[Solution] SDOI2015 title sequence statistics

[Solution] SDOI2015 title sequence statistics

YYB from AFO never recommended

Here is the product, more trouble, but due to the given sequence number is a small number of film quality, it is possible to \ (O (m ^ 2 \ log m) \) find the original with (not actually covering).

Little trouble product, converted into an addition like, and then take the discrete logarithm \ (a_i \ equiv C_i G ^ {} \ MOD m \) , each element is now used in place of the original root index, the problem is transformed into how many kinds of programs such that the product of each element is equal to \ (\ log X \ MOD m \) .

Constructed directly meaning of the questions
\ [F (x) = \
sum [\ exist \ log a_i = i] x ^ i \] The answer is
\ [F (x) ^ n
\] do?

In fact, to magic change it, the process of multiplication to be continuously let \ (F (x) \) is greater than \ (x ^ m \) coefficient count to \ (x ^ {i \ mod m} \) , the obvious reason ,slightly.

Not so perfunctory, and that time will be at a loss to see their own reasons, to take the film is that we use a combination of polynomial multiplication of meaning, meaning now is to get the combination \ (c_i = \ sum_ \ limits {k + j \ equiv i \} a_kb_j MOD m \) , it is so.

If it wants to continue to let \ (F (x) \) to \ (m x ^ \) to take the film, so the \ (\ ln, \ exp \ ) waste

Direct write two \ (\ log \) fast power run fast

//@winlere
#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<vector>

using namespace std;  typedef long long ll;
inline int qr(){
      register int ret=0,f=0;
      register char c=getchar();
      while(c<48||c>57)f|=c==45,c=getchar();
      while(c>=48&&c<=57) ret=ret*10+c-48,c=getchar();
      return f?-ret:ret;
}

 

int n,m,x,s,g;
vector < int > ve;
const int maxn=8009;
int lg[maxn];
int read[1<<19|1];
int ans[1<<19|1];


namespace poly{
      const int maxn=1<<19|1;
      int a[maxn],b[maxn],r[maxn];
      int savlen;
      inline void getr(const int&len){
        if(len==savlen)return;
        int cnt=0;
        for(register int t=1;t<len;t<<=1)++cnt;
        for(register int t=1;t<len;++t)
          r[t]=r[t>>1]>>1|(t&1)<<cnt>>1;
      }
      const int mod=1004535809;
      const int g=3;
      inline int ksm(int base,ll p){
        register int ret=1;
        for(base%=mod;p;p>>=1,base=1ll*base*base%mod)
          if(p&1) ret=1ll*ret*base%mod;
        return ret;
      }
      const int gi=ksm(3,mod-2);
      
      
      inline void NTT(int*a,const int&len,const int&tag){
        getr(len);
        for(register int t=1;t<len;++t)
          if(r[t]>t) swap(a[t],a[r[t]]);
        int *a1,*a0,s=g;
        if(tag!=1) s=gi;
        for(register int t=1,wn;t<len;t<<=1){
          wn=ksm(s,(mod-1)/(t<<1));
          for(register int i=0;i<len;i+=t<<1){
            a1=(a0=a+i)+t;
            for(register int j=0,w=1,tm;j<t;++j,++a1,++a0,w=1ll*w*wn%mod){
                  tm=1ll**a1*w%mod;
                  *a1=(*a0-tm)%mod;
                  *a0=(*a0+tm)%mod;
                  if(*a1<0)*a1+=mod;
            }
          }
        }
        if(tag!=1)
          for(register int t=0,in=ksm(len,mod-2);t<len;++t)
            a[t]=1ll*a[t]*in%mod;
      }
      
      
      inline void print(int*a,int len){
        for(register int t=0;t<len;++t)
          printf("%d ",a[t]);
        putchar('\n');
      }


      inline void KSM(int*a,int*b,const int&len,int p){
        static int ret[maxn],base[maxn];
        memset(ret,0,sizeof ret);
        memset(base,0,sizeof base);
        ret[0]=1;
        for(register int t=0;t<len;++t) base[t]=a[t];
        while(p){
          NTT(base,len<<1,1);
          if(p&1){
            NTT(ret,len<<1,1);
            for(register int t=0;t<len<<1;++t) ret[t]=1ll*ret[t]*base[t]%mod;
            NTT(ret,len<<1,-1);
            for(register int t=(len<<1)-1;t-m+1>=0;--t) ret[t-m+1]=(ret[t-m+1]+ret[t])%mod,ret[t]=0;
          }
          for(register int t=0;t<len<<1;++t) base[t]=1ll*base[t]*base[t]%mod;
          NTT(base,len<<1,-1);
          for(register int t=(len<<1)-1;t-m+1>=0;--t) base[t-m+1]=(base[t-m+1]+base[t])%mod,base[t]=0;
          p>>=1;
        }
        for(int t=0;t<len;++t) b[t]=ret[t];
      }      
}


inline int ksm(const int&base,const int&p,const int&mod=m){
      register int ret=1;
      for(register int t=p,b=base%mod;t;t>>=1,b=1ll*b*b%mod)
        if(t&1) ret=1ll*ret*b%mod;
      return ret;
}

inline void findg(){
#define mod m
      int k=m-1;
      for(register int t=2;1ll*t*t<=k;++t){
        if(k%t==0){
          ve.push_back(t);
          if(k/t!=t) ve.push_back(k/t);
        }
      }
      //for(auto f:ve) cout<<"fac="<<f<<endl;
      for(register int t=2;;++t){
        int l=1;
        for(auto f:ve)
          if(ksm(t,f)==1) l=0;
        if(l) {g=t;return;}
      }
      
#undef mod
}


int main(){

      freopen("sdoi2015_sequence.in","r",stdin);
      freopen("sdoi2015_sequence.out","w",stdout);
      
      n=qr();m=qr();x=qr();s=qr();
      findg();
      
      for(register int t=1,k=g;t<m-1;++t,k=1ll*k*g%m) lg[k]=t;
      for(register int t=1;t<=s;++t) {int t1=qr();if(t1)read[lg[t1]]=1;}
      
      int k=1;
      while(k<=m) k<<=1;
      poly::KSM(read,read,k,n);
      //for(register int t=0;t<k;++t) cout<<read[t]<<' ';
      //cout<<endl;
      int ans=read[lg[x]];
      printf("%d\n",ans);
      return 0;
}