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「BZOJ 3638/3272」「Codeforces 172」k – Maximum Subsequence Sum/Zgg吃东西 - 线段树增广费用流
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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>    给定数列${a_n}$，要求维护以下操作和询问： </p>
<ul>
<li>将$a_i$赋值为$val$ </li>
<li>在区间$[l,r]$中选出最多$k$个互不相交的子段列，最大化这些选中的数的和，输出这个最大值<br>    操作和询问共$m$个。</li>
</ul>
<p>    $1\le n\le 10^5,1\le m\le 10^5,\left|a_i\right|\le500,\left|val\right|\le500,1\le k\le20$</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>首先，可以根据题目设置出这样的网络流图：<br><img src="/bzoj3272-eat/1.svg" alt> </p>
<p>但是若使用费用流，每次都需要重新构图跑，显然时间复杂度很劣。<br>发现模型相当简单，实际上我们可以用线段树来模拟增广。<br>在网络流中，每条增广路的流量一定为$1$，增广后边反向且费用取反。<br>相当于线段树选出一段最大的区间，并把这段区间取反。 </p>
<p>故用线段树模拟即可。 </p>
<hr>
<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">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br><span class="line">102</span><br><span class="line">103</span><br><span class="line">104</span><br><span class="line">105</span><br><span class="line">106</span><br><span class="line">107</span><br><span class="line">108</span><br><span class="line">109</span><br><span class="line">110</span><br><span class="line">111</span><br><span class="line">112</span><br><span class="line">113</span><br><span class="line">114</span><br><span class="line">115</span><br><span class="line">116</span><br><span class="line">117</span><br><span class="line">118</span><br><span class="line">119</span><br><span class="line">120</span><br><span class="line">121</span><br><span class="line">122</span><br><span class="line">123</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><bits/stdc++.h></span></span></span><br><span class="line"></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="function"><span class="keyword">inline</span> <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> num=<span class="number">0</span>,bj=<span class="number">1</span>;</span><br><span class="line"> <span class="keyword">char</span> x=getchar();</span><br><span class="line"> <span class="keyword">while</span>(!<span class="built_in">isdigit</span>(x)) {<span class="keyword">if</span>(x==<span class="string">'-'</span>)bj=<span class="number">-1</span>;x=getchar();}</span><br><span class="line"> <span class="keyword">while</span>(<span class="built_in">isdigit</span>(x)) {num=num*<span class="number">10</span>+x-<span class="string">'0'</span>;x=getchar();}</span><br><span class="line"> <span class="keyword">return</span> num*bj;</span><br><span class="line">}</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>;</span><br><span class="line"></span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">Tag</span> {</span></span><br><span class="line"> <span class="keyword">int</span> lmax,rmax,max,sum;</span><br><span class="line"> <span class="keyword">int</span> lp,rp,p1,p2;</span><br><span class="line"> Tag(<span class="keyword">int</span> p=<span class="number">0</span>,<span class="keyword">int</span> v=<span class="number">0</span>) {</span><br><span class="line"> lp=rp=p1=p2=p;</span><br><span class="line"> lmax=rmax=max=sum=v;</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">bool</span> <span class="title">check</span><span class="params">()</span> <span class="keyword">const</span> </span>{<span class="keyword">return</span> lmax==<span class="number">0</span>&&rmax==<span class="number">0</span>&&max==<span class="number">0</span>&&sum==<span class="number">0</span>&&lp==<span class="number">0</span>&&rp==<span class="number">0</span>&&p1==<span class="number">0</span>&&p2==<span class="number">0</span>;}</span><br><span class="line"> Tag <span class="keyword">operator</span> + (<span class="keyword">const</span> Tag &b) {</span><br><span class="line"> <span class="keyword">if</span>(check())<span class="keyword">return</span> b;</span><br><span class="line"> <span class="keyword">if</span>(b.check())<span class="keyword">return</span> *<span class="keyword">this</span>;</span><br><span class="line"> Tag ans;</span><br><span class="line"> ans.sum=sum+b.sum;</span><br><span class="line"> ans.lmax=lmax,ans.lp=lp;</span><br><span class="line"> <span class="keyword">if</span>(sum+b.lmax>ans.lmax)ans.lmax=sum+b.lmax,ans.lp=b.lp;</span><br><span class="line"> ans.rmax=b.rmax,ans.rp=b.rp;</span><br><span class="line"> <span class="keyword">if</span>(b.sum+rmax>ans.rmax)ans.rmax=b.sum+rmax,ans.rp=rp;</span><br><span class="line"> ans.max=rmax+b.lmax,ans.p1=rp,ans.p2=b.lp;</span><br><span class="line"> <span class="keyword">if</span>(ans.max<max)ans.max=max,ans.p1=p1,ans.p2=p2;</span><br><span class="line"> <span class="keyword">if</span>(ans.max<b.max)ans.max=b.max,ans.p1=b.p1,ans.p2=b.p2;</span><br><span class="line"> <span class="keyword">return</span> ans;</span><br><span class="line"> }</span><br><span class="line">};</span><br><span class="line"></span><br><span class="line"><span class="keyword">int</span> n,q,a[maxn];</span><br><span class="line"></span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">Segment_Tree</span> {</span></span><br><span class="line"> <span class="class"><span class="keyword">struct</span> <span class="title">Tree</span> {</span></span><br><span class="line"> <span class="keyword">int</span> left,right;</span><br><span class="line"> <span class="keyword">bool</span> rev;</span><br><span class="line"> Tag max,min;</span><br><span class="line"> Tree(<span class="keyword">int</span> l=<span class="number">0</span>,<span class="keyword">int</span> r=<span class="number">0</span>,<span class="keyword">int</span> v=<span class="number">0</span>):left(l),right(r),max(Tag(l,v)),min(Tag(l,-v)) {}</span><br><span class="line"> } tree[maxn<<<span class="number">2</span>];</span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> ls index<<1</span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> rs index<<1|1</span></span><br><span class="line"> <span class="function"><span class="keyword">void</span> <span class="title">build</span><span class="params">(<span class="keyword">int</span> index,<span class="keyword">int</span> Left,<span class="keyword">int</span> Right)</span> </span>{</span><br><span class="line"> tree[index]=Tree(Left,Right);</span><br><span class="line"> <span class="keyword">if</span>(Left==Right) {tree[index]=Tree(Left,Right,a[Left]);<span class="keyword">return</span>;}</span><br><span class="line"> <span class="keyword">int</span> mid=(Left+Right)>><span class="number">1</span>;</span><br><span class="line"> build(ls,Left,mid);</span><br><span class="line"> build(rs,mid+<span class="number">1</span>,Right);</span><br><span class="line"> push_up(index);</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">void</span> <span class="title">push_up</span><span class="params">(<span class="keyword">int</span> index)</span> </span>{</span><br><span class="line"> tree[index].max=tree[ls].max+tree[rs].max;</span><br><span class="line"> tree[index].min=tree[ls].min+tree[rs].min;</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">void</span> <span class="title">reverse</span><span class="params">(<span class="keyword">int</span> index)</span> </span>{</span><br><span class="line"> swap(tree[index].min,tree[index].max);</span><br><span class="line"> tree[index].rev^=<span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">void</span> <span class="title">push_down</span><span class="params">(<span class="keyword">int</span> index)</span> </span>{</span><br><span class="line"> <span class="keyword">if</span>(tree[index].left==tree[index].right||!tree[index].rev)<span class="keyword">return</span>;</span><br><span class="line"> reverse(ls);</span><br><span class="line"> reverse(rs);</span><br><span class="line"> tree[index].rev=<span class="number">0</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">void</span> <span class="title">reverse</span><span class="params">(<span class="keyword">int</span> index,<span class="keyword">int</span> Left,<span class="keyword">int</span> Right)</span> </span>{</span><br><span class="line"> <span class="keyword">if</span>(Right<tree[index].left||Left>tree[index].right)<span class="keyword">return</span>;</span><br><span class="line"> <span class="keyword">if</span>(Left<=tree[index].left&&Right>=tree[index].right) {reverse(index);<span class="keyword">return</span>;}</span><br><span class="line"> push_down(index);</span><br><span class="line"> reverse(ls,Left,Right);</span><br><span class="line"> reverse(rs,Left,Right);</span><br><span class="line"> push_up(index);</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">void</span> <span class="title">modify</span><span class="params">(<span class="keyword">int</span> index,<span class="keyword">int</span> tar,<span class="keyword">int</span> val)</span> </span>{</span><br><span class="line"> <span class="keyword">if</span>(tar<tree[index].left||tar>tree[index].right)<span class="keyword">return</span>;</span><br><span class="line"> <span class="keyword">if</span>(tree[index].left==tree[index].right) {tree[index]=Tree(tree[index].left,tree[index].right,val);<span class="keyword">return</span>;}</span><br><span class="line"> push_down(index);</span><br><span class="line"> modify(ls,tar,val);</span><br><span class="line"> modify(rs,tar,val);</span><br><span class="line"> push_up(index);</span><br><span class="line"> }</span><br><span class="line"> <span class="function">Tag <span class="title">query</span><span class="params">(<span class="keyword">int</span> index,<span class="keyword">int</span> Left,<span class="keyword">int</span> Right)</span> </span>{</span><br><span class="line"> <span class="keyword">if</span>(Right<tree[index].left||Left>tree[index].right)<span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line"> <span class="keyword">if</span>(Left<=tree[index].left&&Right>=tree[index].right)<span class="keyword">return</span> tree[index].max;</span><br><span class="line"> push_down(index);</span><br><span class="line"> <span class="keyword">return</span> query(ls,Left,Right)+query(rs,Left,Right);</span><br><span class="line"> }</span><br><span class="line">} st;</span><br><span class="line"></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> pii pair<span class="meta-string"><int,int></span></span></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">Query</span><span class="params">(<span class="keyword">int</span> Left,<span class="keyword">int</span> Right,<span class="keyword">int</span> k)</span> </span>{</span><br><span class="line"> <span class="built_in">vector</span><pii> reved;</span><br><span class="line"> <span class="keyword">int</span> ans=<span class="number">0</span>;</span><br><span class="line"> <span class="keyword">while</span>(k--) {</span><br><span class="line"> Tag tmp=st.query(<span class="number">1</span>,Left,Right);</span><br><span class="line"> <span class="keyword">if</span>(tmp.max><span class="number">0</span>)ans+=tmp.max;</span><br><span class="line"> <span class="keyword">else</span> <span class="keyword">break</span>;</span><br><span class="line"> st.reverse(<span class="number">1</span>,tmp.p1,tmp.p2);</span><br><span class="line"> reved.push_back(pii(tmp.p1,tmp.p2));</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">for</span>(pii x:reved)st.reverse(<span class="number">1</span>,x.first,x.second);</span><br><span class="line"> <span class="keyword">return</span> ans;</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"> <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>; i<=n; i++)a[i]=Get_Int();</span><br><span class="line"> st.build(<span class="number">1</span>,<span class="number">1</span>,n);</span><br><span class="line"> q=Get_Int();</span><br><span class="line"> <span class="keyword">while</span>(q--) {</span><br><span class="line"> <span class="keyword">int</span> opt=Get_Int(),l=Get_Int(),r=Get_Int();</span><br><span class="line"> <span class="keyword">if</span>(opt==<span class="number">1</span>)<span class="built_in">printf</span>(<span class="string">"%d\n"</span>,Query(l,r,Get_Int()));</span><br><span class="line"> <span class="keyword">else</span> st.modify(<span class="number">1</span>,l,r);</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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