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「JZOJ5511」送你一个DAG - 拓扑排序+自然数幂和/Stirling数
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<h2 id="题目大意"><a href="#题目大意" class="headerlink" title="题目大意"></a>题目大意</h2><blockquote>
<p>    送你一个$n$个点$m$条边的DAG和参数$k$, 定义一条经过$l$条边的路径的权值为$k^l$。对于$i=1\ldots n$, 求出所有$1$到$i$的路径的权值之和, 对$998244353$取模。 </p>
</blockquote>
<a id="more"></a>
<hr>
<h2 id="题目分析"><a href="#题目分析" class="headerlink" title="题目分析"></a>题目分析</h2><p>考虑，若$l=1$，这就是一个拓扑排序的问题。<br>若$l\gt1$，前$k\le30$的数据可以使用二项式定理展开$O(nk^2)$解决。<br>对于$k\le500$就要使用<a href="/bzoj2159-civilization/">这里</a>讲过的一个套路了。 </p>
<blockquote>
<script type="math/tex; mode=display">x^k=\sum_{i=1}^k {k \brace i} P_x^i</script></blockquote>
<p>使用组合数替换下降阶乘幂。 </p>
<script type="math/tex; mode=display">x^k=\sum_{i=1}^k {k \brace i} P_x^i=\sum_{i=1}^k {k \brace i}\binom{x}{i}i!</script><p>因为组合数有Pascal递推公式，则设二维状态$f[i,j]$表示到达$i$结点，经过$j$条边的权值$j$次方和。<br>预处理一下斯特林数与阶乘即可。<br>时间复杂度$O(nk)$。 </p>
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<h2 id="代码"><a href="#代码" class="headerlink" title="代码"></a>代码</h2><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span 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class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><algorithm></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><iostream></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><iomanip></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cstring></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cstdlib></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><climits></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><vector></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cstdio></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cmath></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><queue></span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">namespace</span> FastIO {</span><br><span class="line"> <span class="keyword">const</span> <span class="keyword">int</span> L=<span class="number">1</span><<<span class="number">15</span>;</span><br><span class="line"> <span class="keyword">char</span> buf[L],*S,*T;</span><br><span class="line"> <span class="function"><span class="keyword">char</span> <span class="title">getchar</span><span class="params">()</span> </span>{</span><br><span class="line"> <span class="keyword">if</span>(S==T) {T=(S=buf)+fread(buf,<span class="number">1</span>,L,<span class="built_in">stdin</span>);<span class="keyword">if</span>(S==T)<span class="keyword">return</span> EOF;}</span><br><span class="line"> <span class="keyword">return</span> *S++;</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">int</span> <span class="title">Get_Int</span><span class="params">()</span> </span>{</span><br><span class="line"> <span class="keyword">int</span> res=<span class="number">0</span>,bj=<span class="number">1</span>;<span class="keyword">char</span> c=getchar();</span><br><span class="line"> <span class="keyword">while</span>(!<span class="built_in">isdigit</span>(c)) {<span class="keyword">if</span>(c==<span class="string">'-'</span>)bj=<span class="number">-1</span>;c=getchar();}</span><br><span class="line"> <span class="keyword">while</span>(<span class="built_in">isdigit</span>(c)) {res=res*<span class="number">10</span>+c-<span class="string">'0'</span>;c=getchar();}</span><br><span class="line"> <span class="keyword">return</span> res*bj;</span><br><span class="line"> }</span><br><span class="line">}</span><br><span class="line"><span class="keyword">using</span> FastIO::Get_Int;</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> maxn=<span class="number">100005</span>,maxk=<span class="number">505</span>;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> mod=<span class="number">998244353</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">int</span> n,m,k,InDegree[maxn];</span><br><span class="line"><span class="keyword">int</span> S[maxk][maxk],fac[maxk],f[maxn][maxk];</span><br><span class="line"><span class="built_in">vector</span><<span class="keyword">int</span>> edges[maxn];</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">AddEdge</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> y)</span> </span>{</span><br><span class="line"> edges[x].push_back(y);</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span> </span>{</span><br><span class="line"> n=Get_Int();</span><br><span class="line"> m=Get_Int();</span><br><span class="line"> k=Get_Int();</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>; i<=m; i++) {</span><br><span class="line"> <span class="keyword">int</span> x=Get_Int(),y=Get_Int();</span><br><span class="line"> AddEdge(x,y);</span><br><span class="line"> InDegree[y]++;</span><br><span class="line"> }</span><br><span class="line"> <span class="built_in">queue</span><<span class="keyword">int</span>> Q;</span><br><span class="line"> Q.push(<span class="number">1</span>);</span><br><span class="line"> f[<span class="number">1</span>][<span class="number">0</span>]=<span class="number">1</span>;</span><br><span class="line"> <span class="keyword">while</span>(!Q.empty()) {</span><br><span class="line"> <span class="keyword">int</span> Now=Q.front();</span><br><span class="line"> Q.pop();</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>; i<edges[Now].size(); i++) {</span><br><span class="line"> <span class="keyword">int</span> Next=edges[Now][i];</span><br><span class="line"> f[Next][<span class="number">0</span>]=(f[Next][<span class="number">0</span>]+f[Now][<span class="number">0</span>])%mod;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">1</span>; j<=k; j++)f[Next][j]=(<span class="number">1l</span>l*f[Next][j]+f[Now][j]+f[Now][j<span class="number">-1</span>])%mod;</span><br><span class="line"> InDegree[Next]--;</span><br><span class="line"> <span class="keyword">if</span>(!InDegree[Next])Q.push(Next);</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> S[<span class="number">0</span>][<span class="number">0</span>]=fac[<span class="number">0</span>]=<span class="number">1</span>;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>; i<=k; i++) {</span><br><span class="line"> fac[i]=<span class="number">1l</span>l*fac[i<span class="number">-1</span>]*i%mod;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">1</span>; j<=i; j++)S[i][j]=(<span class="number">1l</span>l*S[i<span class="number">-1</span>][j]*j%mod+S[i<span class="number">-1</span>][j<span class="number">-1</span>])%mod;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>; i<=n; i++) {</span><br><span class="line"> <span class="keyword">int</span> ans=<span class="number">0</span>;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">1</span>; j<=k; j++)ans=(ans+<span class="number">1l</span>l*fac[j]*S[k][j]%mod*f[i][j]%mod)%mod;</span><br><span class="line"> <span class="built_in">printf</span>(<span class="string">"%d\n"</span>,ans);</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>
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