【LOJ6363】「地底蔷薇」

其他 2019-04-13 19:22:21 阅读次数: 0
版权声明：本文为博主原创文章，未经博主允许不得转载。 https://blog.csdn.net/qq_39972971/article/details/89259527

【题目链接】

【思路要点】

  • 记 i i 个点任意图个数 a b t i abt_i 的指数型生成函数为 A b t ( x ) Abt(x) ，连通图个数 c o n i con_i 的指数型生成函数为 C o n ( x ) Con(x) ，有根连通图个数 c o n r i conr_i 的指数型生成函数为 C o n r ( x ) Conr(x) ，点双联通图个数 s t r i str_i 的指数型生成函数为 S t r ( x ) Str(x) 。
  • 那么首先有 A b t ( x ) = e C o n ( x ) , C o n ( x ) = l n ( A b t ( x ) ) , c o n r i = i × c o n i Abt(x)=e^{Con(x)},Con(x)=ln(Abt(x)),conr_i=i\times con_i
  • 其次，考虑有根连通图的圆方树，方点的生成函数应当为
    ∑ i ≥ 1 s t r i + 1 i ! C o n r ( x ) i = S t r ′ ( C o n r ( x ) ) \sum_{i\geq1}\frac{str_{i+1}}{i!}Conr(x)^i=Str'(Conr(x))
  • 因此圆点的生成函数满足 C o n r ( x ) = x e S t r ′ ( C o n r ( x ) ) Conr(x)=xe^{Str'(Conr(x))}
  • 用 C o n r ( x ) Conr(x) 的复合逆 C o n r − 1 ( x ) Conr^{-1}(x) 代替上式中的 x x ，有
    x = C o n r − 1 ( x ) e S t r ′ ( x ) x=Conr^{-1}(x)e^{Str'(x)}
    S t r ′ ( x ) = l n ( x C o n r − 1 ( x ) ) Str'(x)=ln(\frac{x}{Conr^{-1}(x)})
  • 若记 F ( x ) = l n ( C o n r ( x ) x ) F(x)=ln(\frac{Conr(x)}{x}) ，那么 S t r ′ ( x ) = F ( C o n r − 1 ( x ) ) Str'(x)=F(Conr^{-1}(x))
  • 可以用拓展形式的拉格朗日反演公式在 O ( N L o g N ) O(NLogN) 的时间内求出 S t r ′ ( x ) Str'(x) 的第 N N 项系数： [ x N ] S t r ′ ( x ) = 1 N [ x N − 1 ] F ′ ( x ) ( x C o n r ( x ) ) N [x^N]Str'(x)=\frac{1}{N}[x^{N-1}]F'(x)(\frac{x}{Conr(x)})^N
  • 令数列 i n s i = [ i ∈ S ] s t r i ins_i=[i\in S]str_i 的指数型生成函数为 I n s ( x ) Ins(x) ，答案 a n s i ans_i 的指数型生成函数为 A n s ( x ) Ans(x) ，类似地，有
    A n s ( x ) = x e I n s ′ ( A n s ( x ) ) Ans(x)=xe^{Ins'(Ans(x))}
  • 用 A n s ( x ) Ans(x) 的复合逆 A n s − 1 ( x ) Ans^{-1}(x) 代替上式中的 x x ，有
    x = A n s − 1 ( x ) e I n s ′ ( x ) x=Ans^{-1}(x)e^{Ins'(x)}
    A n s − 1 ( x ) = x e I n s ′ ( x ) Ans^{-1}(x)=\frac{x}{e^{Ins'(x)}}
  • 求出 A n s − 1 ( x ) Ans^{-1}(x) ，再使用拉格朗日反演公式求出 a n s n ans_n 即可。
  • 时间复杂度 O ( N L o g N + ∑ x i L o g   x i ) O(NLogN+\sum x_iLog\ x_i) 。

【代码】

#include<bits/stdc++.h>
using namespace std;
const int MAXN = 262144;
const int P = 998244353;
typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
template <typename T> void chkmax(T &x, T y) {x = max(x, y); }
template <typename T> void chkmin(T &x, T y) {x = min(x, y); } 
template <typename T> void read(T &x) {
	x = 0; int f = 1;
	char c = getchar();
	for (; !isdigit(c); c = getchar()) if (c == '-') f = -f;
	for (; isdigit(c); c = getchar()) x = x * 10 + c - '0';
	x *= f;
}
template <typename T> void write(T x) {
	if (x < 0) x = -x, putchar('-');
	if (x > 9) write(x / 10);
	putchar(x % 10 + '0');
}
template <typename T> void writeln(T x) {
	write(x);
	puts("");
}
namespace Poly {
	const int MAXN = 262144;
	const int P = 998244353;
	const int LOG = 25;
	const int G = 3;
	int power(int x, int y) {
		if (y == 0) return 1;
		int tmp = power(x, y / 2);
		if (y % 2 == 0) return 1ll * tmp * tmp % P;
		else return 1ll * tmp * tmp % P * x % P;
	}
	int invn[MAXN], tmpa[MAXN], tmpb[MAXN];
	int N, Log, home[MAXN]; bool initialized;
	int forward[MAXN], bckward[MAXN], inv[LOG];
	void init() {
		initialized = true;
		forward[0] = bckward[0] = inv[0] = invn[1] = 1;
		for (int len = 2, lg = 1; len <= MAXN; len <<= 1, lg++)
			inv[lg] = power(len, P - 2);
		for (int i = 2; i < MAXN; i++)
			invn[i] = (P - 1ll * (P / i) * invn[P % i] % P) % P;
		int delta = power(G, (P - 1) / MAXN);
		for (int i = 1; i < MAXN; i++)
			forward[i] = bckward[MAXN - i] = 1ll * forward[i - 1] * delta % P;
	}
	void NTTinit() {
		for (int i = 0; i < N; i++) {
			int ans = 0, tmp = i;
			for (int j = 1; j <= Log; j++) {
				ans <<= 1;
				ans += tmp & 1;
				tmp >>= 1;
			}
			home[i] = ans;
		}
	}
	void NTT(int *a, int mode) {
		assert(initialized);
		for (int i = 0; i < N; i++)
			if (home[i] < i) swap(a[i], a[home[i]]);
		int *g;
		if (mode == 1) g = forward;
		else g = bckward;
		for (int len = 2, lg = 1; len <= N; len <<= 1, lg++) {
			for (int i = 0; i < N; i += len) {
				for (int j = i, k = i + len / 2; k < i + len; j++, k++) {
					int tmp = a[j];
					int tnp = 1ll * a[k] * g[MAXN / len * (j - i)] % P;
					a[j] = (tmp + tnp > P) ? (tmp + tnp - P) : (tmp + tnp);
					a[k] = (tmp - tnp < 0) ? (tmp - tnp + P) : (tmp - tnp);
				}
			}
		}
		if (mode == -1) {
			for (int i = 0; i < N; i++)
				a[i] = 1ll * a[i] * inv[Log] % P;
		}
	}
	void times(vector <int> &a, vector <int> &b, vector <int> &c) {
		assert(a.size() >= 1), assert(b.size() >= 1);
		int goal = a.size() + b.size() - 1;
		N = 1, Log = 0;
		while (N < goal) {
			N <<= 1;
			Log++;
		}
		for (unsigned i = 0; i < a.size(); i++)
			tmpa[i] = a[i];
		for (int i = a.size(); i < N; i++)
			tmpa[i] = 0;
		for (unsigned i = 0; i < b.size(); i++)
			tmpb[i] = b[i];
		for (int i = b.size(); i < N; i++)
			tmpb[i] = 0;
		NTTinit();
		NTT(tmpa, 1);
		NTT(tmpb, 1);
		for (int i = 0; i < N; i++)
			tmpa[i] = 1ll * tmpa[i] * tmpb[i] % P;
		NTT(tmpa, -1);
		c.resize(goal);
		for (int i = 0; i < goal; i++)
			c[i] = tmpa[i];
	}
	void timesabb(vector <int> &a, vector <int> &b, vector <int> &c) {
		assert(a.size() >= 1), assert(b.size() >= 1);
		int goal = a.size() + b.size() * 2 - 2;
		N = 1, Log = 0;
		while (N < goal) {
			N <<= 1;
			Log++;
		}
		for (unsigned i = 0; i < a.size(); i++)
			tmpa[i] = a[i];
		for (int i = a.size(); i < N; i++)
			tmpa[i] = 0;
		for (unsigned i = 0; i < b.size(); i++)
			tmpb[i] = b[i];
		for (int i = b.size(); i < N; i++)
			tmpb[i] = 0;
		NTTinit();
		NTT(tmpa, 1);
		NTT(tmpb, 1);
		for (int i = 0; i < N; i++)
			tmpa[i] = 1ll * tmpa[i] * tmpb[i] % P * tmpb[i] % P;
		NTT(tmpa, -1);
		c.resize(goal);
		for (int i = 0; i < goal; i++)
			c[i] = tmpa[i];
	}
	void getinv(vector <int> &a, vector <int> &b) {
		assert(a.size() >= 1), assert(a[0] != 0);
		b.clear(), b.push_back(power(a[0], P - 2));
		while (b.size() < a.size()) {
			vector <int> c, ta = a;
			ta.resize(b.size() * 2);
			timesabb(ta, b, c);
			b.resize(b.size() * 2);
			for (unsigned i = 0; i < b.size(); i++)
				b[i] = (2ll * b[i] - c[i] + P) % P;
		}
		b.resize(a.size());
	}
	void getder(vector <int> &a, vector <int> &b) {
		assert(a.size() >= 1);
		if (a.size() == 1) {
			b.clear();
			b.resize(1);
		} else {
			b.resize(a.size() - 1);
			for (unsigned i = 0; i < b.size(); i++)
				b[i] = (i + 1ll) * a[i + 1] % P;
		}
	}
	void getint(vector <int> &a, vector <int> &b) {
		b.resize(a.size() + 1), b[0] = 0;
		for (unsigned i = 0; i < a.size(); i++)
			b[i + 1] = 1ll * invn[i + 1] * a[i] % P;
	}
	void getlog(vector <int> &a, vector <int> &b) {
		assert(a.size() >= 1), assert(a[0] == 1);
		vector <int> da, inva, db;
		getder(a, da), getinv(a, inva);
		times(da, inva, db), getint(db, b);
		b.resize(a.size());
	}
	void getexp(vector <int> &a, vector <int> &b) {
		assert(a.size() >= 1), assert(a[0] == 0);
		b.clear(), b.push_back(1);
		while (b.size() < a.size()) {
			vector <int> lnb, res;
			b.resize(b.size() * 2), getlog(b, lnb);
			for (unsigned i = 0; i < lnb.size(); i++)
				if (i == 0) lnb[i] = (P + 1 + a[i] - lnb[i]) % P;
				else if (i < a.size()) lnb[i] = (P + a[i] - lnb[i]) % P;
				else lnb[i] = (P - lnb[i]) % P;
			times(lnb, b, res);
			res.resize(b.size());
			swap(res, b);
		}
		b.resize(a.size());
	}
	void getshl(vector <int> &a, vector <int> &b, ull bits) {
		if (a.size() < bits) bits = a.size();
		b.clear(), b.resize(bits);
		for (unsigned i = 0; b.size() < a.size(); i++)
			b.push_back(a[i]);
	}
	void getshr(vector <int> &a, vector <int> &b, ull bits) {
		if (a.size() < bits) bits = a.size(); b.clear();
		for (unsigned i = bits; i < a.size(); i++)
			b.push_back(a[i]);
		b.resize(a.size());
	}
	void getpowk(vector <int> &a, vector <int> &b, int k) {
		assert(k >= 1);
		unsigned pos = a.size();
		for (unsigned i = 0; i < a.size(); i++)
			if (a[i]) {
				pos = i;
				break;
			}
		if (pos == a.size()) {
			b = a;
			return;
		}
		int val = power(a[pos], k), inv = power(a[pos], P - 2);
		vector <int> lntmp, tmp;
		getshr(a, tmp, pos);
		for (unsigned i = 0; i < tmp.size(); i++)
			tmp[i] = 1ll * tmp[i] * inv % P;
		getlog(tmp, lntmp);
		for (unsigned i = 0; i < lntmp.size(); i++)
			lntmp[i] = 1ll * lntmp[i] * k % P;
		getexp(lntmp, tmp);
		for (unsigned i = 0; i < tmp.size(); i++)
			tmp[i] = 1ll * tmp[i] * val % P;
		getshl(tmp, b, 1ull * pos * k);
	}
	int getinvfunc(vector <int> &f, unsigned n) {
		assert(f[0] == 0 && f[1] != 0);
		assert(n >= 1 && n < f.size());
		int inv = power(n, P - 2);
		vector <int> tmp;
		getshr(f, tmp, 1);
		vector <int> invf;
		getinv(tmp, invf);
		vector <int> res;
		getpowk(invf, res, n);
		return 1ll * inv * res[n - 1] % P;
	}
	int getinvfunc(vector <int> &f, vector <int> &g, unsigned n) {
		assert(f[0] == 0 && f[1] != 0 && g[0] == 0);
		assert(n >= 1 && n < f.size());
		int inv = power(n, P - 2);
		vector <int> tmp;
		getshr(f, tmp, 1);
		vector <int> invf;
		getinv(tmp, invf);
		vector <int> res;
		getpowk(invf, res, n);
		vector <int> dg;
		getder(g, dg);
		vector <int> final;
		times(res, dg, final);
		return 1ll * inv * final[n - 1] % P;
	}
}
int fac[MAXN], inv[MAXN];
int power(int x, int y) {
	if (y == 0) return 1;
	int tmp = power(x, y / 2);
	if (y % 2 == 0) return 1ll * tmp * tmp % P;
	else return 1ll * tmp * tmp % P * x % P;
}
void init(int n) {
	Poly :: init();
	fac[0] = 1;
	for (int i = 1; i <= n; i++)
		fac[i] = 1ll * fac[i - 1] * i % P;
	inv[n] = power(fac[n], P - 2);
	for (int i = n - 1; i >= 0; i--)
		inv[i] = inv[i + 1] * (i + 1ll) % P;
}
int n, m;
vector <int> abt, con, str, fun;
int getstr(int x) {
	assert(x >= 2);
	vector <int> f, g;
	for (int i = 0; i <= x - 1; i++) {
		f.push_back(con[i]);
		g.push_back(fun[i]);
	}
	return Poly :: getinvfunc(f, g, x - 1);
}
int main() {
	read(n), read(m), init(n);
	for (int i = 0; i <= n; i++)
		abt.push_back(1ll * power(2, i * (i - 1ll) / 2 % (P - 1)) * inv[i] % P);
	Poly :: getlog(abt, con);
	for (int i = 0; i <= n; i++)
		con[i] = 1ll * con[i] * i % P;
	vector <int> tmp;
	Poly :: getshr(con, tmp, 1);
	tmp.resize(n);
	Poly :: getlog(tmp, fun);
	str.resize(n + 1);
	for (int i = 1; i <= m; i++) {
		int x; read(x);
		str[x - 1] = getstr(x);
	}
	vector <int> fra, xinv, inv;
	Poly :: getexp(str, fra);
	Poly :: getinv(fra, inv);
	Poly :: getshl(inv, xinv, 1);
	printf("%lld\n", 1ll * Poly :: getinvfunc(xinv, n) * fac[n - 1] % P);
	return 0;
}

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