[生成函数][DFT][NTT] Hdu P6067 Big Integer

 题解

代码

#include <cstdio>
#include <istream>
#define ll long long
using namespace std;
const ll N=200010,mo=786433;
int T,n,m,k,x,y,sum,num[N],f[N],len,L,rev[N],v[10][N];
ll a[10][N],inv[mo],fac[N],ny[N],ans[N],p[N],P[N];
char s[N];
ll ksm(ll a,ll b) { ll r=1; for (;b;b>>=1,a=a*a%mo) if (b&1) r=r*a%mo; return r; }
void ntt(ll *a,int len,int f)
{
    for (int i=0;i<len;i++) if (rev[i]>i) swap(a[i],a[rev[i]]);
    for (int i=2;i<=len;i<<=1)
    {
        ll r,m=i>>1;
        if (f==1) r=p[i]; else r=P[i];
        for (int j=0;j<len;j+=i)
        {
            ll R=1;
            for (int k=0;k<m;k++,R=R*r%mo)
            {
                ll x=a[j+k],y=a[j+k+m]*R%mo;
                a[j+k]=(x+y)%mo,a[j+k+m]=(x-y+mo)%mo;
            }
        }
    }
    if (f==-1) { ll r=ksm(len,mo-2); for (int i=0;i<len;i++) a[i]=a[i]*r%mo; }
}
int main()
{
    
    fac[0]=ny[0]=1;
    for (int i=1;i<N;i++) p[i]=ksm(10,(mo-1)/i),P[i]=ksm(p[i],mo-2);
    for (int i=0;i<mo;i++) inv[i]=ksm(i,mo-2);
    for (int i=1;i<N;i++) fac[i]=fac[i-1]*(ll)i%mo;
    ny[N-1]=ksm(fac[N-1],mo-2);
    for (int i=N-2;i;i--) ny[i]=ny[i+1]*(ll)(i+1)%mo;
    scanf("%d",&T);
    while (T--) 
    {
        scanf("%d%d%d",&n,&m,&k),n--,sum=0;
        for (int i=0;i<n;i++) 
        {
            scanf("%s",s);
            for (int j=0;j<=m;j++) v[i][j]=s[j]-'0';
        }
        for (len=1,L=-1;len<=n*m;len*=2) L++;
        for (int i=0;i<len;i++) rev[i]=(rev[i>>1]>>1)|((i&1)<<L);
        for (int i=0;i<len;i++) num[i]=0,f[i]=1;
        for (int i=0;i<n;i++) 
        {
            for (int j=0;j<len;j++) a[i][j]=0;
            for (int j=0;j<=m;j++) if (v[i][j]) a[i][j]=ny[j];
            ntt(a[i],len,1); 
            for (int j=0;j<len;j++) if (a[i][j]) f[j]=f[j]*a[i][j]%mo; else num[j]++;
        }
        for (int i=0;i<len;i++) ans[i]=num[i]?0:f[i];
        for (int x,y;k;k--)
        {
            scanf("%d%d",&x,&y),x--;
            for (int i=0;i<len;i++) if (a[x][i]) f[i]=f[i]*inv[a[x][i]]%mo; else num[i]--;
            ll flag;
            if (v[x][y]==1) flag=-1; else flag=1;
            v[x][y]^=1;
            ll r=1,R=ksm(p[len],y);
            for (int i=0;i<len;i++,r=r*R%mo) a[x][i]=(a[x][i]+flag*inv[fac[y]]*r%mo+mo)%mo;
            for (int i=0;i<len;i++) if (a[x][i]) f[i]=f[i]*a[x][i]%mo; else num[i]++;
            for (int i=0;i<len;i++) ans[i]=(ans[i]+(num[i]?0:f[i]))%mo;
        }
        ntt(ans,len,-1);
        for (int i=1;i<len;i++) sum=(sum+ans[i]*fac[i]%mo)%mo;
        printf("%lld
",sum);
    }
}