白兔之舞

Posted 2021-03-09 bestlxm

题意：白兔初始在((0,x))的位置，每次跳跃要求第一维单增，第二维从(u)到(v)共(w_{u,v})条边。第一维不能跳超过(L)。第二维取值在(1)到(n)之间。

设步数为(m)，询问(m;mod;k=t)时的跳跃方案对(p)取模的结果。

题解：

单位根反演的式子为：(frac{1}{n}sumlimits_{i=0}^{n-1}omega_{n}^{ik}=[n|k])

当(n=1)时，设(W)为(w_{1,1})的结果，则答案为：

(sumlimits_{i=0}^L [k|(i-t)]W^iinom{L}{i})

(=sumlimits_{i=0}^Lfrac{1}{k}sumlimits_{j=0}^{k-1}omega_k^{j(i-t)}W^iinom{L}{i})

(=frac{1}{k}sumlimits_{i=0}^LW^iinom{L}{i}sumlimits_{j=0}^{k-1}omega_k^{-jt+ij})

(=frac{1}{k}sumlimits_{i=0}^{k-1}omega_k^{-it}sumlimits_{j=0}^LW^jinom{L}{j}omega_k^{ij})

(=frac{1}{k}sumlimits_{i=0}^{k-1}omega^{-it}_k(Womega _k^i+1)^L)

设(C_i=(Womega_k^i+1)^L)

则原式(=frac{1}{k}sumlimits_{i=0}^{k-1}omega_k^{inom{i}{2}+inom{t}{2}-inom{i+t}{2}}C_i)

(=frac{1}{k}omega_k^{inom{t}{2}}sumlimits_{i=0}^{k-1}omega_k^{inom{i}{2}}C_iomega_k^{-inom{i+t}{2}})

发现为卷积的形式。可以用任意模数NTT解决。

当(n>1)时，设(Begin)为初始矩阵，（只有((1,x))位置为1），(W)为转移矩阵，

则(C_i)为(Begin imes (Womega_k^i+1)^L)这个矩阵的((1,y))位置的值。

代码：

#include <cstdio>
#include <cstring>
#include <cmath>
#include <iostream>
#include <algorithm>
#define mod1 998244353
#define mod2 1004535809
#define mod3 469762049
#define M 1000005
#define g 3
using namespace std;
int max_L,max_len,rev[M],C[M],W_k[M],n,k,L,x,y,p;
int read(){
    char c=getchar();int ans=0;bool flag=1;
    while (c<'0'||c>'9') flag&=(c!='-'),c=getchar();
    while (c>='0'&&c<='9') ans=ans*10+c-'0',c=getchar();
    return flag?ans:-ans;
}
void Write(long long x){
    if (x<10) putchar(x^48);
    else Write(x/10),putchar((x%10)^48);
    return;
}
int power(int x,int y,int mod){
    long long ans=1,now=x;
    for (int i=y;i;i>>=1,now=now*now%mod)
        if (i&1) ans=ans*now%mod;
    return ans;
}
int add(int u,int v,int mod){u+=v-mod;return u+((u>>31)&mod);}
int sub(int u,int v,int mod){u-=v;return u+((u>>31)&mod);}
int invs(int x,int mod){return x==1?1:(mod-mod/x+0ll)*invs(mod%x,mod)%mod;}
const long long inv1=invs(mod1%mod2,mod2);
const long long mod1_2=mod1*(mod2+0ll);
const long long inv2=invs(mod1_2%mod3,mod3);
int get_root(int p){
    static int S[105];int tmp=p-1;S[0]=0;
    for (register int i=2;i*i<=tmp;i++){
        if (tmp%i) continue;S[++S[0]]=i;
        while (tmp%i==0) tmp/=i;
    }
    if (tmp!=1) S[++S[0]]=tmp;
    for (register int i=2;;i++){
        bool flag=1;
        for (register int j=1;j<=S[0];j++)
            if (power(i,(p-1)/S[j],p)==1){flag=0;break;}
        if (flag) return i;
    }
    return 0;
}
struct number{
    int num1,num2,num3;
    number operator+ (const number b) const{
        return (number){add(num1,b.num1,mod1),add(num2,b.num2,mod2),add(num3,b.num3,mod3)};
    }
    number operator+= (const number &b){return *this=*this+b;}
    number operator* (const number b) const{
        return (number){(int)((num1+0ll)*b.num1%mod1),(int)((num2+0ll)*b.num2%mod2),(int)((num3+0ll)*b.num3%mod3)};
    }
    number operator*= (const number &b){return *this=*this*b;}
    number operator- (const number b) const{
        return (number){sub(num1,b.num1,mod1),sub(num2,b.num2,mod2),sub(num3,b.num3,mod3)};
    }
    number operator-= (const number &b){return *this=*this-b;}
    number operator^ (const int b) const{
        number ans=(number){1,1,1},now=*this;
        for (register int i=b;i;i>>=1,now*=now)
            if (i&1) ans*=now;
        return ans;
    }
}W[M],inv[M],A[M],B[M];
number make_number (int x){return (number){x%mod1,x%mod2,x%mod3};}
long long get_ans(number now){
    long long x4=(now.num2-now.num1+mod2)*inv1%mod2*mod1+now.num1;
    return (x4+mod1_2%p*((now.num3+mod3-x4%mod3)*inv2%mod3))%p;
}
void prepare(){
    max_len=1<<19,max_L=19;
    for (register int i=0;i<max_len;i++) rev[i]=(rev[i>>1]>>1)|((i&1)<<max_L-1);
    number wn=(number){power(g,(mod1-1)/max_len,mod1),power(g,(mod2-1)/max_len,mod2),power(g,(mod3-1)/max_len,mod3)};
    W[max_len>>1]=(number){1,1,1};inv[1]=(number){1,1,1};
    for (register int i=2;i<=max_len;i++){
        int X=(mod1-mod1/i+0ll)*inv[mod1%i].num1%mod1;
        int Y=(mod2-mod2/i+0ll)*inv[mod2%i].num2%mod2;
        int Z=(mod3-mod3/i+0ll)*inv[mod3%i].num3%mod3;
        inv[i]=(number){X,Y,Z};
    }
    for (register int i=(max_len>>1)+1;i<max_len;i++) W[i]=W[i-1]*wn;
    for (register int i=(max_len>>1)-1;i;i--) W[i]=W[i<<1];
    return;
}
void NTT(number *A,int len,int L,int flag){
    static number F[M];
    for (register int i=0,s=max_L-L;i<len;i++) F[rev[i]>>s]=A[i];
    for (register int l=1;l<len;l<<=1)
        for (register int i=0;i<len;i+=(l<<1))
            for (register int j=0;j<l;j++){
                number u=F[i|j],v=F[i|j|l]*W[j|l];
                F[i|j]=u+v,F[i|j|l]=u-v;
            }
    if (flag){
        A[0]=F[0]*inv[len];
        for (register int i=1;i<len;i++) A[i]=F[len-i]*inv[len];
    }
    else for (register int i=0;i<len;i++) A[i]=F[i];
    return;
}
struct Matrix{
    int a[3][3];
    Matrix operator* (const Matrix b) const{
        Matrix c;
        for (register int i=0;i<3;i++)
            for (register int j=0;j<3;j++)
                c.a[i][j]=(a[i][0]*(b.a[0][j]+0ll)+a[i][1]*(b.a[1][j]+0ll)+a[i][2]*(b.a[2][j]+0ll))%p;
        return c;
    }
    Matrix operator*= (const Matrix &b){return *this=*this*b;}
    Matrix operator* (const int b) const{
        Matrix c=*this;
        for (register int i=0;i<3;i++)
            for (register int j=0;j<3;j++)
                c.a[i][j]=c.a[i][j]*(b+0ll)%p;
        return c;
    }
    Matrix operator*= (const int &b){return *this=*this*b;}
    Matrix operator+ (const Matrix b) const{
        Matrix c;
        for (register int i=0;i<3;i++)
            for (register int j=0;j<3;j++) c.a[i][j]=add(a[i][j],b.a[i][j],p);
        return c;
    }
    Matrix operator+= (const Matrix &b){return *this=*this+b;}
    Matrix operator^ (const int b) const{
        Matrix ans=*this,now=*this;
        if (!b){
            ans.a[0][0]=ans.a[1][1]=ans.a[2][2]=1;
            ans.a[0][1]=ans.a[1][2]=ans.a[2][0]=0;
            ans.a[0][2]=ans.a[1][0]=ans.a[2][1]=0;
            return ans;
        }
        for (register int i=b-1;i;i>>=1,now*=now)
            if (i&1) ans*=now;
        return ans;
    }
}G,Begin,I;
void mul(number *A,number *B,int len){
    int rlen=1,L=0;while (rlen<=(len<<1)) rlen<<=1,++L;
    NTT(A,rlen,L,0),NTT(B,rlen,L,0);
    for (register int i=0;i<rlen;i++) A[i]*=B[i];
    NTT(A,rlen,L,1);
    for (register int i=len;i<rlen;i++) A[i]=make_number(0);
    return;
}
int main(){
    n=read(),k=read(),L=read(),x=read()-1,y=read()-1,p=read();
    for (register int i=0;i<n;i++)
        for (register int j=0;j<n;j++) G.a[i][j]=read();
    I.a[0][0]=I.a[1][1]=I.a[2][2]=1;
    I.a[0][1]=I.a[1][2]=I.a[2][0]=0;
    I.a[0][2]=I.a[1][0]=I.a[2][1]=0;
    Begin.a[0][x]=1;W_k[0]=1;W_k[1]=power(get_root(p),(p-1)/k,p);prepare();
    for (register int i=2;i<=k;i++) W_k[i]=W_k[i-1]*(W_k[1]+0ll)%p;
    for (register int i=0;i<k;i++) C[i]=(Begin*((G*W_k[i]+I)^L)).a[0][y];
    for (register int i=0;i<k;i++) A[i]=make_number(W_k[i*(i-1ll)/2%k]*(C[i]+0ll)%p);
    for (register int i=0;i<k+k;i++) B[k+k-1-i]=make_number(W_k[k-i*(i-1ll)/2%k]%p);
    mul(A,B,k<<1);int invk=invs(k,p);
    for (register int i=0;i<k;i++)
        Write(get_ans(A[k+k-i-1])*invk%p*W_k[i*(i-1ll)/2%k]%p),putchar('
');
    return 0;
}