LOJ166 拉格朗日插值 2【卷积，NTT】

题目链接：LOJ

题目描述：输入多项式的次数$n$，一个整数$m$和$f(0),f(1),f(2),\ldots,f(n)$，输出$f(m),f(m+1),f(m+2),\ldots,f(m+n)$

数据范围：$1\leq n\leq 10^5,n<m\leq 10^8$

一道披着拉格朗日插值模板的外衣的数论题。。。

$$f(m+i)=\sum_{j=0}^nf(j)\dfrac{\prod_{k\not=j}(m+i-k)}{\prod_{k\not=j}(j-k)}$$

$$=\dfrac{(m+i)!}{(m-n+i-1)!}\sum_{j=0}^n\dfrac{f(j)*(-1)^{n-j}}{j!*(n-j)!}*\frac{1}{m+i-j}$$

预处理阶乘，逆元，阶乘逆元，和$mfac[i]=\prod_{j=0}^{i-1}(m-n+j)$，$m-n+i$的逆元和$mfac[i]$的逆元。

使用NTT计算，时间复杂度为$O(n\log n)$

 #include<bits/stdc++.h>
 #define Rint register int
 using namespace std;
 typedef long long LL;
 const int N =  << , mod = , G = , Gi = ;
 inline int kasumi(int a, int b){
     int res = ;
     while(b){
         if(b & ) res = (LL) res * a % mod;
         a = (LL) a * a % mod;
         b >>= ;
     }
     return res;
 }
 int rev[N];
 inline int calrev(int len){
     int L = -, limit = ;
     while(limit <= len){limit <<= ; L ++;}
     for(Rint i = ;i < limit;i ++) rev[i] = (rev[i >> ] >> ) | ((i & ) << L);
     return limit;
 }
 inline void NTT(int *A, int limit, int type){
     for(Rint i = ;i < limit;i ++) if(i < rev[i]) swap(A[i], A[rev[i]]);
     for(Rint mid = ;mid < limit;mid <<= ){
         int Wn = kasumi(type ==  ? G : Gi, (mod - ) / (mid << ));
         for(Rint j = ;j < limit;j += mid << ){
             int w = ;
             for(Rint k = ;k < mid;k ++, w = (LL) w * Wn % mod){
                 int x = A[j + k], y = (LL) w * A[j + k + mid] % mod;
                 A[j + k] = (x + y) % mod;
                 A[j + k + mid] = (x - y + mod) % mod;
             }
         }
     }
     if(type == -){
         int inv = kasumi(limit, mod - );
         for(Rint i = ;i < limit;i ++) A[i] = (LL) A[i] * inv % mod;
     }
 }
 int fac[N], invfac[N], inv[N], mfac[N], minvfac[N], minv[N];
 inline void init(int n, int m){
     fac[] = mfac[] = ;
     for(Rint i = ;i <= (n <<  | );i ++){
         fac[i] = (LL) fac[i - ] * i % mod;
         mfac[i] = (LL) mfac[i - ] * (m - n + i - ) % mod;
     }
     invfac[n <<  | ] = kasumi(fac[n <<  | ], mod - );
     minvfac[n <<  | ] = kasumi(mfac[n <<  | ], mod - );
     for(Rint i = (n <<  | );i;i --){
         invfac[i - ] = (LL) invfac[i] * i % mod;
         minvfac[i - ] = (LL) minvfac[i] * (m - n + i - ) % mod;
         inv[i] = (LL) invfac[i] * fac[i - ] % mod;
         minv[i] = (LL) minvfac[i] * mfac[i - ] % mod;
     }
     minv[] = ;
 }
 int n, m, f[N], A[N], B[N];
 int main(){
     scanf("%d%d", &n, &m);
     for(Rint i = ;i <= n;i ++) scanf("%d", f + i);
     init(n, m);
     int limit = calrev(n * );
     for(Rint i = ;i <= n;i ++){
         A[i] = (LL) f[i] * invfac[i] % mod * invfac[n - i] % mod;
         if(n - i & ) A[i] = mod - A[i];
     }
     for(Rint i = ;i <= (n << );i ++) B[i] = minv[i + ];
     NTT(A, limit, ); NTT(B, limit, );
     for(Rint i = ;i < limit;i ++) A[i] = (LL) A[i] * B[i] % mod;
     NTT(A, limit, -);
     for(Rint i = n;i <= (n << );i ++)
         printf("%d ", (LL) mfac[i + ] * minvfac[i - n] % mod * A[i] % mod);
 }

LOJ166