CF809E Surprise me!

题面

题解

这道题目的话，推式子比较休闲，写起来。。。

首先上套路，根据\(\varphi\)的一些性质，我们可以证明\(\varphi(ij) = \frac{\varphi(i)\varphi(j)\gcd(i, j)}{\varphi(\gcd(i,j))}\)。

开推：首先设\(\textbf{f}(x) = \frac x {\varphi(x)}, \textbf g = \textbf f * \mu, p_{a_i} = i\)，

那么有（我的这种推法是学的rqy的比较简单的，建议参阅）：

\[\begin{aligned} &\sum_{i=1}^n\sum_{j=1}^n\varphi(a_i*a_j)\cdot \mathrm{dist}(i,j) \\ =& \sum_{i=1}^n\sum_{j=1}^n\varphi(i)\varphi(j)\mathrm{dist}(p_i, p_j)\textbf f(\gcd(i,j)) \\ =& \sum_{i=1}^n\sum_{j=1}^n\varphi(i)\varphi(j)\mathrm{dist}(p_i, p_j)\sum_{d\mid i, d\mid j} \textbf g(d) \\ =& \sum_{d=1}^n\textbf g(d)\sum_{d\mid i}\sum_{d\mid j}\varphi(i)\varphi(j)\mathrm{dist}(p_{i},p_{j}) \end{aligned} \]

\(\textbf g\)可以枚举倍数求，复杂度调和级数。

然后后面那一坨可以枚举倍数，将所有是\(d\)的倍数的点全部拉出来建一棵虚树，做一遍树形\(\mathrm{dp}\)就可以了。

虚树的总点数在\(\mathrm{O}(n\log n)\)级别，所以总复杂度约为\(\mathrm{O}(n\log^2n)\)

代码

#include<cstdio>
#include<cstring>
#include<cctype>
#include<algorithm>
#define RG register
#define file(x) freopen(#x".in", "r", stdin), freopen(#x".out", "w", stdout)
#define clear(x, y) memset(x, y, sizeof(x))

inline int read()
{
	int data = 0, w = 1; char ch = getchar();
	while(ch != '-' && (!isdigit(ch))) ch = getchar();
	if(ch == '-') w = -1, ch = getchar();
	while(isdigit(ch)) data = data * 10 + (ch ^ 48), ch = getchar();
	return data * w;
}

const int maxn(2e5 + 10), Mod(1e9 + 7);
int fastpow(int x, int y)
{
	int ans = 1;
	for(; y; y >>= 1, x = 1ll * x * x % Mod)
		if(y & 1) ans = 1ll * ans * x % Mod;
	return ans;
}

namespace Numbers
{
	int prime[maxn], cnt, mu[maxn], phi[maxn], not_prime[maxn], iphi[maxn];
	void Init(int n)
	{
		mu[1] = phi[1] = not_prime[1] = 1;
		for(RG int i = 2; i <= n; i++)
		{
			if(!not_prime[i]) prime[++cnt] = i, phi[i] = i - 1, mu[i] = Mod - 1;
			for(RG int j = 1; j <= cnt && i * prime[j] <= n; j++)
			{
				not_prime[i * prime[j]] = 1;
				if(i % prime[j]) mu[i * prime[j]] = Mod - mu[i],
					phi[i * prime[j]] = phi[i] * phi[prime[j]];
				else { phi[i * prime[j]] = phi[i] * prime[j]; break; }
			}
		}
		for(RG int i = 1; i <= n; i++)
			iphi[i] = fastpow(phi[i], Mod - 2);
	}
}

namespace Vtree
{
	struct edge { int next, to, dis; } e[maxn << 1];
	int head[maxn], e_num, fa[maxn], size[maxn], heavy[maxn], stk[maxn << 1];
	int pos[maxn], end_pos[maxn], belong[maxn], cnt, p[maxn << 1], K, dep[maxn];
	inline void add_edge(int from, int to, int dis = 0)
	{
		e[++e_num] = (edge) {head[from], to, dis};
		head[from] = e_num;
	}

	void dfs(int x)
	{
		size[x] = 1;
		for(RG int i = head[x]; i; i = e[i].next)
		{
			int to = e[i].to; if(to == fa[x]) continue;
			fa[to] = x, dep[to] = dep[x] + 1; dfs(to); size[x] += size[to];
			if(size[heavy[x]] < size[to]) heavy[x] = to;
		}
	}

	void dfs(int x, int chain)
	{
		belong[x] = chain, pos[x] = ++cnt;
		if(heavy[x]) dfs(heavy[x], chain);
		for(RG int i = head[x]; i; i = e[i].next)
		{
			int to = e[i].to;
			if(to == fa[x] || to == heavy[x]) continue;
			dfs(to, to);
		}
		end_pos[x] = cnt;
	}

	int LCA(int x, int y)
	{
		for(; belong[x] != belong[y]; x = fa[belong[x]])
			if(pos[belong[x]] < pos[belong[y]]) std::swap(x, y);
		return pos[x] < pos[y] ? x : y;
	}

	int Dis(int x, int y) { return dep[x] + dep[y] - 2 * dep[LCA(x, y)]; }
	inline bool cmp(int x, int y) { return pos[x] < pos[y]; }
	void build()
	{
		e_num = 0; std::sort(p + 1, p + K + 1, cmp);
		for(RG int i = K; i > 1; i--) p[++K] = LCA(p[i], p[i - 1]);
		p[++K] = 1; std::sort(p + 1, p + K + 1, cmp);
		K = std::unique(p + 1, p + K + 1) - p - 1;
		for(RG int i = 1, top = 0; i <= K; i++)
		{
			while(top && end_pos[stk[top]] < pos[p[i]]) --top;
			add_edge(stk[top], p[i], Dis(stk[top], p[i])); stk[++top] = p[i];
		}
	}
}

int f[maxn], g[maxn], s[maxn], a[maxn], b[maxn], c[maxn];
void dfs(int x)
{
	f[x] = g[x] = 0, s[x] = b[x];
	for(RG int i = Vtree::head[x]; i; i = Vtree::e[i].next)
	{
		int to = Vtree::e[i].to, dis = Vtree::e[i].dis; dfs(to);
		g[x] = (((g[x] + g[to]) % Mod + 1ll * s[to] * (f[x] +
			1ll * s[x] * dis % Mod) % Mod) % Mod
				+ 1ll * f[to] * s[x] % Mod) % Mod;
		s[x] = (s[x] + s[to]) % Mod;
		f[x] = ((f[x] + f[to]) % Mod + 1ll * s[to] * dis % Mod) % Mod;
	}
	Vtree::head[x] = 0, b[x] = 0;
}

int n;
int main()
{
	n = read();
	for(RG int i = 1; i <= n; i++) a[read()] = i;
	for(RG int i = 1, x, y; i < n; i++)
		x = read(), y = read(), Vtree::add_edge(x, y), Vtree::add_edge(y, x);
	Vtree::dfs(1); Vtree::dfs(1, 1); memset(Vtree::head, 0, sizeof Vtree::head);
	Numbers::Init(n); using Numbers::mu; using Numbers::iphi;
	for(RG int d = 1; d <= n; d++)
		for(RG int i = d; i <= n; i += d)
			c[i] = (c[i] + 1ll * d * mu[i / d] % Mod * iphi[d] % Mod) % Mod;
	int ans = 0;
	for(RG int i = 1; i <= n; i++)
	{
		Vtree::K = 0;
		for(RG int j = i; j <= n; j += i)
			Vtree::p[++Vtree::K] = a[j],
			b[a[j]] = Numbers::phi[j];
		Vtree::build(); dfs(1);
		ans = (ans + 2ll * c[i] * g[1] % Mod) % Mod;
	}
	ans = 1ll * ans * fastpow(n, Mod - 2) % Mod * fastpow(n - 1, Mod - 2) % Mod;
	printf("%d\n", ans);
	return 0;
}