【Learning】虚树题目汇总

这里主要是我虚树的练习记录。
关于虚树的建树，参考了 https://www.cnblogs.com/chenhuan001/p/5639482.html 。感谢！至于为什么虚树建树要打那么长，只是为了方便理解。

套路：建虚树后树形DP，或者在建虚树过程中统计答案。

fread好快啊！
T1 xsy1633
建树时直接统计即可。

#include<cstdio>
#include<algorithm>
using namespace std;
const int N=100005;
int n,m,k,u,v,d,cnt,idx,pos,p[N],head[N],to[N*2],nxt[N*2],dd[N*2];
int siz[N],son[N],fa[N],dep[N],dis[N],dfn[N],top[N],stk[N];
char ch[20000005];
inline int rd(){int ret=0;while(ch[pos]<'0'||ch[pos]>'9'){pos++;}while(ch[pos]>='0'&&ch[pos]<='9'){ret=ret*10+ch[pos]-'0';pos++;}return ret;
}
bool cmp(int a,int b){return dfn[a]<dfn[b];
}
void adde(int u,int v,int d){to[++cnt]=v;nxt[cnt]=head[u];dd[cnt]=d;head[u]=cnt;
}
void dfs(int u){siz[u]=1;int v;for(int i=head[u];i;i=nxt[i]){v=to[i];if(v!=fa[u]){fa[v]=u;dep[v]=dep[u]+1;dis[v]=dis[u]+dd[i];dfs(v);siz[u]+=siz[v];if(!son[u]||siz[son[u]]<siz[v]){son[u]=v;}}}
}
void dfs(int u,int tp){dfn[u]=++idx;top[u]=tp;if(son[u]){dfs(son[u],tp);}int v;for(int i=head[u];i;i=nxt[i]){v=to[i];if(v!=fa[u]&&v!=son[u]){dfs(v,v);}}
}
int lca(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;}
}
void solve(){int ans=0;stk[stk[0]=1]=1;for(int i=1;i<=k;i++){int tmp=lca(stk[stk[0]],p[i]);if(tmp!=stk[stk[0]]){while(stk[0]>1){if(dfn[stk[stk[0]-1]]>dfn[tmp]){ans+=dis[stk[stk[0]]]-dis[stk[stk[0]-1]];stk[0]--;}else if(stk[stk[0]-1]==tmp){ans+=dis[stk[stk[0]]]-dis[tmp];stk[0]--;break;}else if(dfn[stk[stk[0]-1]]<dfn[tmp]){ans+=dis[stk[stk[0]]]-dis[tmp];stk[0]--;stk[++stk[0]]=tmp;break;}}}stk[++stk[0]]=p[i];}while(stk[0]>1){ans+=dis[stk[stk[0]]]-dis[stk[stk[0]-1]];stk[0]--;}printf("%d\n",ans);
}
int main(){fread(ch,20000000,1,stdin);n=rd();for(int i=1;i<n;i++){u=rd(),v=rd(),d=rd();adde(u,v,d);adde(v,u,d);}dfs(1);dfs(1,1);m=rd();for(int i=1;i<=m;i++){k=rd();for(int j=1;j<=k;j++){p[j]=rd();}sort(p+1,p+k+1,cmp);solve();}return 0;
}

T2 bzoj2286 虚树+树形dp

#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;
const int N=250005;
typedef long long ll;
int n,m,k,u,v,d,cnt,idx,pos,h[N],stk[N],head[N],to[N*2],nxt[N*2],dd[N*2];
int siz[N],son[N],fa[N],dep[N],top[N],dfn[N];
ll mind[N],f[N];
bool ck[N];
char ch[20000005];
inline int rd(){register int ret=0;while(ch[pos]<'0'||ch[pos]>'9'){pos++;}while(ch[pos]>='0'&&ch[pos]<='9'){ret=ret*10+ch[pos]-'0';pos++;}return ret;
}
bool cmp(int a,int b){return dfn[a]<dfn[b];
}
void adde(int u,int v,int d){to[++cnt]=v;nxt[cnt]=head[u];dd[cnt]=d;head[u]=cnt;
}
void dfs(int u){siz[u]=1;int v;for(int i=head[u];i;i=nxt[i]){v=to[i];if(v!=fa[u]){fa[v]=u;dep[v]=dep[u]+1;mind[v]=min(mind[u],1LL*dd[i]);dfs(v);siz[u]+=siz[v];if(!son[u]||siz[son[u]]<siz[v]){son[u]=v;}}}
}
void dfs(int u,int tp){dfn[u]=++idx;top[u]=tp;if(son[u]){dfs(son[u],tp);}int v;for(int i=head[u];i;i=nxt[i]){v=to[i];if(v!=fa[u]&&v!=son[u]){dfs(v,v);}}
}
int lca(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;}
}
void build(){stk[stk[0]=1]=1;for(int i=1;i<=k;i++){int tmp=lca(stk[stk[0]],h[i]);if(tmp!=stk[stk[0]]){while(stk[0]>1){if(dfn[stk[stk[0]-1]]>dfn[tmp]){adde(stk[stk[0]-1],stk[stk[0]],0);stk[0]--;}else if(stk[stk[0]-1]==tmp){adde(tmp,stk[stk[0]],0);stk[0]--;break;}else if(dfn[stk[stk[0]-1]]<dfn[tmp]){adde(tmp,stk[stk[0]],0);stk[0]--;stk[++stk[0]]=tmp;break;}}}stk[++stk[0]]=h[i];}while(stk[0]>1){adde(stk[stk[0]-1],stk[stk[0]],0);stk[0]--;}
}
void dp(int u){ll sum=0,v;f[u]=mind[u];for(int i=head[u];i;i=nxt[i]){v=to[i];dp(v);sum+=f[v];}if(!ck[u]){f[u]=min(sum,f[u]);}head[u]=ck[u]=0;
}
int main(){fread(ch,20000000,1,stdin);n=rd();for(int i=1;i<n;i++){u=rd(),v=rd(),d=rd();adde(u,v,d);adde(v,u,d);}mind[1]=1e15;dfs(1);dfs(1,1);m=rd();memset(head,0,sizeof(head));for(int i=1;i<=m;i++){k=rd();for(int j=1;j<=k;j++){h[j]=rd();ck[h[j]]=true;}sort(h+1,h+k+1,cmp);cnt=0;build();dp(1);printf("%lld\n",f[1]);}return 0;
}

T3 bzoj3611 虚树+树形dp

#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;
const int N=1000005,inf=0x7f7f7f7f;
int n,m,k,u,v,cnt,idx,pos,p[N],head[N],to[N*2],nxt[N*2];
int siz[N],son[N],fa[N],dep[N],top[N],dfn[N],stk[N];
long long sum,maxn,minn,mx[N],mi[N];
bool ck[N];
char ch[100000005];
inline int rd(){int ret=0;while(ch[pos]<'0'||ch[pos]>'9'){pos++;}while(ch[pos]>='0'&&ch[pos]<='9'){ret=ret*10+ch[pos]-'0';pos++;}return ret;
}
bool cmp(int a,int b){return dfn[a]<dfn[b];
}
void adde(int u,int v){to[++cnt]=v;nxt[cnt]=head[u];head[u]=cnt;
}
void dfs(int u){siz[u]=1;int v;for(int i=head[u];i;i=nxt[i]){v=to[i];if(v!=fa[u]){fa[v]=u;dep[v]=dep[u]+1;dfs(v);siz[u]+=siz[v];if(!son[u]||siz[son[u]]<siz[v]){son[u]=v;}}}
}
void dfs(int u,int tp){dfn[u]=++idx;top[u]=tp;if(son[u]){dfs(son[u],tp);}int v;for(int i=head[u];i;i=nxt[i]){v=to[i];if(v!=fa[u]&&v!=son[u]){dfs(v,v);}}
}
int lca(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;}
}
void dfssize(int u){siz[u]=ck[u];int v;for(int i=head[u];i;i=nxt[i]){v=to[i];dfssize(v);siz[u]+=siz[v];}
}
void dp(int u){int v;mx[u]=ck[u]?0:-inf;mi[u]=ck[u]?0:inf;for(int i=head[u];i;i=nxt[i]){v=to[i];dp(v);sum+=1LL*siz[v]*(siz[1]-siz[v])*(dep[v]-dep[u]);maxn=max(maxn,mx[u]+mx[v]+dep[v]-dep[u]);minn=min(minn,mi[u]+mi[v]+dep[v]-dep[u]);mx[u]=max(mx[u],mx[v]+dep[v]-dep[u]);mi[u]=min(mi[u],mi[v]+dep[v]-dep[u]);}head[u]=ck[u]=0;
}
void solve(){stk[stk[0]=1]=1;for(int i=1;i<=k;i++){int tmp=lca(stk[stk[0]],p[i]);if(tmp!=stk[stk[0]]){while(stk[0]>1){if(dfn[stk[stk[0]-1]]>dfn[tmp]){adde(stk[stk[0]-1],stk[stk[0]]);stk[0]--;}else if(stk[stk[0]-1]==tmp){adde(tmp,stk[stk[0]]);stk[0]--;break;}else if(dfn[stk[stk[0]-1]]<dfn[tmp]){adde(tmp,stk[stk[0]]);stk[stk[0]]=tmp;break;}}}if(stk[stk[0]]!=p[i]){stk[++stk[0]]=p[i];}}while(stk[0]>1){adde(stk[stk[0]-1],stk[stk[0]]);stk[0]--;}sum=0;minn=inf;maxn=-inf;dfssize(1);dp(1);printf("%lld %lld %lld\n",sum,minn,maxn);
}
int main(){fread(ch,100000000,1,stdin);n=rd();for(int i=1;i<n;i++){u=rd(),v=rd();adde(u,v);adde(v,u);}dfs(1);dfs(1,1);memset(head,0,sizeof(head));m=rd();for(int i=1;i<=m;i++){k=rd();for(int j=1;j<=k;j++){p[j]=rd();ck[p[j]]=true;}sort(p+1,p+k+1,cmp);cnt=0;solve();}return 0;
}

T4 bzoj3572 虚树+树形dp+倍增

#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;
const int N=300005;
int n,m,k,u,v,cnt,idx,pos,p[N],head[N],to[N*2],nxt[N*2],dep[N],fa[N][20],dfn[N],siz[N],stk[N];
int bel[N],left[N],sum[N],tmp[N];
char ch[20000005];
inline int rd(){register int ret=0;while(ch[pos]<'0'||ch[pos]>'9'){pos++;}while(ch[pos]>='0'&&ch[pos]<='9'){ret=ret*10+ch[pos]-'0';pos++;}return ret;
}
bool cmp(int a,int b){return dfn[a]<dfn[b];
}
void adde(int u,int v){to[++cnt]=v;nxt[cnt]=head[u];head[u]=cnt;
}
void dfs(int u){dfn[u]=++idx;siz[u]=1;for(int i=1;(1<<i)<=dep[u];i++){fa[u][i]=fa[fa[u][i-1]][i-1];}int v;for(int i=head[u];i;i=nxt[i]){v=to[i];if(v!=fa[u][0]){fa[v][0]=u;dep[v]=dep[u]+1;dfs(v);siz[u]+=siz[v];}}
}
int lca(int u,int v){if(dep[u]<dep[v]){swap(u,v);}int d=dep[u]-dep[v];for(int i=0;i<=19;i++){if(d&(1<<i)){u=fa[u][i];}}if(u==v){return u;}for(int i=19;i>=0;i--){if(fa[u][i]!=fa[v][i]){u=fa[u][i];v=fa[v][i];}}return fa[u][0];
}
int dis(int u,int v){return dep[u]+dep[v]-2*dep[lca(u,v)];
}
void build(){stk[stk[0]=1]=1;for(int i=1;i<=k;i++){int tmp=lca(stk[stk[0]],p[i]);if(tmp!=stk[stk[0]]){while(stk[0]>1){if(dfn[stk[stk[0]-1]]>dfn[tmp]){adde(stk[stk[0]-1],stk[stk[0]]);stk[0]--;}else if(stk[stk[0]-1]==tmp){adde(tmp,stk[stk[0]]);stk[0]--;break;}else if(dfn[stk[stk[0]-1]]<dfn[tmp]){adde(tmp,stk[stk[0]]);stk[stk[0]]=tmp;break;}}}if(p[i]!=stk[stk[0]]){stk[++stk[0]]=p[i];}}while(stk[0]>1){adde(stk[stk[0]-1],stk[stk[0]]);stk[0]--;}
}
void dp1(int u){left[u]=siz[u];int v,d1,d2;for(int i=head[u];i;i=nxt[i]){v=to[i];dp1(v);if(bel[v]){if(!bel[u]){bel[u]=bel[v];}else{d1=dis(u,bel[u]),d2=dis(u,bel[v]);if(d2<d1||(d2==d1&&bel[v]<bel[u])){bel[u]=bel[v];}}}}
}
void dp2(int u){int v,d1,d2;for(int i=head[u];i;i=nxt[i]){v=to[i];if(!bel[v]){bel[v]=bel[u];}else{d1=dis(v,bel[v]),d2=dis(v,bel[u]);if(d2<d1||(d2==d1&&bel[u]<bel[v])){bel[v]=bel[u];}}dp2(v);}
}
void work(int u,int v){int tmp=v,mid=v,nxt,d1,d2;for(int i=19;i>=0;i--){if(dep[fa[tmp][i]]>dep[u]){tmp=fa[tmp][i];}}left[u]-=siz[tmp];if(bel[u]==bel[v]){sum[bel[u]]+=siz[tmp]-siz[v];return;}for(int i=19;i>=0;i--){nxt=fa[mid][i];if(dep[nxt]>dep[u]){d1=dis(bel[u],nxt);d2=dis(bel[v],nxt);if(d2<d1||(d2==d1&&bel[v]<bel[u])){mid=nxt;}}}sum[bel[u]]+=siz[tmp]-siz[mid];sum[bel[v]]+=siz[mid]-siz[v];
}
void dp3(int u){int v;for(int i=head[u];i;i=nxt[i]){v=to[i];work(u,v);dp3(v);}sum[bel[u]]+=left[u];
}
void clear(int u){for(int i=head[u];i;i=nxt[i]){clear(to[i]);}bel[u]=sum[u]=head[u]=left[u]=0;
}
int main(){fread(ch,20000000,1,stdin);n=rd();for(int i=1;i<n;i++){u=rd(),v=rd();adde(u,v);adde(v,u);}dfs(1);m=rd();memset(head,0,sizeof(head));for(int i=1;i<=m;i++){k=rd();for(int j=1;j<=k;j++){p[j]=rd();bel[p[j]]=tmp[j]=p[j];}sort(p+1,p+k+1,cmp);cnt=0;build();dp1(1);dp2(1);dp3(1);for(int j=1;j<=k;j++){printf("%d ",sum[tmp[j]]);}puts("");clear(1);}return 0;
}

T5 codeforces613D 虚树+树形dp
终于成功注册codeforces，辛辛苦苦把验证码搞出来了

#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;
const int N=100005;
int n,m,k,u,v,cnt,idx,p[N],head[N],to[N*2],nxt[N*2];
int dfn[N],fa[N][20],dep[N],stk[N];
bool ck[N],exi[N];
bool cmp(int a,int b){return dfn[a]<dfn[b];
}
void adde(int u,int v){to[++cnt]=v;nxt[cnt]=head[u];head[u]=cnt;
}
void dfs(int u){dfn[u]=++idx;for(int i=1;(1<<i)<=dep[u];i++){fa[u][i]=fa[fa[u][i-1]][i-1];}int v;for(int i=head[u];i;i=nxt[i]){v=to[i];if(v!=fa[u][0]){fa[v][0]=u;dep[v]=dep[u]+1;dfs(v);}}
}
int lca(int u,int v){if(dep[u]<dep[v]){swap(u,v);}int d=dep[u]-dep[v];for(int i=0;(1<<i)<=d;i++){if(d&(1<<i)){u=fa[u][i];}}if(u==v){return u;}for(int i=18;i>=0;i--){if(fa[u][i]!=fa[v][i]){u=fa[u][i];v=fa[v][i];}}return fa[u][0];
}
void build(){stk[stk[0]=1]=1;for(int i=1;i<=k;i++){int tmp=lca(stk[stk[0]],p[i]);if(tmp!=stk[stk[0]]){while(stk[0]>1){if(dfn[stk[stk[0]-1]]>dfn[tmp]){adde(stk[stk[0]-1],stk[stk[0]]);stk[0]--;}else if(stk[stk[0]-1]==tmp){adde(tmp,stk[stk[0]]);stk[0]--;break;}else if(dfn[stk[stk[0]-1]]<dfn[tmp]){adde(tmp,stk[stk[0]]);stk[stk[0]]=tmp;break;}}}if(p[i]!=stk[stk[0]]){stk[++stk[0]]=p[i];}}while(stk[0]>1){adde(stk[stk[0]-1],stk[stk[0]]);stk[0]--;}
}
int dp(int u){int sum=0,tot=0,v;for(int i=head[u];i;i=nxt[i]){v=to[i];sum+=dp(v);tot+=exi[v];}if(ck[u]){exi[u]=1;return sum+tot;}else{exi[u]=tot==1;return sum+(tot>1);   }
}
void clear(int u){for(int i=head[u];i;i=nxt[i]){clear(to[i]);}head[u]=ck[u]=exi[u]=0;
}
bool judge(){for(int i=1;i<=k;i++){if(ck[fa[p[i]][0]]){return false;}}return true;
}
int main(){scanf("%d",&n);for(int i=1;i<n;i++){scanf("%d%d",&u,&v);adde(u,v);adde(v,u);}dfs(1);memset(head,0,sizeof(head));scanf("%d",&m);for(int i=1;i<=m;i++){scanf("%d",&k);for(int j=1;j<=k;j++){scanf("%d",&p[j]);ck[p[j]]=true;}if(!judge()){puts("-1");for(int j=1;j<=k;j++){ck[p[j]]=false;}continue;}sort(p+1,p+k+1,cmp);cnt=0;build();printf("%d\n",dp(1));clear(1);}return 0;
}

T6 bzoj3991 乱入的一道dfs序+平衡树

#include<cstdio>
#include<algorithm>
using namespace std;
typedef long long ll;
const int N=100005;
int n,m,u,v,cnt,idx,pos,dfn[N],fa[N][20],dep[N],head[N],to[N*2],nxt[N*2];
ll d,ans,dd[N*2],dis[N];
bool ck[N];
char ch[20000005];
inline int rd(){register int ret=0;while(ch[pos]<'0'||ch[pos]>'9'){pos++;}while(ch[pos]>='0'&&ch[pos]<='9'){ret=ret*10+ch[pos]-'0';pos++;}return ret;
}
void adde(int u,int v,ll d){to[++cnt]=v;nxt[cnt]=head[u];dd[cnt]=d;head[u]=cnt;
}
void dfs(int u){dfn[u]=++idx;for(int i=1;(1<<i)<=dep[u];i++){fa[u][i]=fa[fa[u][i-1]][i-1];}int v;for(int i=head[u];i;i=nxt[i]){v=to[i];if(v!=fa[u][0]){fa[v][0]=u;dep[v]=dep[u]+1;dis[v]=dis[u]+dd[i];dfs(v);}}
}
int lca(int u,int v){if(dep[u]<dep[v]){swap(u,v);}int d=dep[u]-dep[v];for(int i=0;i<=17;i++){if(d&(1<<i)){u=fa[u][i];}}if(u==v){return u;}for(int i=17;i>=0;i--){if(fa[u][i]!=fa[v][i]){u=fa[u][i];v=fa[v][i];}}return fa[u][0];
}
ll dist(int u,int v){return dis[u]+dis[v]-2*dis[lca(u,v)];
}
struct ScapegoatTree{int root,cnt,*goat,val[N],ch[N][2],siz[N],tot[N],del[N],mmp[N],pos[N];int rnk(int x){int k=root,ret=1;while(k){if(dfn[x]<=dfn[val[k]]){k=ch[k][0];}else{ret+=siz[ch[k][0]]+del[k];k=ch[k][1];}}return ret;}int kth(int x){if(x==0){x=siz[root];}else if(x==siz[root]+1){x=1;}int k=root;while(k){if(del[k]&&x==siz[ch[k][0]]+1){return val[k];}else if(x<=siz[ch[k][0]]+del[k]){k=ch[k][0];}else{x-=siz[ch[k][0]]+del[k];k=ch[k][1];}}}int newnode(int &k,int x){if(mmp[0]){k=mmp[mmp[0]--];}else{k=++cnt;}ch[k][0]=ch[k][1]=0;siz[k]=tot[k]=del[k]=1;val[k]=x;}void dfs(int k){if(!k){return;}dfs(ch[k][0]);if(del[k]){pos[++pos[0]]=k;}else{mmp[++mmp[0]]=k;}dfs(ch[k][1]);}void build(int &k,int l,int r){if(l>r){k=0;return;}int mid=(l+r)/2;k=pos[mid];build(ch[k][0],l,mid-1);build(ch[k][1],mid+1,r);siz[k]=siz[ch[k][0]]+siz[ch[k][1]]+1;tot[k]=tot[ch[k][0]]+tot[ch[k][1]]+1;}void rebuild(int &k){pos[0]=0;dfs(k);build(k,1,pos[0]);}void insert(int &k,int x){if(!k){newnode(k,x);return;}siz[k]++;tot[k]++;insert(ch[k][dfn[x]>dfn[val[k]]],x);if(tot[k]*0.75<max(tot[ch[k][0]],tot[ch[k][1]])){goat=&k;}}void insert(int x){goat=NULL;insert(root,x);if(goat){rebuild(*goat);}}void remove(int &k,int x){siz[k]--;if(del[k]&&x==siz[ch[k][0]]+1){del[k]=0;return;}if(x<=siz[ch[k][0]]+del[k]){remove(ch[k][0],x);}else{remove(ch[k][1],x-siz[ch[k][0]]-del[k]);}}void remove(int x){remove(root,rnk(x));if(siz[root]<tot[root]*0.75){rebuild(root);}}
}sgt;
int main(){fread(ch,20000000,1,stdin);n=rd(),m=rd();for(int i=1;i<n;i++){u=rd(),v=rd(),d=rd();adde(u,v,d);adde(v,u,d);}dfs(1);for(int i=1;i<=m;i++){u=rd();if(!ck[u]){if(sgt.siz[sgt.root]==0){sgt.insert(u);}else{sgt.insert(u);int tmp=sgt.rnk(u),pre=sgt.kth(tmp-1),sub=sgt.kth(tmp+1);ans+=dist(u,pre)+dist(u,sub)-dist(pre,sub);}ck[u]=true;}else{if(sgt.siz[sgt.root]==1){sgt.remove(u);}else{int tmp=sgt.rnk(u),pre=sgt.kth(tmp-1),sub=sgt.kth(tmp+1);ans-=dist(u,pre)+dist(u,sub)-dist(pre,sub);sgt.remove(u);}ck[u]=false;}printf("%lld\n",ans);}return 0;
}

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【Learning】虚树题目汇总

套路：建虚树后树形DP，或者在建虚树过程中统计答案。

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