付公主的背包——生成函数

题面

　　  洛谷P4389

解析

　　每一个物品可以看作一个多项式$f(x)=\sum_{i=0}^{\infty}[i\%V==0]x^i=\frac{1}{1-x^V}$，暴力把$n$个多项式乘起来是$O(NM\log M)$的复杂度，显然不能接受。

　　于是看了下题解，学了一手奇技淫巧。

　　设$h(x)=\ln(x)$, $g(x)=\ln(f(x))=h(f(x))$，有：$$\begin{align*}g'(x)&=h'(f(x))f'(x)\\ &=\frac{f'(x)}{f(x)}\\&=\frac{\sum_{i=1}^{\infty}[i\%V==0]i \cdot x^{i-1}}{\frac{1}{1-x^V}}\\ &=(1-x^V)\sum_{i=1}^{\infty}Vi \cdot x^{Vi-1}\\ &= \sum_{i=1}^{\infty}Vi \cdot x^{Vi-1}-\sum_{i=1}^{\infty}Vi \cdot x^{V(i+1)-1}\\ &= \sum_{i=1}^{\infty}Vi \cdot x^{Vi-1}-\sum_{i=2}^{\infty}V(i-1) \cdot x^{Vi-1}\\ &= \sum_{i=1}^{\infty}Vi \cdot x^{Vi-1}-\sum_{i=2}^{\infty}Vi \cdot x^{Vi-1} + \sum_{i=2}^{\infty}V\cdot x^{Vi-1}\\ &= \sum_{i=1}^{\infty}V\cdot x^{Vi-1}\end{align*}$$

　　则：$$g(x)=\sum_{i=1}^{\infty}\frac{1}{i}x^{Vi}$$

　　而$f(x)=e^{g(x)}$，最终答案为：$$ans=\prod_{i=1}^n f_i(x)\\ ans= \prod_{i=1}^n e^{g_i(x)}\\ ans = e^{\sum_{i=1}^n g_i(x)}$$

　　对每个不同体积求$g(x)$之和的时间是$O(M \ln M)$，最后$exp$的时间是$O(M \log M)$，因此总时间为$O(M\ln M+M\log M)$

　代码：

#include<cstdio>
#include<iostream>
#include<algorithm>
#include<cstring>
using namespace std;
typedef long long ll;
const int maxn = 200005, mod = 998244353, g = 3;

inline int read()
{
    int ret, f=1;
    char c;
    while((c=getchar())&&(c<'0'||c>'9'))if(c=='-')f=-1;
    ret=c-'0';
    while((c=getchar())&&(c>='0'&&c<='9'))ret=(ret<<3)+(ret<<1)+c-'0';
    return ret*f;
}

int add(int x, int y)
{
    return x + y < mod? x + y: x + y - mod;
}

int rdc(int x, int y)
{
    return x - y < 0? x - y + mod: x - y;
}

ll qpow(ll x, int y)
{
    ll ret = 1;
    while(y)
    {
        if(y&1)
            ret = ret * x % mod;
        x = x * x % mod;
        y >>= 1;
    }
    return ret;
}

int n, m, num[maxn], lim, bit, rev[maxn<<1];
ll ginv, ln[maxn<<1], iv[maxn<<1], inv[maxn], F[maxn<<1], G[maxn<<1], c[maxn<<1];

void init()
{
    ginv = qpow(g, mod - 2);
    inv[0] = inv[1] = 1;
    for(int i = 2; i <= m; ++i)
        inv[i] = (mod - mod / i) * inv[mod%i] % mod;
}

void NTT_init(int x)
{
    lim = 1;
    bit = 0;
    while(lim <= x)
    {
        lim <<= 1;
        ++ bit;
    }
    for(int i = 1; i < lim; ++i)
        rev[i] = (rev[i>>1] >> 1) | ((i & 1) << (bit - 1));
}

void NTT(ll *x, int y)
{
    for(int i = 1; i < lim; ++i)
        if(i < rev[i])
            swap(x[i], x[rev[i]]);
    ll wn, w, u, v;
    for(int i = 1; i < lim; i <<= 1)
    {
        wn = qpow((y == 1)? g: ginv, (mod - 1) / (i << 1));
        for(int j = 0; j < lim; j += (i << 1))
        {
            w = 1;
            for(int k = 0; k < i; ++k)
            {
                u = x[j+k];
                v = x[j+k+i] * w % mod;
                x[j+k] = add(u, v);
                x[j+k+i] = rdc(u, v);
                w = w * wn % mod;
            }
        }
    }
    if(y == -1)
    {
        ll linv = qpow(lim, mod - 2);
        for(int i = 0; i < lim; ++i)
            x[i] = x[i] * linv % mod;
    }
}

void get_inv(ll *x, ll *y, int len)
{
    if(len == 1)
    {
        x[0] = qpow(y[0], mod - 2);
        return ;
    }
    get_inv(x, y, (len + 1) >> 1);
    for(int i = 0; i < len; ++i)
        c[i] = y[i];
    NTT_init(len << 1);
    NTT(x, 1);
    NTT(c, 1);
    for(int i = 0; i < lim; ++i)
    {
        x[i] = rdc(add(x[i], x[i]), (c[i] * x[i] % mod) * x[i] % mod);
        c[i] = 0;
    }
    NTT(x, -1);
    for(int i = len; i < lim; ++i)
        x[i] = 0;
}

void get_ln(ll *x, ll *y, int len)
{
    for(int i = 0; i < len - 1; ++i)
        x[i] = y[i+1] * (i + 1) % mod;
    get_inv(iv, y, len);
    NTT_init(len << 1);
    NTT(x, 1);
    NTT(iv, 1);
    for(int i = 0; i < lim; ++i)
    {
        x[i] = x[i] * iv[i] % mod;
        iv[i] = 0;
    }
    NTT(x, -1);
    for(int i = len - 1; i >= 1; --i)
        x[i] = x[i-1] * qpow(i, mod - 2) % mod;
    x[0] = 0;
    for(int i = len; i < lim; ++i)
        x[i] = 0;
}

void get_exp(ll *x, ll *y, int len)
{
    if(len == 1)
    {
        x[0] = 1;
        return ;
    }
    get_exp(x, y, (len + 1) >> 1);
    get_ln(ln, x, len);
    for(int i = 0; i < len; ++i)
    {
        c[i] = rdc(add(i == 0, y[i]), ln[i]);
        ln[i] = 0;
    }
    NTT_init(len << 1);
    NTT(x, 1);
    NTT(c, 1);
    for(int i = 0; i < lim; ++i)
    {
        x[i] = x[i] * c[i] % mod;
        c[i] = 0;
    }
    NTT(x, -1);
    for(int i = len; i < lim; ++i)
        x[i] = 0;
}

int main()
{
    n = read(); m = read();
    init();
    int x;
    for(int i = 1; i <= n; ++i)
    {
        x = read();
        ++ num[x];
    }
    for(int i = 1; i <= m; ++i)
        if(num[i])
        {
            for(int j = 1; i * j <= m; ++j)
                G[i*j] = add(G[i*j], num[i] * inv[j] % mod);
        }
    get_exp(F, G, m + 1);
    for(int i = 1; i <= m; ++i)
        printf("%lld\n", F[i]);
    return 0;
}