颜色与之前每一次抽出的颜色均不同，则进行下一次抽取操作。</p></li>
</ul>
<p>求高桥抽取袜子次数的期望值，对 <span
class="math inline">\(998244353\)</span> 取模。</p>
<h2 id="数据范围">数据范围</h2>
<ul>
<li><span class="math inline">\(1 \le N \le 3\times 10^{5}\)</span></li>
<li><span class="math inline">\(2 \le A_i \le 3000\)</span></li>
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<h2 id="题目链接">题目链接</h2>
<p><a
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<h2 id="题目大意">题目大意</h2>
<p>给定一个 <span class="math inline">\(N \times N\)</span> 的方阵 <span
class="math inline">\(A_{i, j}\)</span> 和一个正整数 <span
class="math inline">\(d\)</span>。每次可以花费 <span
class="math inline">\(x\)</span> 的代价将矩阵的某一个位置加上 <span
class="math inline">\(x\)</span> 或减去 <span
class="math inline">\(x\)</span>。请构造一种方案使得方阵中所有相邻位置的差的绝对值大于等于
<span class="math inline">\(d\)</span>，且代价之和小于等于 <span
class="math inline">\(\frac{1}{2}N^{2}d\)</span>。</p>
<h2 id="数据范围">数据范围</h2>
<ul>
<li><span class="math inline">\(2\le N \le 500\)</span></li>
<li><span class="math inline">\(1\le d \le 1000\)</span></li>
<li><span class="math inline">\(-1000 \le A_{i, j} \le
1000\)</span></li>
</ul>
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<h2 id="题目链接">题目链接</h2>
<p><a
target="_blank" rel="noopener" href="https://codeforces.com/contest/1942/problem/E">https://codeforces.com/contest/1942/problem/E</a></p>
<h2 id="题目大意">题目大意</h2>
<p>小 N 和小 J 各有 <span class="math inline">\(n\)</span> 头奶牛，小 N
的奶牛位置为 <span class="math inline">\(a_1, a_2, \cdots,
a_n\)</span>，小 J 的奶牛位置为 <span class="math inline">\(b_1, b_2,
\cdots, b_n\)</span>。其中小 N 和小 J 的奶牛交替出现，且位置在 <span
class="math inline">\(1\)</span> 与 <span
class="math inline">\(l\)</span> 之间，即满足 <span
class="math inline">\(0 < a_1 < b_1 < a_2 < b_2 < \cdots
< a_n < b_n < l + 1\)</span> 或 <span class="math inline">\(0
< b_1 < a_1 < b_2 < a_2 < \cdots < b_n < a_n < l
+ 1\)</span>。</p>
<p>现在小 J 和小 N 轮流进行以下操作，小 J 先手：</p>
<ul>
<li>当前操作的人任意选中自己的 <span class="math inline">\(k(1 \le k \le
n)\)</span>
头奶牛，并且让这些奶牛一起向左或向右移动<strong>一格</strong>。移动后，位置不能与另外一个人的奶牛重合，也不能超出边界（即位置不能小于
<span class="math inline">\(1\)</span> 也不能大于 <span
class="math inline">\(n\)</span>）。</li>
<li>若无法进行上述操作，当前操作的人判负。</li>
</ul>
<p>问有多少种合法的序列组合 <span class="math inline">\((a,
b)\)</span>，使得小 J 必胜（两人均足够聪明）。</p>
<h2 id="数据范围">数据范围</h2>
<p>多组测试数据，保证</p>
<ul>
<li><span class="math inline">\(2 \le l \le 10^6 , \sum l \le
10^{6}\)</span></li>
<li><span class="math inline">\(1 \le n \le
\left\lfloor\frac{l}{2}\right\rfloor\)</span></li>
</ul>
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<h2 id="题目链接">题目链接</h2>
<p><a