【LOJ #2541】「PKUWC2018」猎人杀（容斥+分治NTT）

传送门

能发现对于一个死的人

计算的时候不忽略他的概率而看做遇到他继续射下一个不会影响答案

考虑活、死的人的概率之和为 w 1 , w 2 w1,w2 w1,w2

那么下一次射杀 i i i的概率为 w i w 1 \frac{w_i}{w1} w1wi​​

如果把死的人也算进去

那么 x = w 2 w 1 + w 2 x + w i w 1 + w 2 x=\frac{w2}{w1+w2}x+\frac{w_i}{w1+w2} x=w1+w2w2​x+w1+w2wi​​

x = w i w 1 x=\frac{w_i}{w1} x=w1wi​​

仍然相同

考虑容斥

强行令某些人在 1 1 1之后死

那么 a n s = ∑ S ( − 1 ) ∣ S ∣ ∑ i = 0 ∞ ( 1 − w s + w 1 W ) i w 1 W = ∑ S ( − 1 ) ∣ S ∣ w 1 w 1 + w s ans=\sum_{S}(-1)^{|S|}\sum_{i=0}^{\infty}(1-\frac{w_s+w_1}{W})^i\frac{w_1}{W}=\sum_{S}(-1)^{|S|}\frac{w_1}{w_1+w_s} ans=∑S​(−1)∣S∣∑i=0∞​(1−Wws​+w1​​)iWw1​​=∑S​(−1)∣S∣w1​+ws​w1​​

考虑 ∑ w \sum w ∑w很小

对 i i i构建生成函数 ( 1 − x w i ) (1-x^{w_i}) (1−xwi​)

乘起来即可

#include<bits/stdc++.h>
using namespace std;
#define re register
#define cs const
#define pb push_back
#define ll long long
#define pii pair<int,int>
#define fi first
#define se second
#define bg begin
#define poly vector<int>
cs int RLEN=1<<20|1;
inline char gc(){
	static char ibuf[RLEN],*ib,*ob;
	(ib==ob)&&(ob=(ib=ibuf)+fread(ibuf,1,RLEN,stdin));
	return ib==ob?EOF:*ib++;
}
inline int read(){
	char ch=gc();
	int res=0;bool f=1;
	while(!isdigit(ch))f^=ch=='-',ch=gc();
	while(isdigit(ch))res=(res+(res<<2)<<1)+(ch^48),ch=gc();
	return f?res:-res;
}
cs int mod=998244353,G=3;
inline int add(int a,int b){a+=b-mod;return a+(a>>31&mod);}
inline int dec(int a,int b){a-=b;return a+(a>>31&mod);}
inline int mul(int a,int b){return 1ll*a*b%mod;}
inline void Add(int &a,int b){a+=b-mod,a+=a>>31&mod;}
inline void Dec(int &a,int b){a-=b,a+=a>>31&mod;}
inline void Mul(int &a,int b){a=1ll*a*b%mod;}
inline int ksm(int a,int b,int res=1){for(;b;b>>=1,Mul(a,a))(b&1)&&(Mul(res,a),1);return res;}
inline int Inv(int x){return ksm(x,mod-2);}
cs int C=18;
poly w[C+1];
int rev[(1<<C)|5];
inline void init_w(){
	for(int i=1;i<=C;i++)w[i].resize(1<<(i-1));
	int wn=ksm(G,(mod-1)/(1<<C));
	w[C][0]=1;
	for(int i=1;i<(1<<(C-1));i++)w[C][i]=mul(w[C][i-1],wn);
	for(int i=C-1;i;i--)
	for(int j=0;j<(1<<(i-1));j++)w[i][j]=w[i+1][j<<1];
}
inline void init_rev(int lim){
	for(int i=0;i<lim;i++)rev[i]=(rev[i>>1]>>1)|((i&1)*(lim>>1));
}
inline void ntt(poly &f,int lim,int kd){
	for(int i=0;i<lim;i++)if(i>rev[i])swap(f[i],f[rev[i]]);
	for(int mid=1,l=1,a0,a1;mid<lim;mid<<=1,l++)
	for(int i=0;i<lim;i+=(mid<<1))
	for(int j=0;j<mid;j++)
	a0=f[i+j],a1=mul(f[i+j+mid],w[l][j]),f[i+j]=add(a0,a1),f[i+j+mid]=dec(a0,a1);
	if(kd==-1){
		reverse(f.bg()+1,f.bg()+lim);
		for(int i=0,iv=Inv(lim);i<lim;i++)Mul(f[i],iv);
	}
}
inline poly operator *(poly a,poly b){
	int deg=a.size()+b.size()-1,lim=1;
	if(deg<=32){
		poly c(deg,0);
		for(int i=0;i<a.size();i++)
		for(int j=0;j<b.size();j++)
		Add(c[i+j],mul(a[i],b[j]));
		return c;
	}
	while(lim<deg)lim<<=1;
	init_rev(lim);
	a.resize(lim),ntt(a,lim,1);
	b.resize(lim),ntt(b,lim,1);
	for(int i=0;i<lim;i++)Mul(a[i],b[i]);
	ntt(a,lim,-1),a.resize(deg);
	return a;
}
cs int N=100005;
int n,W[N],inv[N];
inline void init_inv(){
	inv[0]=inv[1]=1;
	for(int i=2;i<N;i++)inv[i]=mul(mod-mod/i,inv[mod%i]);
}
#define lc (u<<1)
#define rc ((u<<1)|1)
#define mid ((l+r)>>1)
poly f[N<<2];
void build(int u,int l,int r){
	if(l==r){f[u].resize(W[l]+1),f[u][0]=1,f[u][W[l]]=mod-1;return;}
	build(lc,l,mid),build(rc,mid+1,r);
	f[u]=f[lc]*f[rc];
}
#undef lc 
#undef rc
#undef mid
int main(){
	#ifdef Stargazer
	freopen("lx.cpp","r",stdin);
	#endif
	n=read(),init_w(),init_inv();int w1=read();
	for(int i=1;i<n;i++)W[i]=read();
	build(1,1,n-1);
	poly now=f[1];
	int res=0;
	for(int i=0;i<now.size();i++)Add(res,mul(now[i],mul(w1,inv[i+w1])));
	cout<<res;
}