P3412 仓鼠找sugar II

这种随机游走题用 \(DP\) 求的一般思路就是固定一个方向，然后考虑走到那个方向或从那个方向走过来的期望。
设 \(f_i\) 表示 \(i\) 走向自己的父亲的期望步数， \(g_i\) 表示 \(i\) 的父亲走向自己的期望步数。
列一下方程 （设 \(deg_u\) 表示 \(u\) 的度数）

\[f_u = 1 + \Sigma_{fa_v = u} (f_v+f_u) \times \frac{1}{deg_u} \]

解一下方程

\[f_u = deg_u + \Sigma_{fa_v=u} f_v \]

\(g_i\) 也解下方程

\[g_u = 1 + \frac{1}{deg_{fa_u}} \times (g_{fa_u} + g_u + (\Sigma_{fa_v = fa_u} f_{v} + g_u) - f_{u}) \]

\[g_u = f_{fa_u} - f_u + g_{fa_u} \]

然后计算一下每个 \(f_u\) 和 \(g_u\) 被用到的次数（\(sz_u * (n - sz_u)\)）。

#include<bits/stdc++.h>
#define RG register
#define LL long long
#define U(x, y, z) for(RG int x = y; x <= z; ++x)
#define D(x, y, z) for(RG int x = y; x >= z; --x)
#define update(x, y) (x = x + y >= mod ? x + y - mod : x + y)
using namespace std;
const int mod = 998244353;
namespace FastIO {
#define il inline
const int iL = 1 << 25;
char ibuf[iL], *iS = ibuf + iL, *iT = ibuf + iL;
#define GC() (iS == iT) ? \
  (iT = (iS = ibuf) + fread(ibuf, 1, iL, stdin), (iS == iT) ? EOF : *iS++) : *iS++
void read(){}
template<typename _Tp, typename... _Tps>
void read(_Tp &x, _Tps &...Ar) {
    x = 0; char ch = GC(); bool flg = 0;
    for (; !isdigit(ch); ch = GC()) flg |= (ch == '-');
    for (; isdigit(ch); ch = GC()) x = (x << 1) + (x << 3) + (ch ^ 48);
    if (flg) x = -x;
    read(Ar...);    
}
char Out[iL], *iter = Out;
#define Flush() fwrite(Out, 1, iter - Out, stdout); iter = Out
template <class T>il void write(T x, char LastChar = '\n') {
    int c[35], len = 0;
    if (x < 0) {*iter++ = '-'; x = -x;}
    do {c[++len] = x % 10; x /= 10;} while (x);
    while (len) *iter++ = c[len--] + '0';
    *iter++ = LastChar; Flush();
}
template <typename T> inline void writeln(T n){write(n, '\n');}
template <typename T> inline void writesp(T n){write(n, ' ');}
inline char Getchar(){ char ch; for (ch = GC(); !isalpha(ch); ch = GC()); return ch;}
inline void readstr(string &s) { s = ""; static char c = GC(); while (isspace(c)) c = GC(); while (!isspace(c)) s = s + c, c = GC();}
}
using namespace FastIO;
struct modint{
    int x;
    modint(int o=0){x=o;}
    modint &operator = (int o){return x=o,*this;}
    modint &operator +=(modint o){return x=x+o.x>=mod?x+o.x-mod:x+o.x,*this;}
    modint &operator -=(modint o){return x=x-o.x<0?x-o.x+mod:x-o.x,*this;}
    modint &operator *=(modint o){return x=1ll*x*o.x%mod,*this;}
    modint &operator ^=(int b){
        if(b<0)return x=0,*this;
        b%=mod-1;
        modint a=*this,c=1;
        for(;b;b>>=1,a*=a)if(b&1)c*=a;
        return x=c.x,*this;
    }
    modint &operator /=(modint o){return *this *=o^=mod-2;}
    modint &operator +=(int o){return x=x+o>=mod?x+o-mod:x+o,*this;}
    modint &operator -=(int o){return x=x-o<0?x-o+mod:x-o,*this;}
    modint &operator *=(int o){return x=1ll*x*o%mod,*this;}
    modint &operator /=(int o){return *this *= ((modint(o))^=mod-2);}
    template<class I>friend modint operator +(modint a,I b){return a+=b;}
    template<class I>friend modint operator -(modint a,I b){return a-=b;}
    template<class I>friend modint operator *(modint a,I b){return a*=b;}
    template<class I>friend modint operator /(modint a,I b){return a/=b;}
    friend modint operator ^(modint a,int b){return a^=b;}
    friend bool operator ==(modint a,int b){return a.x==b;}
    friend bool operator !=(modint a,int b){return a.x!=b;}
    bool operator ! () {return !x;}
    modint operator - () {return x?mod-x:0;}
};
template <typename T> inline void chkmin(T &x, T y){x = x < y ? x : y;}
template <typename T> inline void chkmax(T &x, T y){x = x > y ? x : y;}
template <typename T> inline T Min(T x, T y){return x < y ? x : y;}
template <typename T> inline T Max(T x, T y){return x > y ? x : y;}
inline void FO(string s){freopen((s + ".in").c_str(), "r", stdin); freopen((s + ".out").c_str(), "w", stdout);}

const int N = 100010;
modint f[N], g[N], ans;
int n, sz[N], deg[N];
vector<int> e[N];

inline modint Qpow(modint a, int b) {
	modint res = 1;
	for (; b; b >>= 1) {
		if (b & 1) (res *= a);
		(a *= a);
	}
	return res;
}

inline void dfs(int u, int fa) {
	sz[u] = 1;
	f[u] = deg[u];
	for (auto v: e[u]) {
		if (v ^ fa) {
			dfs(v, u);
			sz[u] += sz[v];
			f[u] += f[v];
		}
	}
}

inline void dp(int u, int fa) {
	if (u != 1) g[u] = f[fa] - f[u] + g[fa];
	// cerr << u << " " << g[u].x << "\n";
	ans += (f[u] + g[u]) * sz[u] * (n - sz[u]);
	// cerr << sz[u] << " " << n - sz[u] << " " << f[u].x << ' ' << g[u].x << " " << ans.x << "\n";
	for (auto v: e[u]) {
		if (v ^ fa) {
			dp(v, u);
		}
	}
}

int main(){
	// FO("P3412");
	read(n);
	U(i, 2, n) {
		int u, v;
		read(u, v);
		e[u].push_back(v);
		e[v].push_back(u);
		deg[u]++;
		deg[v]++;
	}

	dfs(1, 0);
	dp(1, 0);
	modint B = n;
	B = B * n;
	// cerr << ans.x << "\n";
	// cerr << B.x + mod << " " << (LL) n * n % mod << "\n";
	writeln((Qpow(B, mod - 2) * ans).x);
	return 0;
}