• 【LOJ #2325】「清华集训 2017」小 Y 和恐怖的奴隶主(矩阵快速幂)


    传送门

    沙雕题

    打一下发现状态很少
    可以矩乘优化
    但是复杂度不对
    考虑一个行向量乘矩阵是n2n^2
    于是把2k2^k的矩阵预处理出来每次乘即可

    有些卡常

    #include<bits/stdc++.h>
    using  namespace std;
    #define re register
    #define pb push_back
    #define cs const
    #define ll long long
    #define pii pair<int,int>
    #define fi first
    #define se second
    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,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;
    }
    inline ll readl(){
    	char ch=gc();
    	ll 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;
    }
    template<class tp>inline void chemx(tp &a,tp b){a<b?a=b:0;}
    template<class tp>inline void chemn(tp &a,tp b){a>b?a=b:0;}
    cs int mod=998244353;
    cs ll lim=1ll*mod*mod*5;
    inline int add(int a,int b){return (a+=b)>=mod?a-mod:a;}
    inline int dec(int a,int b){return (a-=b)<0?a+mod:a;}
    inline int mul(int a,int b){return 1ll*a*b%mod;}
    inline void Add(int &a,int b){(a+=b)>=mod?a-=mod:0;}
    inline void Dec(int &a,int b){(a-=b)<0?a+=mod:0;}
    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 M=166;
    int all,m,k;
    ll ss[M][M];
    struct mat{
    	int a[M][M];
    	inline void clear(){memset(a,0,sizeof(a));}
    	friend inline mat operator *(cs mat &a,cs mat &b){
    		mat c;
    		memset(ss,0,sizeof(ss));
    		for(int i=0;i<=all;i++)
    		for(int k=0;k<=all;k++)
    		for(int j=0;j<=all;j++)
    		ss[i][j]+=1ll*a.a[i][k]*b.a[k][j],(ss[i][j]>=lim)&&(ss[i][j]-=lim);
    		for(int i=0;i<=all;i++)
    		for(int j=0;j<=all;j++)
    		c.a[i][j]=ss[i][j]%mod;
    	//	Add(c.a[i][j],mul(a.a[i][k],b.a[k][j]));
    		return c;
    	}
    }trans,pw[70];
    int inv[M],id[10][10],id2[10][10][10];
    int now[M],tp[M];
    inline void init_inv(){
    	inv[0]=inv[1]=1;
    	for(int i=2;i<M;i++)inv[i]=mul(mod-mod/i,inv[mod%i]);
    }
    inline void Mult(int *tp,int p){
    	static ll tmp[M];
    	memset(tmp,0,sizeof(ll)*(all+1));
    	for(int i=0;i<=all;i++)
    	for(int j=0;j<=all;j++){
    		tmp[i]+=1ll*tp[j]*pw[p].a[j][i];
    		if(tmp[i]>=lim)tmp[i]-=lim;
    	}
    //	Add(tmp[i],mul(tp[j],pw[p].a[j][i]));
    	for(int i=0;i<=all;i++)tp[i]=tmp[i]%mod;
    }
    inline void solve(){
    	ll n=readl();
    	memcpy(now,tp,sizeof(tp));
    //	for(int i=0;i<=all;i++)cout<<now[i]<<" ";puts("");
    	for(int i=60;i>=0;i--)if(n&(1ll<<i))Mult(now,i);
    	cout<<now[all]<<'
    ';
    }
    int main(){
    	#ifdef Stargazer
    	freopen("lx.cpp","r",stdin);
    //	freopen("my.out","w",stdout);
    	#endif
    	int T=read();m=read(),k=read();
    	init_inv(),trans.clear();
    	if(m==1){
    		tp[1]=1;
    		all=k+1;
    		for(int i=0;i<=k;i++){
    			trans.a[i][i]=inv[i+1],trans.a[i][all]=inv[i+1];
    			if(i)trans.a[i][i-1]=mul(i,inv[i+1]);
    		}
    		trans.a[all][all]=1;
    	}
    	else if(m==2){
    		for(int i=0;i<=k;i++)
    		for(int j=0;i+j<=k;j++)id[i][j]=all++;
    		tp[id[0][1]]=1;
    		for(int i=0;i<=k;i++)
    		for(int j=0;i+j<=k;j++){
    			int p=id[i][j];
    			Add(trans.a[p][p],inv[i+j+1]);
    			Add(trans.a[p][all],inv[i+j+1]);
    			if(i)Add(trans.a[p][id[i-1][j]],mul(i,inv[i+j+1]));
    			if(j)Add(trans.a[p][id[i+1][j-(i+j==k)]],mul(j,inv[i+j+1]));
    		}
    		trans.a[all][all]=1;
    	}
    	else if(m==3){
    		for(int i=0;i<=k;i++)
    		for(int j=0;i+j<=k;j++)
    		for(int p=0;i+j+p<=k;p++)
    		id2[i][j][p]=all++;
    		tp[id2[0][0][1]]=1;
    		for(int i=0;i<=k;i++)
    		for(int j=0;i+j<=k;j++)
    		for(int p=0;i+j+p<=k;p++){
    			int id=id2[i][j][p];
    			Add(trans.a[id][id],inv[i+j+p+1]);
    			Add(trans.a[id][all],inv[i+j+p+1]);
    			if(i)Add(trans.a[id][id2[i-1][j][p]],mul(inv[i+j+p+1],i));
    			if(j)Add(trans.a[id][id2[i+1][j-1][p+(i+j+p<k)]],mul(inv[i+j+p+1],j));
    			if(k)Add(trans.a[id][id2[i][j+1][p-(i+j+p==k)]],mul(inv[i+j+p+1],p));
    		}
    		trans.a[all][all]=1;
    	}
    	pw[0]=trans;
    	for(int i=1;i<=60;i++)pw[i]=pw[i-1]*pw[i-1];
    	while(T--)solve();
    }
    
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  • 原文地址:https://www.cnblogs.com/stargazer-cyk/p/12328377.html
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