[權限題][https://www.lydsy.com/JudgeOnline/status.php?problem_id=4712&jresult=4]

這道動態\(dp\)終於不是獨立集/覆蓋集了

我們來看看暴力轉移

設\(dp_x\)表示把\(x\)子樹內部全部切斷的最小花費

顯然有
\[ dp_x=min(a_x,\sum_{(x,v)\in e}dp_v ) \]
於是我們可以考慮把所有輕兒子的\(dp_v\)存下來

之後就能搞了

#include<algorithm>
#include<iostream>
#include<cstring>
#include<cstdio>
#define max(a,b) ((a)>(b)?(a):(b))
#define min(a,b) ((a)<(b)?(a):(b))
#define LL long long
#define re register
#define maxn 200005
#define inf 99999999999999
inline int read() {
    char c=getchar();int x=0;while(c<'0'||c>'9') c=getchar();
    while(c>='0'&&c<='9') x=(x<<3)+(x<<1)+c-48,c=getchar();return x;
}
int n,m,num,__;
struct E{int v,nxt;}e[maxn<<1];
struct mat{LL a[2][2];}d[maxn<<2];
int l[maxn<<2],r[maxn<<2];
int deep[maxn],fa[maxn],pos[maxn],sum[maxn],son[maxn];LL dp[maxn];
int id[maxn],dfn[maxn],top[maxn],bot[maxn],head[maxn],a[maxn];
inline void C(int x,int y) {e[++num].v=y;e[num].nxt=head[x];head[x]=num;}
void dfs1(int x) {
    sum[x]=1;int maxx=-1;
    for(re int i=head[x];i;i=e[i].nxt) {
        if(deep[e[i].v]) continue;
        deep[e[i].v]=deep[x]+1;fa[e[i].v]=x;
        dfs1(e[i].v);sum[x]+=sum[e[i].v];
        if(sum[e[i].v]>maxx) maxx=sum[e[i].v],son[x]=e[i].v;
    }
}
int dfs2(int x,int topf) {
    top[x]=topf;dfn[x]=++__,id[__]=x;
    if(!son[x]) return bot[x]=x;
    bot[x]=dfs2(son[x],topf);
    for(re int i=head[x];i;i=e[i].nxt) {
        if(top[e[i].v]) continue;
        dfs2(e[i].v,e[i].v);
    }
    return bot[x];
}
inline mat operator*(mat a,mat b) {
    mat c;
    c.a[0][0]=min(a.a[0][0]+b.a[0][0],a.a[0][1]+b.a[1][0]);
    c.a[0][1]=min(a.a[0][1]+b.a[1][1],a.a[0][0]+b.a[0][1]);
    c.a[1][0]=min(a.a[1][0]+b.a[0][0],a.a[1][1]+b.a[1][0]);
    c.a[1][1]=min(a.a[1][1]+b.a[1][1],a.a[1][0]+b.a[0][1]);
    return c;
}
inline void pushup(int i) {d[i]=d[i<<1]*d[i<<1|1];}
inline void change(int i,int val) {d[i].a[0][1]=val;while(i) {i>>=1;pushup(i);}}
inline void push(int i,mat val) {d[i]=val;while(i) {i>>=1;pushup(i);}}
void build(int x,int y,int i) {
    l[i]=x,r[i]=y;
    if(x==y) {
        int now=id[x];
        for(re int j=head[now];j;j=e[j].nxt) {
            if(deep[e[j].v]<deep[now]||son[now]==e[j].v) continue;
            dp[now]+=dp[e[j].v];
        }
        if(son[now]) d[i].a[0][0]=dp[now];else d[i].a[0][0]=inf;
        d[i].a[0][1]=a[now];
        d[i].a[1][0]=inf,d[i].a[1][1]=0;
        if(son[now]) dp[now]+=dp[son[now]];
        if(!dp[now]||dp[now]>a[now]) dp[now]=a[now];pos[now]=i;
        return;
    }
    int mid=x+y>>1;
    build(mid+1,y,i<<1|1),build(x,mid,i<<1);
    pushup(i);
}
mat query(int x,int y,int i) {
    if(x<=l[i]&&y>=r[i]) return d[i];
    int mid=l[i]+r[i]>>1;
    if(y<=mid) return query(x,y,i<<1);
    if(x>mid) return query(x,y,i<<1|1);
    return query(x,y,i<<1)*query(x,y,i<<1|1);
}
int main() {
    n=read();
    for(re int i=1;i<=n;i++) a[i]=read();
    for(re int x,y,i=1;i<n;i++) x=read(),y=read(),C(x,y),C(y,x);
    deep[1]=1,dfs1(1),dfs2(1,1),build(1,n,1);
    m=read();int x,y;char opt[2];
    while(m--) {
        scanf("%s",opt);x=read();
        if(opt[0]=='Q') {
            mat now=query(dfn[x],dfn[bot[x]],1);
            printf("%lld\n",min(now.a[0][0],now.a[0][1]));
        }
        if(opt[0]=='C') {
            y=read();a[x]+=y;
            mat pre=query(dfn[top[x]],dfn[bot[x]],1);
            change(pos[x],a[x]);
            while(1) {
                if(top[x]==1) break;
                mat now=query(dfn[top[x]],dfn[bot[x]],1);
                mat t=d[pos[fa[top[x]]]];
                t.a[0][0]-=min(pre.a[0][0],pre.a[0][1]);
                t.a[0][0]+=min(now.a[0][0],now.a[0][1]);
                x=fa[top[x]];
                pre=query(dfn[top[x]],dfn[bot[x]],1);push(pos[x],t);
            }
        }
    }
    return 0;
}
 

bzoj 4712: 洪水