虚树初探

虚树其实没什么的。。

只是因为点太多了不能全开于是只开那些需要用到的点。

一棵虚树包括要求点以及它们的lca。。

虚树的构建。。。(其实感觉如果会虚树的构建的话接下来就是树dp啦没什么的。。。

首先我们应该对整棵树dfs,求出它的dfs序列。然后对于给的点,按dfs排序。。

因为我们是按dfs序排列的,所以虚树一定是由一条条链构成的。。

扫一遍给的点,如果这个点在当前的这条链上,那加在栈顶就可以了。

如果不是的话,那就不断地退栈使的原来的那条链上面的边全部被加到边集中。。

    rep(i,1,n){
        int t=a[i],f=0; 
        while (top){
            f=lca(t,s[top]);
            if (top>1&&dep[f]<dep[s[top-1]]) insert(s[top-1],s[top],dis(s[top-1],s[top])),top--;
            else if (dep[f]<dep[s[top]]) {insert(f,s[top],dis(f,s[top])); top--;break;}
            else break;
        }
        if (f!=s[top]) s[++top]=f;
        s[++top]=t;
    }
    while (top>1) insert(s[top-1],s[top],dis(s[top-1],s[top])),top--;

【bzoj3611】[Heoi2014]大工程

两两都要连一条边嘛,记f[i]=i子树上所有点到i这个点的距离和,sz[i]表示i子树上要求点个数。

那么 sum+=(sz[u]*e[j].c+f[u])*sz[v]+f[v]*sz[u]  f[u]+=e[j].c*sz[v]+f[v] 

#include<cstring>
#include<iostream>
#include<cstdio>
#include<algorithm>
#define rep(i,l,r) for (int i=l;i<=r;i++)
#define down(i,l,r) for (int i=l;i>=r;i--)
#define clr(x,y) memset(x,y,sizeof(x))
#define maxn 1005000
#define ll long long
#define inf 1152921504606846976
using namespace std;
struct data{int obj,pre;ll c;
}e[maxn*2];
int head[maxn],dfn[maxn],dep[maxn],fa[maxn][22],bin[22],a[maxn],b[maxn],bel[maxn],sz[maxn],s[maxn];
ll mx[maxn],mn[maxn],sum[maxn],f[maxn],ans,ans1,ans2;
int n,Q,tot,top,idx;
void insert(int x,int y,ll z){
    e[++tot].obj=y; e[tot].pre=head[x]; head[x]=tot; e[tot].c=z;
}
int read(){
    int x=0,f=1; char ch=getchar();
    while (!isdigit(ch)){if (ch=='-') f=-1; ch=getchar();}
    while (isdigit(ch)){x=x*10+ch-'0'; ch=getchar();}
    return x*f;
}
void dfs(int u){
    dfn[u]=++idx;
    rep(i,1,20) if (dep[u]>=bin[i]) fa[u][i]=fa[fa[u][i-1]][i-1];
    for (int j=head[u];j;j=e[j].pre){
        int v=e[j].obj; 
        if (v!=fa[u][0]) {
            fa[v][0]=u; dep[v]=dep[u]+1;
            dfs(v);
        }
    }
}
int lca(int x,int y){
    if (dep[x]<dep[y]) swap(x,y);
    int t=dep[x]-dep[y];
    rep(i,0,20) if (t&bin[i]) x=fa[x][i];
    down(i,20,0) if (fa[x][i]!=fa[y][i]) x=fa[x][i],y=fa[y][i];
    if (x==y) return x;
    return fa[x][0];
}
ll dis(int x,int y){
    return 1LL*(dep[x]+dep[y]-2*dep[lca(x,y)]);
}
bool cmp(int x,int y){
    return dfn[x]<dfn[y];
}
void dp(int u){
    sz[u]=bel[u]; sum[u]=0; f[u]=0;
    if (bel[u]) mn[u]=0,mx[u]=0;
    else mn[u]=inf,mx[u]=-inf;
    for (int j=head[u];j;j=e[j].pre){
        int v=e[j].obj;
        dp(v);
        ans+=(sz[u]*e[j].c+f[u])*sz[v]+f[v]*sz[u];
        f[u]+=e[j].c*sz[v]+f[v];
        ans1=min(ans1,mn[u]+e[j].c+mn[v]);
        ans2=max(ans2,mx[u]+e[j].c+mx[v]);
        mn[u]=min(mn[u],mn[v]+e[j].c);
        mx[u]=max(mx[u],mx[v]+e[j].c);
        sz[u]+=sz[v];
    }
    head[u]=0;
}
void solve(){
    tot=top=idx=0;
    n=read();
    rep(i,1,n) a[i]=b[i]=read(),bel[a[i]]=1;
    sort(a+1,a+1+n,cmp);
    if (bel[1]!=1) s[++top]=1; 
    rep(i,1,n){
        int t=a[i],f=0; 
        while (top){
            f=lca(t,s[top]);
            if (top>1&&dep[f]<dep[s[top-1]]) insert(s[top-1],s[top],dis(s[top-1],s[top])),top--;
            else if (dep[f]<dep[s[top]]) {insert(f,s[top],dis(f,s[top])); top--;break;}
            else break;
        }
        if (f!=s[top]) s[++top]=f;
        s[++top]=t;
    }
    while (top>1) insert(s[top-1],s[top],dis(s[top-1],s[top])),top--;
    ans=0; ans1=inf; ans2=-inf;
    dp(1);
    printf("%lld %lld %lld\n",ans,ans1,ans2);
    rep(i,1,n) bel[b[i]]=0;
}
int main(){
    bin[0]=1; rep(i,1,20) bin[i]=bin[i-1]*2;
    n=read();
    rep(i,1,n-1){
        int x=read(),y=read(); insert(x,y,0); insert(y,x,0);
    }
    dfs(1);
    clr(head,0);
    Q=read(); 
    while (Q--) solve();
    return 0;
}
View Code

3572: [Hnoi2014]世界树

先把虚树建出来,然后跑两边dfs得到虚树上每个点的belong,对于虚树上每条边a,b,可以通过二分得到一个点x使得a到x-1的都让a管,x到b的都让b管。。

#include<cstring>
#include<iostream>
#include<algorithm>
#include<cstdio>
#include<queue>
#include<set>
#include<cmath>
#include<vector>
#include<map>
#define rep(i,l,r) for (int i=l;i<=r;i++)
#define down(i,l,r) for (int i=l;i>=r;i--)
#define clr(x,y) memset(x,y,sizeof(x))
#define maxn 300500
#define maxm 1005000
#define inf int(1e9)
#define mm 1000000007
#define eps 1e-7
typedef long long ll;
using namespace std;
struct data{int obj,pre;
}e[maxn*2];
int head[maxn],c[maxn],dfn[maxn],sz[maxn],fa[maxn][22],bin[22],rem[maxn],s[maxn],dep[maxn],bel[maxn],f[maxn],a[maxn],b[maxn];
int tot,idx,n,top,Q;
int read(){
    int x=0,f=1; char ch=getchar();
    while (!isdigit(ch)){if (ch=='-') f=-1; ch=getchar();}
    while (isdigit(ch)){x=x*10+ch-'0'; ch=getchar();}
    return x*f;
}   
void insert(int x,int y){
    e[++tot].obj=y; e[tot].pre=head[x]; head[x]=tot; 
}
void dfs(int u){
    dfn[u]=++idx; sz[u]=1;
    rep(i,1,20) if (dep[u]>=bin[i]) fa[u][i]=fa[fa[u][i-1]][i-1];
    for (int j=head[u];j;j=e[j].pre){
        int v=e[j].obj;
        if (v!=fa[u][0]){
            fa[v][0]=u; dep[v]=dep[u]+1; 
            dfs(v);
            sz[u]+=sz[v];
        }
    }
}
int lca(int x,int y){
    if (dep[x]<dep[y]) swap(x,y);
    int t=dep[x]-dep[y];
    rep(i,0,20) if (t&bin[i]) x=fa[x][i];
    down(i,20,0) if (fa[x][i]!=fa[y][i]) x=fa[x][i],y=fa[y][i];
    if (x==y) return x;
    return fa[x][0];
}
int dis(int x,int y){
    return dep[x]+dep[y]-2*dep[lca(x,y)];
}
void dfs1(int u){
    c[++idx]=u; rem[u]=sz[u];
    for (int j=head[u];j;j=e[j].pre){
        int v=e[j].obj;
        dfs1(v);
        if (!bel[v]) continue;
        int t1=dis(bel[u],u),t2=dis(bel[v],u);
        if (t1>t2||(t1==t2&&bel[u]>bel[v])||!bel[u]) bel[u]=bel[v];
    }
}
void dfs2(int u){
    for (int j=head[u];j;j=e[j].pre){
        int v=e[j].obj;
        int t1=dis(bel[u],v),t2=dis(bel[v],v);
        if (t1<t2||(t1==t2&&bel[u]<bel[v])||!bel[v]) bel[v]=bel[u]; 
        dfs2(v);
    }
}
void solve(int a,int b){
    int x=b,mid=b;
    down(i,20,0) if (dep[fa[x][i]]>dep[a]) x=fa[x][i];
    rem[a]-=sz[x];
    if (bel[a]==bel[b]) {f[bel[a]]+=sz[x]-sz[b]; return;}
    down(i,20,0) {
        int nxt=fa[mid][i];
        if (dep[nxt]<=dep[a]) continue;
        int t1=dis(bel[a],nxt),t2=dis(bel[b],nxt);
        if (t1>t2||(t1==t2&&bel[b]<bel[a])) mid=nxt;
    }
    f[bel[a]]+=sz[x]-sz[mid];
    f[bel[b]]+=sz[mid]-sz[b];
}
bool cmp(int x,int y){
    return dfn[x]<dfn[y];
}
void query(){
    idx=tot=top=0;  
    n=read();
    rep(i,1,n) {a[i]=read(); b[i]=a[i];bel[a[i]]=a[i];}
    sort(a+1,a+1+n,cmp);
    if (bel[1]!=1) s[++top]=1;
    rep(i,1,n){
        int t=a[i],f=0; 
        while (top){
            f=lca(t,s[top]);
            if (top>1&&dep[f]<dep[s[top-1]]) {insert(s[top-1],s[top]),top--;}
            else if (dep[f]<dep[s[top]]) {insert(f,s[top]);top--; break;}
            else break;
        }
        if (f!=s[top]) s[++top]=f;
        s[++top]=t;
    }
    while (top>1) insert(s[top-1],s[top]),top--;
    dfs1(1); dfs2(1);
    rep(i,1,idx) 
        for (int j=head[c[i]];j;j=e[j].pre) 
            solve(c[i],e[j].obj); 
    rep(i,1,idx) f[bel[c[i]]]+=rem[c[i]];
    rep(i,1,n) printf("%d ",f[b[i]]); puts("");
    rep(i,1,idx) head[c[i]]=rem[c[i]]=f[c[i]]=bel[c[i]]=0,c[i]=0;
}
int main(){
    bin[0]=1; rep(i,1,20) bin[i]=bin[i-1]*2;
    n=read();
    rep(i,1,n-1){
        int x=read(),y=read();
        insert(x,y); insert(y,x);
    }
    dfs(1);
    clr(head,0);
    Q=read(); 
    while (Q--) 
        query();
    return 0;
}
View Code

2286: [Sdoi2011]消耗战

#include<cstring>
#include<cstdio>
#include<algorithm>
#include<iostream>
#include<queue>
#include<vector>
#include<cmath>
#define low(i) (i&(-i))
#define pa pair<int,int>
#define rep(i,l,r) for(int i=l;i<=r;i++)
#define maxn 250250
#define inf 1152921504606846976
#define mm 12345678910
typedef long long ll;
using namespace std;
struct data{int obj,pre;ll c;
}e[maxn*2],ed[maxn*2];
int st[maxn],h[maxn],mark[maxn],head[maxn],head2[maxn],dep[maxn],top[maxn],s[maxn],son[maxn],fa[maxn];
ll f[maxn],mn[maxn];
int n,m,tt,tot,idx;
ll read()
{
    ll x=0, f=1; char ch=getchar();
    while (!isdigit(ch)) {if (ch=='-') f=-1; ch=getchar();}
    while (isdigit(ch)) x=x*10+ch-'0', ch=getchar();
    return x*f;
}
void insert(int x,int y,int z){
    e[++tot].obj=y; e[tot].c=z; e[tot].pre=head[x]; head[x]=tot; 
}
void insert2(int x,int y){
    if (x==y) return;
    ed[++tot].obj=y; ed[tot].pre=head2[x]; head2[x]=tot;
}
void dfs(int u){
    mark[u]=++idx;
    s[u]=1;
    for (int j=head[u];j;j=e[j].pre){
        int v=e[j].obj;
        if (v!=fa[u]){
            fa[v]=u; dep[v]=dep[u]+1;mn[v]=min(mn[u],e[j].c);
            dfs(v);
            s[u]+=s[v];
            if (s[v]>s[son[u]]) son[u]=v;
        }
    }
}
void make(int u,int ff){
    top[u]=ff;
    if (son[u]) make(son[u],ff); 
    for (int j=head[u];j;j=e[j].pre){
        int v=e[j].obj;
        if (v!=son[u]&&v!=fa[u]) make(v,v);
    }
}
int lca(int x,int y){
    while (top[x]!=top[y]){
        if (dep[top[x]]<dep[top[y]]) swap(x,y); 
        x=fa[top[x]];
    }
    return dep[x]<dep[y]?x:y;
}
bool cmp(int x,int y){
    return mark[x]<mark[y];
}
void dp(int u){
    ll tmp=0;
    f[u]=mn[u];
    for (int j=head2[u];j;j=ed[j].pre){
        int v=ed[j].obj;
        dp(v);
        tmp+=f[v];
    }
    head2[u]=0;
    if (tmp==0) f[u]=mn[u];
    else f[u]=min(f[u],tmp);
}
void solve(){
    m=read();
    rep(i,1,m) h[i]=read();
    sort(h+1,h+1+m,cmp); 
    int cnt=0;
    h[++cnt]=h[1]; 
    rep(i,2,m) {
        int t=lca(h[cnt],h[i]);
        if (t!=h[cnt]) h[++cnt]=h[i];
    } 
    tot=0;
    int top=0;
    st[++top]=1;
    rep(i,1,cnt){
        int now=h[i],t=lca(st[top],now);
        while (1){
            if (dep[t]>=dep[st[top-1]]){
                insert2(t,st[top--]);
                if (st[top]!=t) st[++top]=t;
                break;
            }
            insert2(st[top-1],st[top]); top--;
        }
        if (st[top]!=now) st[++top]=now;
    }
    while (--top) insert2(st[top],st[top+1]);
    dp(1);
    printf("%lld\n",f[1]);
}
int main(){
    n=read();
    rep(i,1,n-1){
        int x=read(),y=read(); ll z=read();
        insert(x,y,z);
        insert(y,x,z);
    }
    tt=read();
    dep[1]=1; mn[1]=inf; dfs(1);
    make(1,1);
    while (tt--) solve();
    return 0;
}
View Code

 

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