Luogu4191

前言

性能优化……经典题。

这题可以看到它给你暴力算了循环卷积，于是想到 FFT 单位根处点值乘法对应循环卷积的本质。

可是 FFT 长度是 \(2^n\) 的，会挂掉。

于是我们可以考虑 Bluestein 算法。

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Bluestein 算法

前置知识：任意模数 Chirp Z-Transform，确保你能过板以免被卡常。

以下设 \(\omega_n\) 为 \(n\) 次本原单位根。

设一个长度为 \(n\) 的数列 \(\{f_n\}\) 的循环卷积点值数列 \(\{g_n\}\) 为

\[g_t=\sum_{k=0}^{n-1}f_k\omega_n^{kt} \]

由单位根反演，有

\[f_t=\frac1n\sum_{k=0}^{n-1}g_k\omega_n^{-kt} \]

以上两项均可用 CZT 加速。

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回到本题

由于单位根点值点乘本质即数列循环卷积，这题变得可做。

把 \(a,b\) 点值搞一下，点值按要求乘起来，回演系数，输出，做完了？！

于是你写，交上去，发现被卡常了……

卡卡常就过了。

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Code

\(c\) 能对 \(n\) 取模是因为在点值计算时能使用费马小定理。

// Problem: P4191 [CTSC2010]性能优化
// Contest: Luogu
// URL: https://www.luogu.com.cn/problem/P4191
// Memory Limit: 250 MB
// Time Limit: 6000 ms

#include <algorithm>
#include <math.h>
#include <stdio.h>
#include <vector>
typedef long long llt;
typedef unsigned uint;typedef unsigned long long ullt;
typedef bool bol;typedef char chr;typedef void voi;
typedef double dbl;
template<typename T>bol _max(T&a,T b){return(a<b)?a=b,true:false;}
template<typename T>bol _min(T&a,T b){return(b<a)?a=b,true:false;}
template<typename T>T power(T base,T index,T mod){return((index<=1)?(index?base:1):(power(base*base%mod,index>>1,mod)*power(base,index&1,mod)))%mod;}
template<typename T>T lowbit(T n){return n&-n;}
template<typename T>T gcd(T a,T b){return b?gcd(b,a%b):a;}
template<typename T>T lcm(T a,T b){return(a!=0||b!=0)?a/gcd(a,b)*b:(T)0;}
template<typename T>T exgcd(T a,T b,T&x,T&y){if(!b)return y=0,x=1,a;T ans=exgcd(b,a%b,y,x);y-=a/b*x;return ans;}
const dbl Pi=acos(-1);
class cpx
{
    public:
        dbl a,b;
        cpx():a(0),b(0){}
        cpx(dbl a):a(a),b(0){}
        cpx(dbl a,dbl b):a(a),b(b){}
        voi unit(dbl alpha){a=cos(alpha),b=sin(alpha);}
        cpx friend operator+(cpx a,cpx b){return cpx(a.a+b.a,a.b+b.b);}
        cpx friend operator-(cpx a,cpx b){return cpx(a.a-b.a,a.b-b.b);}
        cpx operator-(){return cpx(-a,-b);}
        cpx friend operator*(cpx a,cpx b){return cpx(a.a*b.a-a.b*b.b,a.b*b.a+b.b*a.a);}
        cpx friend operator/(cpx a,ullt v){return cpx(a.a/v,a.b/v);}
        cpx conj(){return cpx(a,-b);}
        cpx mul_i(){return cpx(-b,a);}
        cpx div_i(){return cpx(b,-a);}
    public:
        cpx&operator=(ullt v){return a=v,b=0,*this;}
        cpx&operator+=(cpx v){return*this=*this+v;}
        cpx&operator-=(cpx v){return*this=*this-v;}
        cpx&operator*=(cpx v){return*this=*this*v;}
        cpx&operator/=(ullt v){return a/=v,b/=v,*this;}
        dbl&real(){return a;}
        dbl&imag(){return b;}
};
ullt Mod;
ullt chg(ullt v){return(v<Mod)?v:v-Mod;}
class poly
{
    private:
        std::vector<ullt>V;
    public:
        class FFT
        {
            private:
                std::vector<uint>V;std::vector<cpx>G;uint len;
            public:
                uint length(){return len;}
                voi bzr(uint length)
                {
                    uint p=0;len=1,V.clear(),G.clear();
                    while(length){p++,len<<=1,length>>=1;}
                    V.resize(len),G.resize(len);
                    for(uint i=0;i<len;++i)V[i]=((i&1)?(V[i>>1]|len)>>1:(V[i>>1]>>1)),G[i].unit(Pi*2/len*i);
                }
                voi fft(std::vector<cpx>&y,bol op)
                {
                    if(y.size()<len)y.resize(len);
                    for(uint i=0;i<len;i++)if(V[i]<i)std::swap(y[i],y[V[i]]);
                    for(uint h=2;h<=len;h<<=1)for(uint j=0;j<len;j+=h)for(uint k=j;k<j+(h>>1);k++){cpx u=y[k],t=G[len/h*(k-j)]*y[k+h/2];y[k]=u+t,y[k+h/2]=u-t;}
                    if(op){uint l=1,r=len-1;while(l<r)std::swap(y[l++],y[r--]);for(uint i=0;i<len;i++)y[i]/=len;}
                }
                voi fft_fft(std::vector<cpx>&a,std::vector<cpx>&b,bol op)
                {
                    if(a.size()<len)a.resize(len);
                    if(b.size()<len)b.resize(len);
                    for(uint i=0;i<len;i++)a[i]+=b[i].mul_i();
                    fft(a,op),b[0]=a[0].conj();for(uint i=1;i<len;i++)b[i]=a[len-i].conj();
                    for(uint i=0;i<len;i++){cpx p=a[i],q=b[i];a[i]=(p+q)/2llu,b[i]=(p-q).div_i()/2llu;}
                }
        };
    public:
        poly(){V.clear();}
        poly(std::vector<ullt>V){for(uint i=0;i<V.size();i++)push(V[i]%Mod);cut_zero();}
        bol empty(){return cut_zero(),!size();}
        voi bzr(){V.clear();}
        voi push(ullt v){V.push_back(v%Mod);}
        voi pop(){V.pop_back();}
        ullt val(uint n){return(n<V.size())?V[n]:0;}
        uint deg(){return V.size()-1;}
        uint size(){return V.size();}
        voi add(uint p,ullt v)
        {
            if(deg()<p)chg_deg(p);
            V[p]=(V[p]+v)%Mod;
        }
        poly friend operator+(poly a,ullt v){a.add(0,v);return a;}
        poly friend operator+(poly a,poly b)
        {
            uint len=std::max(a.size(),b.size());
            a.chg_siz(len),b.chg_siz(len);
            for(uint i=0;i<len;i++)a[i]=chg(a[i]+b[i]);
            a.cut_zero();
            return a;
        }
        poly friend operator-(poly a,poly b)
        {
            uint len=std::max(a.size(),b.size());
            a.chg_siz(len),b.chg_siz(len);
            for(uint i=0;i<len;i++)a[i]=chg(a[i]+Mod-b[i]);
            a.cut_zero();
            return a;
        }
        poly operator-()
        {
            cut_zero();uint len=size();
            poly ans;ans.chg_siz(len);
            for(uint i=0;i<len;i++)ans[i]=chg(Mod-V[i]);
            return ans;
        }
        poly friend operator*(poly a,poly b)
        {
            FFT s;poly p;
            uint n=a.deg(),m=b.deg(),len;
            s.bzr(n+m+1),len=s.length();
            std::vector<cpx>v1(len),v2(len),v3(len),v4(len);
            for(uint i=0;i<len;i++)v3[i]=cpx(a.val(i)&32767),v1[i]=cpx(a.val(i)>>15),v4[i]=cpx(b.val(i)&32767),v2[i]=cpx(b.val(i)>>15);
            s.fft_fft(v1,v2,0),s.fft_fft(v3,v4,0);
            for(uint i=0;i<len;i++)v4[i]=(v3[i]+v1[i].mul_i())*v4[i],v2[i]=(v3[i]+v1[i].mul_i())*v2[i];
            s.fft(v2,1),s.fft(v4,1),p.chg_deg(n+m);for(uint i=0;i<=n+m;i++)p[i]=(((ullt)(v2[i].b+.5)%Mod<<30)+((ullt)(v2[i].a+v4[i].b+.5)%Mod<<15)+(ullt)(v4[i].a+.5))%Mod;
            p.cut_zero();
            return p;
        }
        poly inv(){return inv(size());}
        poly inv(uint prec)
        {
            poly ans,f,tmp,w;
            llt x,y;
            exgcd<llt>(val(0),Mod,x,y);
            ans.push(x%(llt)Mod+(llt)Mod),f.push(val(0));
            for(uint k=1;k<prec;k<<=1)
            {
                for(uint i=k;i<(k<<1);++i)f.push(val(i));
                tmp=f*ans,tmp.chg_siz(k<<1),w.bzr();for(uint i=0;i<k;++i)w.push(tmp[i+k]);
                w*=ans;for(uint i=0;i<k;++i)ans.push(Mod-w[i]);
            }
            return ans;
        }
        poly diff(){uint n=size();poly ans;for(uint i=1;i<n;++i)ans.push(V[i]*i);return ans;}
        poly inte()
        {
            uint n=size();
            poly ans;
            ans.chg_deg(n);
            ullt k=1;llt x,y;
            std::vector<ullt>W;W.push_back(1),W.push_back(1);
            for(uint i=2;i<n;++i)W.push_back(k=(k*i)%Mod);
            exgcd<llt>(k*n%Mod,Mod,x,y);
            k=chg(x%(llt)Mod+(llt)Mod);
            for(uint i=n;i;--i)ans[i]=V[i-1]*k%Mod*W[i-1]%Mod,k=k*i%Mod;
            return ans;
        }
        poly ln(){return(this->diff()*this->inv()).inte().chg_deg_ed(deg());}
        poly exp(){return exp(size());}
        poly exp(uint prec)
        {
            poly m;m.push(1);
            if(empty())return m;
            uint tp=1;
            while(tp<prec)m*=*this-(m.diff()*m.inv(tp<<=1)).inte()+1,m.chg_siz(tp);
            m.chg_siz(prec);
            return m;
        }
        poly reverse(){poly ans;for(uint i=deg();~i;--i)ans.push(V[i]);return ans;}
        poly operator/(poly b)
        {
            cut_zero(),b.cut_zero();uint m=size(),n=b.deg();if(m<=n)return poly();
            poly f=this->reverse()*b.reverse().inv(m-n);f.chg_siz((m>n)?m-n:0);return f.reverse();
        }
        poly operator%(poly b){poly f=*this-*this/b*b;f.cut_zero();return f;}
        voi cut_zero(){while(!V.empty()&&!V.back())V.pop_back();}
        voi chg_siz(const uint siz){while(V.size()<siz)V.push_back(0);while(V.size()>siz)V.pop_back();}
        voi chg_deg(const uint d){chg_siz(d+1);}
        poly chg_deg_ed(const uint d){poly ans=*this;return ans.chg_deg(d),ans;}
    public:
        ullt&operator[](uint num){return V[num];}
        poly&operator=(std::vector<ullt>V){bzr();for(uint i=0;i<V.size();i++)push(V[i]%Mod);cut_zero();return*this;}
        poly&operator=(std::vector<cpx>V){bzr();for(uint i=0;i<V.size();i++)push((llt)(V[i].a+.5)%(llt)Mod+(llt)(Mod));cut_zero();return*this;}
        poly&operator+=(poly b){return*this=*this+b;}
        poly&operator-=(poly b){return*this=*this-b;}
        poly&operator*=(poly b){return*this=*this*b;}
        poly&operator/=(poly b){return*this=*this/b;}
        poly&operator%=(poly b){return*this=*this%b;}
};
ullt gotg()
{
    static ullt Fac[15];uint cnt=0;
    ullt v=Mod-1;
    for(ullt i=2;i*i<=v;i++)
        if(!(v%i))
        {
            Fac[cnt++]=i;
            do v/=i;while(!(v%i));
        }
    if(v>1)Fac[cnt++]=v;
    for(ullt ans=2;;ans++)if(power<ullt>(ans,Mod-1,Mod)==1)
    {
        bol b=true;
        for(uint i=0;b&&i<cnt;i++)if(power<ullt>(ans,(Mod-1)/Fac[i],Mod)==1)b=false;
        if(b)return ans;
    }
    return 0;
}
ullt A[500005],B[500005];
uint g_pow[1000005];
uint g_binom_pow[1000005];
uint inv_pow[1000005];
uint inv_binom_pow[1000005];
poly P1,P2;
int main()
{
	uint n;ullt c;scanf("%u%llu",&n,&c),Mod=n+1,c%=n;
	ullt g=gotg();ullt inv=power(g,Mod-2,Mod);
	g_pow[0]=inv_pow[0]=g_binom_pow[0]=inv_binom_pow[0]=1;
	for(uint i=1;i<=n*2;i++)
	{
		g_pow[i]=(ullt)g_pow[i-1]*g%Mod,inv_pow[i]=(ullt)inv_pow[i-1]*inv%Mod,
		g_binom_pow[i]=(ullt)g_binom_pow[i-1]*g_pow[i-1]%Mod,
		inv_binom_pow[i]=(ullt)inv_binom_pow[i-1]*inv_pow[i-1]%Mod;
	}
	for(uint i=0;i<n;i++)scanf("%llu",A+i),A[i]%=Mod;
	for(uint i=0;i<n;i++)scanf("%llu",B+i),B[i]%=Mod;
	P1.chg_siz(n<<1),P2.chg_siz(n);
	for(uint i=0;i<(n<<1);i++)P1[i]=g_binom_pow[i];
	for(uint i=0;i<n;i++)
		P2[i]=A[i]*inv_binom_pow[i]%Mod;
	P2=P2.reverse()*P1;
	for(uint i=0;i<n;i++)
		A[i]=P2.val(n+i-1)*inv_binom_pow[i]%Mod;
	P2.chg_siz(n);
	for(uint i=0;i<n;i++)
		P2[i]=B[i]*inv_binom_pow[i]%Mod;
	P2=P2.reverse()*P1;
	for(uint i=0;i<n;i++)
		A[i]=A[i]*power(P2.val(n+i-1)*inv_binom_pow[i]%Mod,c,Mod)%Mod;
	for(uint i=0;i<(n<<1);i++)P1[i]=inv_binom_pow[i];
	P2.chg_siz(n);
	for(uint i=0;i<n;i++)P2[i]=A[i]*g_binom_pow[i]%Mod;
	P2=P2.reverse()*P1;
	for(uint i=0;i<n;i++)printf("%llu\n",P2.val(n+i-1)*(Mod-g_binom_pow[i])%Mod);
    return 0;
}