Codeforces809E Surprise me!
wdnmd这题做起来刺激😂
\(\text{wdnmd}\)这题做起来刺激😂
转换一下\(a_i\)的定义：权值为\(i\)的点的编号。
这样求的答案就成了\(\sum\limits_{i=1}^n\sum\limits_{j=1}^n\varphi(i*j)dis(a_i,a_j)\)
反演还是很套路的：
\(=\sum\limits_{i=1}^n\sum\limits_{j=1}^n\dfrac{\varphi(i)\varphi(j)gcd(i,j)}{\varphi(gcd(i,j))}dis(a_i,a_j)\)
\(=\sum\limits_{d=1}^n\dfrac{d}{\varphi(d)}\sum\limits_{i=1}^n\varphi(i)\sum\limits_{j=1}^n\varphi(j)[gcd(i,j)=d]dis(a_i,a_j)\)
\(=\sum\limits_{d=1}^n\dfrac{d}{\varphi(d)}\sum\limits_{i=1}^{\lfloor\frac{n}{d}\rfloor}\varphi(id)\sum\limits_{j=1}^{\lfloor\frac{n}{d}\rfloor}\varphi(jd)dis(a_{id},a_{jd})\sum\limits_{a|gcd(i,j)}\mu(a)\)
\(=\sum\limits_{d=1}^n\dfrac{d}{\varphi(d)}\sum\limits_{a=1}^{\lfloor\frac{n}{d}\rfloor}\mu(a)\sum\limits_{i=1}^{\lfloor\frac{n}{ad}\rfloor}\varphi(iad)\sum\limits_{j=1}^{\lfloor\frac{n}{ad}\rfloor}\varphi(jad)dis(a_{iad},a_{jad})\)
令\(T=ad\)，枚举\(T\)：
\(=\sum\limits_{d=1}^n\dfrac{d}{\varphi(d)}\sum\limits_{T=1}^n[d|T]\mu(\dfrac{T}{d})\sum\limits_{i=1}^{\lfloor\frac{n}{T}\rfloor}\varphi(iT)\sum\limits_{j=1}^{\lfloor\frac{n}{T}\rfloor}\varphi(jT)dis(a_{iT},a_{jT})\)
\(=\sum\limits_{T=1}^n\sum\limits_{i=1}^{\lfloor\frac{n}{T}\rfloor}\varphi(iT)\sum\limits_{j=1}^{\lfloor\frac{n}{T}\rfloor}\varphi(jT)dis(a_{iT},a_{jT})\sum\limits_{d|T}\mu(\dfrac{T}{d})\dfrac{d}{\varphi(d)}\)
最后面\(\sum\limits_{d|T}\)的一坨是个卷积形式，可以线筛，不过\(O(n\log n)\)枚举倍数预处理就够了。
看中间那一坨，考虑它的含义，其实就是把编号为\(a_{iT}\)的点拿出来，点权为\(\varphi(iT)\)，求它们两两之间\(value(i)*value(j)*dis(i,j)\)之和。
而这个式子是个调和级数，总点数是\(O(n\log n)\)的，大力对点集\(\{a_{iT}|i\in \mathbb{Z}\}\)上虚树。
记\(siz(i)\)为点\(i\)的子树权值和，考虑每条边\(edge(i,j)\)的贡献，为\(length(edge(i,j))*siz(j)*(siz(root)-siz(j))\)。
于是就能\(O(n\log^2n)\)解决辣。
代码：
#include <iostream>
#include <cstdio>
#include <algorithm>
#include <cmath>
#include <cstring>
#include <vector>
#define maxn 200005
#define inf 0x3f3f3f3f
const int mod = 1e9 + 7;
using namespace std;
inline int read(){
int x=0,y=0;
char ch=getchar();
while(ch<'0'||ch>'9'){if(ch=='-')y=1;ch=getchar();}
while(ch>='0'&&ch<='9')x=(x<<1)+(x<<3)+(ch^48),ch=getchar();
return y?-x:x;
}
int deep[maxn],seg[maxn],a[maxn],all;
vector<int>p;
bool vis[maxn];
namespace origin{
int top[maxn],fa[maxn],son[maxn],siz[maxn],h[maxn],num;
struct edge{
int pre,to;
}e[maxn<<1];
inline void add(int from,int to){
e[++num].pre=h[from],h[from]=num,e[num].to=to;
}
void dfs1(int node=1){
siz[node]=1;
int x;
for(register int i=h[node];i;i=e[i].pre){
x=e[i].to;
if(siz[x])continue;
fa[x]=node,deep[x]=deep[node]+1,dfs1(x),siz[node]+=siz[x];
if(siz[x]>siz[son[node]])son[node]=x;
}
}
void dfs2(int node=1){
seg[node]=++all;
if(son[node]){
int x=son[node];
top[x]=top[node],dfs2(x);
for(register int i=h[node];i;i=e[i].pre){
x=e[i].to;
if(!seg[x])top[x]=x,dfs2(x);
}
}
}
int lca(int x,int y){
while(top[x]!=top[y])deep[top[x]]<deep[top[y]]?y=fa[top[y]]:x=fa[top[x]];
return deep[x]<deep[y]?x:y;
}
}
namespace virt{
int siz[maxn],h[maxn],num,ans;
struct edge{
int pre,to,l;
}e[maxn<<1];
inline void add(int from,int to){
int l=abs(deep[from]-deep[to]);
e[++num].pre=h[from],h[from]=num,e[num].to=to,e[num].l=l;
e[++num].pre=h[to],h[to]=num,e[num].to=from,e[num].l=l;
}
bool cmp(int x,int y){return seg[x]<seg[y];}
struct Monostack{
int sta[maxn],top;
void push(int x){sta[++top]=x;}
void clear(){
for(register int i=2;i<=top;++i)add(sta[i],sta[i-1]);
top=0;
}
void check(int x){
if(x==1)return;
int l=origin::lca(x,sta[top]);
h[x]=0;
if(l!=sta[top]){
while(seg[l]<seg[sta[top-1]])add(sta[top],sta[top-1]),--top;
--top;
if(l!=sta[top])h[l]=0,add(sta[top+1],l),push(l);
else add(sta[top+1],l);
}
push(x);
}
}s;
void build(){
h[1]=0,s.push(1);
for(vector<int>::iterator iter=p.begin();iter!=p.end();++iter)s.check(*iter);
s.clear();
}
void dfs(int node=1,int fa=0){
if(!vis[node])siz[node]=0;
int x;
for(register int i=h[node];i;i=e[i].pre){
x=e[i].to;
if(x!=fa)dfs(x,node),(siz[node]+=siz[x])%=mod;
}
}
void calc(int node=1,int fa=0){
int x;
for(register int i=h[node];i;i=e[i].pre){
x=e[i].to;
if(x!=fa)calc(x,node),(ans+=1ll*e[i].l*siz[x]%mod*(siz[1]-siz[x])%mod)%=mod;
}
}
int solve(){
sort(p.begin(),p.end(),cmp);
num=0,ans=0,build(),dfs(),calc();
for(vector<int>::iterator iter=p.begin();iter!=p.end();++iter)vis[*iter]=siz[*iter]=0;
return ans;
}
}
namespace mobius{
int phi[maxn],mu[maxn],f[maxn],prime[maxn>>2],inv[maxn],cnt;
bool is[maxn];
void solve(int n){
int ans=0;
is[1]=phi[1]=mu[1]=inv[1]=1;
for(register int i=2;i<=n;++i){
inv[i]=1ll*(mod-mod/i)*inv[mod%i]%mod;
if(!is[i])prime[++cnt]=i,phi[i]=i-1,mu[i]=-1;
for(register int j=1;j<=cnt&&i*prime[j]<=n;++j){
is[i*prime[j]]=1;
if(i%prime[j]==0){
mu[i*prime[j]]=0,phi[i*prime[j]]=phi[i]*prime[j];
break;
}
phi[i*prime[j]]=phi[i]*(prime[j]-1),mu[i*prime[j]]=-mu[i];
}
}
for(register int i=1;i<=n;++i)
for(register int j=i;j<=n;j+=i)
(f[j]+=1ll*mu[j/i]*i*inv[phi[i]]%mod)%=mod;
for(register int i=1;i<=n;++i){
p.clear();
for(register int j=i;j<=n;j+=i)
p.push_back(a[j]),vis[a[j]]=1,virt::siz[a[j]]=phi[j];
(ans+=1ll*virt::solve()*f[i]%mod)%=mod;
}
ans=1ll*(ans<<1)*inv[n]%mod*inv[n-1]%mod;
printf("%d\n",(ans+mod)%mod);
}
}
int main(){
using namespace origin;
int n=read(),x,y;
for(register int i=1;i<=n;++i)a[read()]=i;
for(register int i=1;i<n;++i)x=read(),y=read(),add(x,y),add(y,x);
dfs1(),dfs2();
mobius::solve(n);
}
