luogu4389 付公主的背包

题目链接：洛谷

题目大意：现在有$n$个物品，每种物品体积为$v_i$，对任意$s\in [1,m]$，求背包恰好装$s$体积的方案数（完全背包问题）。

数据范围：$n,m\leq 10^5$

---

这道题，看到数据范围就知道是生成函数。
$$Ans=\prod_{i=1}^n\frac{1}{1-x^{v_i}}$$

但是这个式子直接乘会tle，我们考虑进行优化。

看见这个连乘的式子，应该是要上$\ln$.

$$Ans=\exp(\sum_{i=1}^n\ln(\frac{1}{1-x^{v_i}}))$$

接下来的问题就是如何快速计算$\ln(\frac{1}{1-x^{v_i}})$。

$$\ln(f(x))=\int f'f^{-1}dx$$

所以
$$\ln(\frac{1}{1-x^v})=\int\sum_{i=1}^{+\infty}vix^{vi-1}*(1-x^v)dx$$
$$=\int(\sum_{i=1}^{+\infty}vix^{vi-1}-\sum_{i=2}^{+\infty}v(i-1)x^{vi-1})dx$$
$$=\int(\sum_{i=1}^{+\infty}vx^{vi-1})dx$$
$$=\sum_{i=1}^{+\infty}\frac{1}{i}x^{vi}$$

然后就可以直接代公式了。

 #include<cstdio>
 #include<algorithm>
 #define Rint register int
 using namespace std;
 typedef long long LL;
 const int N = , P = , G = , Gi = ;
 int n, m, cnt[N], A[N];
 inline int kasumi(int a, int b){
     int res = ;
     while(b){
         if(b & ) res = (LL) res * a % P;
         a = (LL) a * a % P;
         b >>= ;
     }
     return res;
 }
 int R[N];
 inline void NTT(int *A, int limit, int type){
     for(Rint i = ;i < limit;i ++)
         if(i < R[i]) swap(A[i], A[R[i]]);
     for(Rint mid = ;mid < limit;mid <<= ){
         int Wn = kasumi(type ==  ? G : Gi, (P - ) / (mid << ));
         for(Rint j = ;j < limit;j += mid << ){
             int w = ;
             for(Rint k = ;k < mid;k ++, w = (LL) w * Wn % P){
                 int x = A[j + k], y = (LL) w * A[j + k + mid] % P;
                 A[j + k] = (x + y) % P;
                 A[j + k + mid] = (x - y + P) % P;
             }
         }
     }
     if(type == -){
         int inv = kasumi(limit, P - );
         for(Rint i = ;i < limit;i ++)
             A[i] = (LL) A[i] * inv % P;
     }
 }
 int ans[N];
 inline void poly_inv(int *A, int deg){
     static int tmp[N];
     if(deg == ){
         ans[] = kasumi(A[], P - );
         return;
     }
     poly_inv(A, (deg + ) >> );
     int limit = , L = -;
     while(limit <= (deg << )){limit <<= ; L ++;}
     for(Rint i = ;i < limit;i ++)
         R[i] = (R[i >> ] >> ) | ((i & ) << L);
     for(Rint i = ;i < deg;i ++) tmp[i] = A[i];
     for(Rint i = deg;i < limit;i ++) tmp[i] = ;
     NTT(tmp, limit, ); NTT(ans, limit, );
     for(Rint i = ;i < limit;i ++)
         ans[i] = ( - (LL) tmp[i] * ans[i] % P + P) % P * ans[i] % P;
     NTT(ans, limit, -);
     for(Rint i = deg;i < limit;i ++) ans[i] = ;
 }
 int Ln[N];
 inline void get_Ln(int *A, int deg){
     static int tmp[N];
     poly_inv(A, deg);
     for(Rint i = ;i < deg;i ++)
         tmp[i - ] = (LL) i * A[i] % P;
     tmp[deg - ] = ;
     int limit = , L = -;
     while(limit <= (deg << )){limit <<= ; L ++;}
     for(Rint i = ;i < limit;i ++)
         R[i] = (R[i >> ] >> ) | ((i & ) << L);
     NTT(ans, limit, ); NTT(tmp, limit, );
     for(Rint i = ;i < limit;i ++) Ln[i] = (LL) ans[i] * tmp[i] % P;
     NTT(Ln, limit, -);
     for(Rint i = deg + ;i < limit;i ++) Ln[i] = ;
     for(Rint i = deg;i;i --) Ln[i] = (LL) Ln[i - ] * kasumi(i, P - ) % P;
     for(Rint i = ;i < limit;i ++) tmp[i] = ans[i] = ;
     Ln[] = ;
 }
 int Exp[N];
 inline void get_Exp(int *A, int deg){
     if(deg == ){
         Exp[] = ;
         return;
     }
     get_Exp(A, (deg + ) >> );
     get_Ln(Exp, deg);
     for(Rint i = ;i < deg;i ++) Ln[i] = (A[i] + (i == ) - Ln[i] + P) % P;
     int limit = , L = -;
     while(limit <= (deg << )){limit <<= ; L ++;}
     for(Rint i = ;i < limit;i ++)
         R[i] = (R[i >> ] >> ) | ((i & ) << L);
     NTT(Exp, limit, ); NTT(Ln, limit, );
     for(Rint i = ;i < limit;i ++) Exp[i] = (LL) Exp[i] * Ln[i] % P;
     NTT(Exp, limit, -);
     for(Rint i = deg;i < limit;i ++) Exp[i] = ;
     for(Rint i = ;i < limit;i ++) Ln[i] = ans[i] = ;
 }
 int main(){
     scanf("%d%d", &n, &m);
     for(Rint i = ;i <= n;i ++){
         int x;
         scanf("%d", &x);
         ++ cnt[x];
     }
     for(Rint i = ;i <= m;i ++){
         if(!cnt[i]) continue;
         for(Rint j = i;j <= m;j += i)
             A[j] = (A[j] + (LL) cnt[i] * kasumi(j / i, P - ) % P) % P;
     }
     get_Exp(A, m + );
     for(Rint i = ;i <= m;i ++)
         printf("%d\n", Exp[i]);
 }

luogu4389