标签：phi back ons name log var 莫比乌斯反演 ++i cout
简化题意：给出一棵\(n\)个点的树，编号为\(1\)到\(n\)，第\(i\)个点的点权为\(a_i\)，保证序列\(a_i\)是一个\(1\)到\(n\)的排列，求
\[ \frac{1}{n(n-1)} \sum\limits_{i=1}^n \sum\limits_{j=1}^n \varphi(a_ia_j) dist(i,j)\]
其中\(dist(i,j)\)为树上\(i,j\)两点的距离。
看到\(\varphi\)第一反应推式子
因为序列\(a_i\)是一个\(1\)到\(n\)的排列，设\(t_i\)表示点权为\(i\)的点的编号，那么原式等于$ \frac{1}{n(n-1)} \sum\limits_{i=1}^n \sum\limits_{j=1}^n \varphi(ij) dist(t_i,t_j)$
接下来化简$\sum\limits_{i=1}^n \sum\limits_{j=1}^n\varphi(ij) $
考虑\(gcd(i,j)\)在\(\varphi(ij)\)中的贡献，不难得到\(\varphi(ij) = \frac{\varphi(i) \varphi(j) gcd(i,j)}{\varphi(gcd(i,j))}\)
代入得$ \sum\limits_{i=1}^n \sum\limits_{j=1}^n \varphi(ij) =\sum\limits_{i=1}^n \sum\limits_{j=1}^n \frac{\varphi(i) \varphi(j) gcd(i,j)}{\varphi(gcd(i,j))}$
按照套路枚举\(gcd\)：\(=\sum\limits_{d=1}^n \frac{d}{\varphi(d)} \sum\limits_{i=1}^{\lfloor \frac{n}{d} \rfloor}\sum\limits_{j=1}^{\lfloor \frac{n}{d} \rfloor} \varphi(id) \varphi(jd) [gcd(i,j) == 1]=\sum\limits_{d=1}^n \frac{d}{\varphi(d)} \sum\limits_{i=1}^{\lfloor \frac{n}{d} \rfloor}\sum\limits_{j=1}^{\lfloor \frac{n}{d} \rfloor} \varphi(id) \varphi(jd) \sum\limits_{p | gcd(i,j)} \mu(p)\)
将\(p\)移到前面：\(=\sum\limits_{d=1}^n \sum\limits_{p=1}^{\lfloor \frac{n}{d} \rfloor} \frac{d \mu(p)}{\varphi(d)} \sum\limits_{i=1}^{\lfloor \frac{n}{dp} \rfloor}\sum\limits_{j=1}^{\lfloor \frac{n}{dp} \rfloor} \varphi(idp) \varphi(jdp)\)
\(dp\)太难看了考虑枚举\(T=dp\)：\(=\sum\limits_{T=1}^n \sum\limits_{d | T} \frac{d \mu(\frac{T}{d})}{\varphi(d)} \sum\limits_{i=1}^{\lfloor \frac{n}{T} \rfloor}\sum\limits_{j=1}^{\lfloor \frac{n}{T} \rfloor} \varphi(iT) \varphi(jT)\)
然后将\(dist(t_i,t_j)\)代回来。注意这个时候\(i,j\)的意义发生了变化，我们应该要代入的是\(dist(t_{iT} , t_{jT})\)
所以我们需要求的是\(=\sum\limits_{T=1}^n \sum\limits_{d | T} \frac{d \mu(\frac{T}{d})}{\varphi(d)} \sum\limits_{i=1}^{\lfloor \frac{n}{T} \rfloor}\sum\limits_{j=1}^{\lfloor \frac{n}{T} \rfloor} \varphi(iT) \varphi(jT) dist(t_{iT} , t_{jT})\)
推到这里我们就可以做了，相对来说还是比较好推的……
对于\(\sum\limits_{d | T} \frac{d \mu(\frac{T}{d})}{\varphi(d)}\)，枚举倍数做到\(O(nlogn)\)预处理
对于\(\sum\limits_{i=1}^{\lfloor \frac{n}{T} \rfloor}\sum\limits_{j=1}^{\lfloor \frac{n}{T} \rfloor} \varphi(iT) \varphi(jT) dist(t_{iT} , t_{jT})\)，它等于\(\sum\limits_{i=1}^{\lfloor \frac{n}{T} \rfloor}\sum\limits_{j=1}^{\lfloor \frac{n}{T} \rfloor} \varphi(iT) \varphi(jT) (dep_{t_{iT}} + dep_{t_{jT}} - 2 * dep_{LCA(t_{iT} , t_{jT})})\)。我们把所有点权为\(T\)的倍数的点拿出来建立虚树进行树形DP，对于每一个点维护它子树中满足条件的点的\(\sum \varphi(i) \times dep_i\)与\(\sum \varphi(i)\)，在两个子树合并时计算答案即可。总复杂度约为\(O(nlog^2n)\)。
最后记录一些被坑的细节（大概只有我这种菜鸡才会犯……）
\(1.\)建立虚树的点不是编号为\(T\)的倍数的点，而是点权是\(T\)的倍数的点
\(2.\)\(\frac{d}{\varphi(d)}\)不一定是整数
\(3.\)写程序的时候要保持清醒……发现很多地方\(i\)打成\(j\)，\(a_x\)打成\(x\)之类的……
\(4.\)记得清空数组
#include<bits/stdc++.h>
//This code is written by Itst
using namespace std;
inline int read(){
int a = 0;
char c = getchar();
bool f = 0;
while(!isdigit(c) && c != EOF){
if(c == ‘-‘)
f = 1;
c = getchar();
}
if(c == EOF)
exit(0);
while(isdigit(c)){
a = a * 10 + c - 48;
c = getchar();
}
return f ? -a : a;
}
const int MAXN = 2e5 + 9 , MOD = 1e9 + 7;
int mu[MAXN] , phi[MAXN] , prime[MAXN] , cnt;
bool nprime[MAXN];
struct Edge{
int end , upEd;
}Ed[MAXN << 1];
int N , cntEd , ts , top , cntT , cntST , cur , ans1 , ans;
int head[MAXN] , val[MAXN] , ind[MAXN];
int dfn[MAXN] , dep[MAXN] , fir[MAXN] , ST[21][MAXN << 1] , logg2[MAXN << 1];
int st[MAXN] , tree[MAXN] , dp1[MAXN] , dp2[MAXN] , q[MAXN];
vector < int > ch[MAXN];
inline void addEd(int a , int b){
Ed[++cntEd].end = b;
Ed[cntEd].upEd = head[a];
head[a] = cntEd;
}
void input(){
N = read();
for(int i = 1 ; i <= N ; ++i)
ind[val[i] = read()] = i;
for(int i = 1 ; i < N ; ++i){
int a = read() , b = read();
addEd(a , b);
addEd(b , a);
}
}
inline void init(){
phi[1] = mu[1] = 1;
for(int i = 2 ; i <= N ; ++i){
if(!nprime[i]){
prime[++cnt] = i;
phi[i] = i - 1;
mu[i] = -1;
}
for(int j = 1 ; j <= cnt && prime[j] * i <= N ; ++j){
nprime[i * prime[j]] = 1;
if(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] * -1;
}
}
}
void init_dfs(int x , int p){
dfn[x] = ++ts;
ST[0][++cntST] = x;
fir[x] = cntST;
dep[x] = dep[p] + 1;
for(int i = head[x] ; i ; i = Ed[i].upEd)
if(Ed[i].end != p){
init_dfs(Ed[i].end , x);
ST[0][++cntST] = x;
}
}
inline int cmp(int a , int b){
return dep[a] < dep[b] ? a : b;
}
void init_ST(){
for(int i = 2 ; i <= cntST ; ++i)
logg2[i] = logg2[i >> 1] + 1;
for(int i = 1 ; 1 << i <= cntST ; ++i)
for(int j = 1 ; j + (1 << i) <= cntST + 1 ; ++j)
ST[i][j] = cmp(ST[i - 1][j] , ST[i - 1][j + (1 << (i - 1))]);
}
inline int LCA(int x , int y){
x = fir[x];
y = fir[y];
if(x > y)
swap(x , y);
int t = logg2[y - x + 1];
return cmp(ST[t][x] , ST[t][y - (1 << t) + 1]);
}
inline int poww(long long a , int b){
int times = 1;
while(b){
if(b & 1)
times = times * a % MOD;
a = a * a % MOD;
b >>= 1;
}
return times;
}
bool cmp1(int a , int b){
return dfn[a] < dfn[b];
}
int dfs(int x){
dp1[x] = dp2[x] = 0;
int sum = 0;
for(int i = 0 ; i < ch[x].size() ; ++i){
sum = (sum + dfs(ch[x][i])) % MOD;
sum = (sum + 1ll * dp1[x] * dp2[ch[x][i]] % MOD + 1ll * dp1[ch[x][i]] * dp2[x] % MOD - 2ll * dep[x] * dp1[x] % MOD * dp1[ch[x][i]] % MOD + MOD) % MOD;
dp1[x] = (dp1[x] + dp1[ch[x][i]]) % MOD;
dp2[x] = (dp2[x] + dp2[ch[x][i]]) % MOD;
}
if(val[x] % cur == 0){
sum = (sum + 1ll * dp2[x] * phi[val[x]] % MOD - 1ll * dep[x] * dp1[x] % MOD * phi[val[x]] % MOD + MOD) % MOD;
dp1[x] = (dp1[x] + phi[val[x]]) % MOD;
dp2[x] = (dp2[x] + 1ll * phi[val[x]] * dep[x]) % MOD;
}
return sum;
}
inline void calc(int x){
cntT = 0;
for(int i = 1 ; x * i <= N ; ++i){
ch[ind[x * i]].clear();
tree[++cntT] = ind[x * i];
}
sort(tree + 1 , tree + cntT + 1 , cmp1);
for(int i = 1 ; i <= cntT ; ++i){
if(top){
int t = LCA(st[top] , tree[i]);
while(top - 1 && dep[st[top - 1]] >= dep[t]){
ch[st[top - 1]].push_back(st[top]);
--top;
}
if(t != st[top]){
ch[t].clear();
ch[t].push_back(st[top]);
st[top] = t;
}
}
st[++top] = tree[i];
}
int rt = st[1];
while(top > 1){
ch[st[top - 1]].push_back(st[top]);
--top;
}
top = 0;
ans = (ans + 1ll * q[x] * dfs(rt)) % MOD;
}
int main(){
#ifndef ONLINE_JUDGE
freopen("in","r",stdin);
//freopen("out","w",stdout);
#endif
input();
init();
init_dfs(1 , 0);
init_ST();
for(int i = 1 ; i <= N ; ++i)
for(int j = 1 ; j * i <= N ; ++j)
q[i * j] = (q[i * j] + 1ll * i * mu[j] * poww(phi[i] , MOD - 2) % MOD + MOD) % MOD;
for(cur = 1 ; cur <= N / 2 ; ++cur)
calc(cur);
cout << 2ll * ans * poww(1ll * N * (N - 1) % MOD , MOD - 2) % MOD;
return 0;
}CF809E Surprise me! 莫比乌斯反演、虚树、树形DP
标签：phi back ons name log var 莫比乌斯反演 ++i cout
原文地址：https://www.cnblogs.com/Itst/p/10284347.html
