UOJ268 [清华集训2016] 数据交互 【动态DP】【堆】【树链剖分】【线段树】

题目分析：

不难发现可以用动态DP做。

题目相当于是要我求一条路径，所有与路径有交的链的代价加入进去，要求代价最大。

我们把链的代价分成两个部分：一部分将代价加入$LCA$之中，用$g$数组保存；另一部分将代价加在整条链上，用$d$数组保存。

这时候我们可以发现，一条从$u$到$v$的路径的代价相当于是$d[LCA(u,v)]+\sum_{x \in edge(u,v)}g[x]$。

如果是静态的，可以用树形DP解决。

看过《神奇的子图》的同学都知道，叶子结点是从它的儿子中取两个最大的出来，所以堆维护。

考虑合并。

链从左延申出的最大的$g$的总和记录。链从右延申包括$d$的总和记录，每次向上$update$的时候拼起来与原答案比较即可。

代码：

 #include<bits/stdc++.h>
 using namespace std;

 typedef long long ll;

 const int maxn = ;

 int n,m;

 vector <int> g[maxn];
 int sz[maxn],top[maxn],fa[maxn],dep[maxn],son[maxn],ind[maxn],dr[maxn];
 int tail[maxn],num;

 struct Query{int from,to,w;}Q[maxn];
 struct Priority_Queue{
     priority_queue<ll,vector<ll>,less<ll> > pq,del;
     void push(ll now){pq.push(now);}
     void pop(){
     while(!del.empty() && pq.top() == del.top()) pq.pop(),del.pop();
     pq.pop();
     }
     ll top(){
     while(!del.empty() && pq.top() == del.top()) pq.pop(),del.pop();
     if(pq.empty()) return ;
     else return pq.top();
     }
     ll sec(){
     while(!del.empty() && pq.top() == del.top()) pq.pop(),del.pop();
     if(pq.size() ==) return ;
     ll oop = pq.top(); pq.pop();
     while(!del.empty() && pq.top() == del.top()) pq.pop(),del.pop();
     if(pq.size() == ){pq.push(oop);return ;}
     else {ll ret = pq.top();pq.push(oop);return ret;}
     }
     void Erase(ll now){del.push(now);}
 }Son[maxn],Ans;

 struct segmentTree{
     ll tg,ff,lazy,REC,L,R;
 }T[maxn<<];

 void push_down(int now){
     T[now<<].ff += T[now].lazy; T[now<<|].ff += T[now].lazy;
     T[now<<].REC += T[now].lazy; T[now<<|].REC += T[now].lazy;
     T[now<<].lazy += T[now].lazy; T[now<<|].lazy += T[now].lazy;
     T[now<<].R += T[now].lazy; T[now<<|].R += T[now].lazy;
     T[now].lazy = ;
 }

 segmentTree merge(segmentTree alpha,segmentTree beta){
     segmentTree RES;RES.lazy = ;  RES.ff = ;
     RES.tg = alpha.tg + beta.tg;
     RES.REC = max(alpha.REC,beta.REC);
     RES.REC = max(RES.REC,alpha.R + beta.L);
     RES.L = max(alpha.L,alpha.tg + beta.L);
     RES.R = max(beta.R,alpha.R + beta.tg);
     return RES;
 }

 void dfs1(int now,int f,int dp){
     dep[now] = dp; fa[now] = f;
     int maxx = ;
     for(auto it:g[now]){
     if(it == f) continue;
     dfs1(it,now,dp+);
     sz[now] += sz[it];
     if(maxx ==  || sz[it] > sz[maxx]) maxx = it;
     }
     son[now] = maxx; sz[now]++;
 }

 void dfs2(int now,int tp){
     top[now] = tp; ind[now] = ++num; dr[num] = now;
     if(now == tp) Ans.push();
     if(son[now]) dfs2(son[now],tp);
     else tail[tp] = now;
     for(auto it : g[now]){
     if(it == fa[now] || it == son[now]) continue;
     dfs2(it,it);
     }
 }

 void read(){
     scanf("%d%d",&n,&m);
     for(int i=;i<n;i++){
     int u,v; scanf("%d%d",&u,&v);
     g[u].push_back(v); g[v].push_back(u);
     }
 }

 int QueryLca(int u,int v){
     while(top[u] != top[v]){
     if(dep[top[u]] > dep[top[v]]) u = fa[top[u]];
     else v = fa[top[v]];
     }
     if(dep[u] < dep[v]) return u; else return v;
 }

 segmentTree Querylen(int now,int tl,int tr,int l,int r){
     if(tl >= l && tr <= r) return T[now];
     if(T[now].lazy) push_down(now);
     int mid = (tl+tr)/;
     if(mid < l) return Querylen(now<<|,mid+,tr,l,r);
     if(mid >= r) return Querylen(now<<,tl,mid,l,r);
     segmentTree pp = Querylen(now<<|,mid+,tr,l,r);
     segmentTree qq = Querylen(now<<,tl,mid,l,r);
     return merge(qq,pp);
 }

 void AddG(int now,int tl,int tr,int place,int w){
     if(tl == tr){T[now].tg += w;T[now].L += w;return;}
     if(T[now].lazy) push_down(now);
     int mid = (tl+tr)/;
     if(mid >= place) AddG(now<<,tl,mid,place,w);
     else AddG(now<<|,mid+,tr,place,w);
     T[now] = merge(T[now<<],T[now<<|]);
 }

 void ModifyG(int now,int tl,int tr,int place){
     if(tl == tr){
     tl = dr[tl]; T[now].L = Son[tl].top() + T[now].tg;
     T[now].REC = T[now].ff + Son[tl].top() + Son[tl].sec();
     T[now].R = T[now].ff + Son[tl].top();
     return;
     }
     if(T[now].lazy) push_down(now);
     int mid = (tl+tr)/;
     if(mid >= place) ModifyG(now<<,tl,mid,place);
     else ModifyG(now<<|,mid+,tr,place);
     T[now] = merge(T[now<<],T[now<<|]);
 }

 void ModifyF(int now,int tl,int tr,int l,int r,int w){
     if(tl >= l && tr <= r){
     T[now].lazy += w; T[now].ff += w; T[now].R += w; T[now].REC+=w;
     return;
     }
     if(T[now].lazy) push_down(now);
     int mid = (tl+tr)/;
     if(mid >= l) ModifyF(now<<,tl,mid,l,r,w);
     if(mid+ <= r) ModifyF(now<<|,mid+,tr,l,r,w);
     T[now] = merge(T[now<<],T[now<<|]);
 }

 void SingleModify(int now,int w){
     int hole = tail[top[now]];
     segmentTree fk = Querylen(,,n,ind[top[hole]],ind[hole]);
     AddG(,,n,ind[now],w); now=fa[top[now]];
     while(now){
     segmentTree rl = Querylen(,,n,ind[top[hole]],ind[hole]);
     Ans.Erase(fk.REC); Ans.push(rl.REC);
     Son[now].Erase(fk.L); Son[now].push(rl.L);
     hole = tail[top[now]]; fk = Querylen(,,n,ind[top[hole]],ind[hole]);
     ModifyG(,,n,ind[now]); now = fa[top[now]];
     }
     Ans.Erase(fk.REC);
     fk = Querylen(,,n,ind[top[hole]],ind[hole]);
     Ans.push(fk.REC);
 }

 void WideModify(int now,int LCA,int w){
     while(dep[now] >= dep[LCA]){
     segmentTree fk = Querylen(,,n,ind[top[now]],ind[tail[top[now]]]);
     if(dep[top[now]] < dep[LCA]) ModifyF(,,n,ind[LCA],ind[now],w);
     else ModifyF(,,n,ind[top[now]],ind[now],w);
     Ans.Erase(fk.REC);
     fk = Querylen(,,n,ind[top[now]],ind[tail[top[now]]]);
     Ans.push(fk.REC);
     now = fa[top[now]];
     }
 }

 void Modify(int u,int v,int w){
     int LCA = QueryLca(u,v);
     SingleModify(LCA,w);
     WideModify(u,LCA,w); // u
     WideModify(v,LCA,w); // v
     WideModify(LCA,LCA,-w); // LCA
 }

 void build_tree(int now,int tl,int tr){
     if(tl == tr){
     tl = dr[tl];
     for(auto it : g[tl]){
         if(it == son[tl] || it == fa[tl]) continue;
         Son[tl].push();
     }
     }else{
     int mid = (tl+tr)/;
     build_tree(now<<,tl,mid); build_tree(now<<|,mid+,tr);
     }
 }

 void work(){
     dfs1(,,);
     dfs2(,);
     build_tree(,,n);
     for(int i=;i<=m;i++){
     char ch = getchar(); while(ch != '+' && ch != '-') ch = getchar();
     int fr,t,w;
     if(ch == '+'){
         scanf("%d%d%d",&fr,&t,&w);Q[i].from=fr;Q[i].to=t;Q[i].w=w;
     }else{
         int x; scanf("%d",&x);fr = Q[x].from,t = Q[x].to,w = -Q[x].w;
     }
     Modify(fr,t,w);
     printf("%lld\n",Ans.top());
     }
 }

 int main(){
     read();
     work();
     return ;
 }