洛谷 7440.「KrOI2021」Feux Follets

首先把给定的多项式转成牛顿级数,即转写成 \[ F(x) = \sum\limits_{i=0}^{k-1} a_i \binom xi \]

让我们考虑一个错排,显然它是由非自环的循环置换为基本单位构成的,即 \[ {\rm e}^{-x-\ln(1-x)} \]

然后考虑从中选 \(k\) 个。不难发现这只需要对一个循环置换附加一个因子 \((1+y)\) 即可做到: \[ {\rm e}^{(1+y)(-x-\ln(1-x))} \]

令其为 \(G\),考虑求算 \[ \sum\limits_{i=0}^{k-1} a_i [y^i] G \]

直接计算是困难的;但是我们不妨考虑应用转置原理,首先考虑 \[ \sum\limits_{i=0}^{k-1} a_i [x^i] G \]

\(F_i = [x^i] G\),对 \(x\) 求导可以得到 \[ \frac{\partial G}{\partial x} = (1+y)\frac x{1-x} G \]

\[ iF_i = (i-1)F_{i-1} + (1+y) F_{i-2} \]

考虑构造矩阵满足 \[ \begin{bmatrix} F_i & F_{i-1} \end{bmatrix} = \begin{bmatrix} F_{i-1} & F_{i-2} \end{bmatrix} A_i \]

不妨假定 \(F_{-1}=1\)

那么我们要求的其实就是 \[ \sum\limits_{i=0}^{n-1} \begin{bmatrix}a_i & 0\end{bmatrix} A_1 A_2 \cdots A_i \]

用线性算法描述就是 \[ A \begin{bmatrix} \begin{bmatrix} a_0 & 0 \end{bmatrix} \\ \begin{bmatrix} a_1 & 0 \end{bmatrix} \\ \vdots \\ \begin{bmatrix} a_{n-1} & 0 \end{bmatrix} \end{bmatrix} \]

其中 \[ \def\res{ {\rm res} } A_{ij} = [y^i] A_0 A_1 \cdots A_j \]

我们在计算此问题时,使用分治,同时维护当前区间乘出来的列向量 \(\res\),合并时 \[ \res = \res_L + \Pi_L \res_R \]

而在递归到长度为 \(1\) 的区间时,让 \(\res\) 返回其左乘 \(a_i\) 的向量的结果。

其中 \(\Pi_L\) 表示左区间的 \(A_i\) 之积。

那么我们的转置其实就已经明晰了。分治时传入一个 \(2\) 维行向量的列向量 \(\res\),分治时 \[ \res_L \gets \res,\res_R \gets \res \times^T \Pi_L^T \]

然后递归进入子问题计算。

时间复杂度瓶颈为转牛顿级数,总复杂度 \(O(k \log^2 k + n \log^2 n)\)

代码:

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#include <cstdio>
#include <vector>
#include <cstring>
#include <algorithm>
#define add(a,b) (a + b >= mod ? a + b - mod : a + b)
#define dec(a,b) (a < b ? a - b + mod : a - b)
#define ls (p << 1)
#define rs (ls | 1)
using namespace std;
const int N = 1e5;
const int mod = 998244353;
inline int fpow(int a,int b)
{
int ret = 1;
for(;b;b >>= 1)
(b & 1) && (ret = (long long)ret * a % mod),a = (long long)a * a % mod;
return ret;
}
int n,k;
namespace Poly
{
const int LG = 17;
const int N = 1 << LG + 1;
const int G = 3;
int lg2[N + 5];
int rev[N + 5],fac[N + 5],ifac[N + 5],inv[N + 5];
int rt[N + 5];
inline void init()
{
for(register int i = 2;i <= N;++i)
lg2[i] = lg2[i >> 1] + 1;
rt[0] = 1,rt[1 << LG] = fpow(G,(mod - 1) >> LG + 2);
for(register int i = LG;i;--i)
rt[1 << i - 1] = (long long)rt[1 << i] * rt[1 << i] % mod;
for(register int i = 1;i < N;++i)
rt[i] = (long long)rt[i & i - 1] * rt[i & -i] % mod;
fac[0] = 1;
for(register int i = 1;i <= N;++i)
fac[i] = (long long)fac[i - 1] * i % mod;
ifac[N] = fpow(fac[N],mod - 2);
for(register int i = N;i;--i)
ifac[i - 1] = (long long)ifac[i] * i % mod;
for(register int i = 1;i <= N;++i)
inv[i] = (long long)ifac[i] * fac[i - 1] % mod;
}
struct poly
{
vector<int> a;
inline poly() {}
inline poly(int x)
{
a.push_back(x);
}
inline poly(int x,int y)
{
a.push_back(x),a.push_back(y);
}
inline poly(const vector<int> &o)
{
a = o;
}
inline poly(const poly &o)
{
a = o.a;
}
inline int size() const
{
return a.size();
}
inline void resize(int x)
{
a.resize(x);
}
inline int operator[](int x) const
{
if(x < 0 || x >= size())
return 0;
return a[x];
}
inline void clear()
{
vector<int>().swap(a);
}
inline poly rever() const
{
return poly(vector<int>(a.rbegin(),a.rend()));
}
inline void dif()
{
int n = size();
for(register int i = 0,len = n >> 1;len;++i,len >>= 1)
for(register int j = 0,*w = rt;j < n;j += len << 1,++w)
for(register int k = j,R;k < j + len;++k)
R = (long long)*w * a[k + len] % mod,
a[k + len] = dec(a[k],R),
a[k] = add(a[k],R);
}
inline void dit()
{
int n = size();
for(register int i = 0,len = 1;len < n;++i,len <<= 1)
for(register int j = 0,*w = rt;j < n;j += len << 1,++w)
for(register int k = j,R;k < j + len;++k)
R = add(a[k],a[k + len]),
a[k + len] = (long long)(a[k] - a[k + len] + mod) * *w % mod,
a[k] = R;
reverse(a.begin() + 1,a.end());
for(register int i = 0;i < n;++i)
a[i] = (long long)a[i] * inv[n] % mod;
}
inline void ntt(int type = 1)
{
type == 1 ? dif() : dit();
}
friend inline poly operator+(const poly &a,const poly &b)
{
vector<int> ret(max(a.size(),b.size()));
for(register int i = 0;i < ret.size();++i)
ret[i] = add(a[i],b[i]);
return poly(ret);
}
friend inline poly operator-(const poly &a,const poly &b)
{
vector<int> ret(max(a.size(),b.size()));
for(register int i = 0;i < ret.size();++i)
ret[i] = dec(a[i],b[i]);
return poly(ret);
}
friend inline poly operator*(poly a,poly b)
{
if(a.a.empty() || b.a.empty())
return poly();
if(a.size() < 40 || b.size() < 40)
{
if(a.size() > b.size())
swap(a,b);
poly ret;
ret.resize(a.size() + b.size() - 1);
for(register int i = 0;i < ret.size();++i)
for(register int j = 0;j <= i && j < a.size();++j)
ret.a[i] = (ret[i] + (long long)a[j] * b[i - j]) % mod;
return ret;
}
int lim = 1,tot = a.size() + b.size() - 1;
for(;lim < tot;lim <<= 1);
a.resize(lim),b.resize(lim);
a.ntt(),b.ntt();
for(register int i = 0;i < lim;++i)
a.a[i] = (long long)a[i] * b[i] % mod;
a.ntt(-1),a.resize(tot);
return a;
}
poly &operator+=(const poly &o)
{
resize(max(size(),o.size()));
for(register int i = 0;i < o.size();++i)
a[i] = add(a[i],o[i]);
return *this;
}
poly &operator-=(const poly &o)
{
resize(max(size(),o.size()));
for(register int i = 0;i < o.size();++i)
a[i] = dec(a[i],o[i]);
return *this;
}
poly &operator*=(poly o)
{
return (*this) = (*this) * o;
}
poly deriv() const
{
if(a.empty())
return poly();
vector<int> ret(size() - 1);
for(register int i = 0;i < size() - 1;++i)
ret[i] = (long long)(i + 1) * a[i + 1] % mod;
return poly(ret);
}
poly integ() const
{
if(a.empty())
return poly();
vector<int> ret(size() + 1);
for(register int i = 0;i < size();++i)
ret[i + 1] = (long long)a[i] * inv[i + 1] % mod;
return poly(ret);
}
inline poly modxn(int n) const
{
if(a.empty())
return poly();
n = min(n,size());
return poly(vector<int>(a.begin(),a.begin() + n));
}
inline poly inver(int m) const
{
poly ret(fpow(a[0],mod - 2)),f,g;
for(register int k = 1;k < m;)
{
k <<= 1,f.resize(k),g.resize(k);
for(register int i = 0;i < k;++i)
f.a[i] = (*this)[i],g.a[i] = ret[i];
f.ntt(),g.ntt();
for(register int i = 0;i < k;++i)
f.a[i] = (long long)f[i] * g[i] % mod;
f.ntt(-1);
for(register int i = 0;i < (k >> 1);++i)
f.a[i] = 0;
f.ntt();
for(register int i = 0;i < k;++i)
f.a[i] = (long long)f[i] * g[i] % mod;
f.ntt(-1);
ret.resize(k);
for(register int i = (k >> 1);i < k;++i)
ret.a[i] = dec(0,f[i]);
}
return ret.modxn(m);
}
inline pair<poly,poly> div(poly o) const
{
if(size() < o.size())
return make_pair(poly(),*this);
poly f,g;
f = (rever().modxn(size() - o.size() + 1) * o.rever().inver(size() - o.size() + 1)).modxn(size() - o.size() + 1).rever();
g = (modxn(o.size() - 1) - o.modxn(o.size() - 1) * f.modxn(o.size() - 1)).modxn(o.size() - 1);
return make_pair(f,g);
}
inline poly log(int m) const
{
return (deriv() * inver(m)).integ().modxn(m);
}
inline poly exp(int m) const
{
poly ret(1),iv,it,d = deriv(),itd,itd0,t1;
if(m < 70)
{
ret.resize(m);
for(register int i = 1;i < m;++i)
{
for(register int j = 1;j <= i;++j)
ret.a[i] = (ret[i] + (long long)j * operator[](j) % mod * ret[i - j]) % mod;
ret.a[i] = (long long)ret[i] * inv[i] % mod;
}
return ret;
}
for(register int k = 1;k < m;)
{
k <<= 1;
it.resize(k >> 1);
for(register int i = 0;i < (k >> 1);++i)
it.a[i] = ret[i];
itd = it.deriv(),itd.resize(k >> 1);
iv = ret.inver(k >> 1),iv.resize(k >> 1);
it.ntt(),itd.ntt(),iv.ntt();
for(register int i = 0;i < (k >> 1);++i)
it.a[i] = (long long)it[i] * iv[i] % mod,
itd.a[i] = (long long)itd[i] * iv[i] % mod;
it.ntt(-1),itd.ntt(-1),it.a[0] = dec(it[0],1);
for(register int i = 0;i < k - 1;++i)
itd.a[i % (k >> 1)] = dec(itd[i % (k >> 1)],d[i]);
itd0.resize((k >> 1) - 1);
for(register int i = 0;i < (k >> 1) - 1;++i)
itd0.a[i] = d[i];
itd0 = (itd0 * it).modxn((k >> 1) - 1);
t1.resize(k - 1);
for(register int i = (k >> 1) - 1;i < k - 1;++i)
t1.a[i] = itd[(i + (k >> 1)) % (k >> 1)];
for(register int i = k >> 1;i < k - 1;++i)
t1.a[i] = dec(t1[i],itd0[i - (k >> 1)]);
t1 = t1.integ();
for(register int i = 0;i < (k >> 1);++i)
t1.a[i] = t1[i + (k >> 1)];
for(register int i = (k >> 1);i < k;++i)
t1.a[i] = 0;
t1.resize(k >> 1),t1 = (t1 * ret).modxn(k >> 1),t1.resize(k);
for(register int i = (k >> 1);i < k;++i)
t1.a[i] = t1[i - (k >> 1)];
for(register int i = 0;i < (k >> 1);++i)
t1.a[i] = 0;
ret -= t1;
}
return ret.modxn(m);
}
inline poly sqrt(int m) const
{
poly ret(1),f,g;
for(register int k = 1;k < m;)
{
k <<= 1;
f = ret,f.resize(k >> 1);
f.ntt();
for(register int i = 0;i < (k >> 1);++i)
f.a[i] = (long long)f[i] * f[i] % mod;
f.ntt(-1);
for(register int i = 0;i < k;++i)
f.a[i % (k >> 1)] = dec(f[i % (k >> 1)],(*this)[i]);
g = (2 * ret).inver(k >> 1),f = (f * g).modxn(k >> 1),f.resize(k);
for(register int i = (k >> 1);i < k;++i)
f.a[i] = f[i - (k >> 1)];
for(register int i = 0;i < (k >> 1);++i)
f.a[i] = 0;
ret -= f;
}
return ret.modxn(m);
}
inline poly pow(int m,int k1,int k2 = -1) const
{
if(a.empty())
return poly();
if(k2 == -1)
k2 = k1;
int t = 0;
for(;t < size() && !a[t];++t);
if((long long)t * k1 >= m)
return poly();
poly ret;
ret.resize(m);
int u = fpow(a[t],mod - 2),v = fpow(a[t],k2);
for(register int i = 0;i < m - t * k1;++i)
ret.a[i] = (long long)operator[](i + t) * u % mod;
ret = ret.log(m - t * k1);
for(register int i = 0;i < ret.size();++i)
ret.a[i] = (long long)ret[i] * k1 % mod;
ret = ret.exp(m - t * k1),t *= k1,ret.resize(m);
for(register int i = m - 1;i >= t;--i)
ret.a[i] = (long long)ret[i - t] * v % mod;
for(register int i = 0;i < t;++i)
ret.a[i] = 0;
return ret;
}
};
}
using Poly::init;
using Poly::poly;
inline int C(int n,int m)
{
return n < m ? 0 : (long long)Poly::fac[n] * Poly::ifac[m] % mod * Poly::ifac[n - m] % mod;
}
namespace Newton_Series
{
poly seg[(N << 2) + 5];
inline poly comp_xplusa(const poly &f,int a)
{
int n = f.size();
poly t1,t2,ret;
t1.resize(n),t2.resize(n);
for(register int i = 0,pw = 1;i < n;++i,pw = (long long)pw * a % mod)
t1.a[n - 1 - i] = (long long)Poly::fac[i] * f[i] % mod,
t2.a[i] = (long long)Poly::ifac[i] * pw % mod;
t1 *= t2;
ret.resize(n);
for(register int i = 0;i < n;++i)
ret.a[i] = (long long)Poly::ifac[i] * t1[n - 1 - i] % mod;
return ret;
}
void build(int n,int p)
{
if(n == 1)
{
seg[p].resize(2),seg[p].a[1] = 1;
return ;
}
int mid = n + 1 >> 1;
build(mid,ls),build(n - mid,rs);
seg[p] = seg[ls] * comp_xplusa(seg[rs],dec(0,mid));
}
poly solve(const poly &f,int n,int p)
{
if(n == 1)
return f;
int mid = n + 1 >> 1;
poly ret;
ret.resize(n);
pair<poly,poly> res = f.div(seg[ls]);
poly t1 = solve(res.second,mid,ls);
poly t2 = solve(comp_xplusa(res.first,mid),n - mid,rs);
for(register int i = 0;i < mid;++i)
ret.a[i] = t1[i];
for(register int i = mid;i < n;++i)
ret.a[i] = t2[i - mid];
return ret;
}
poly calc(const poly &f)
{
int n = f.size();
poly ret;
build(n,1),ret = solve(f,n,1);
for(register int i = 0;i < n;++i)
ret.a[i] = (long long)ret[i] * Poly::fac[i] % mod;
return ret;
}
}
inline poly mulT(poly a,poly b,int k = -1)
{
if(a.a.empty() || b.a.empty())
return poly();
int n = a.size(),m = b.size();
if(k == -1)
k = n - m + 1;
if(k <= 0)
return poly();
if(k < 40 || b.size() < 40)
{
poly ret;
ret.resize(k);
for(register int i = 0;i < k;++i)
for(register int j = 0;j < b.size();++j)
ret.a[i] = (ret[i] + (long long)a[i + j] * b[j]) % mod;
return ret;
}
reverse(b.a.begin(),b.a.end());
if(k == n - m + 1)
{
int lim = 1;
for(;lim < n;lim <<= 1);
a.resize(lim),b.resize(lim);
a.ntt(),b.ntt();
for(register int i = 0;i < lim;++i)
a.a[i] = (long long)a[i] * b[i] % mod;
a.ntt(-1);
for(register int i = 0;i < k;++i)
a.a[i] = a[m - 1 + i];
for(register int i = k;i < lim;++i)
a.a[i] = 0;
a.resize(n - m + 1);
}
else
{
a *= b,a.resize(n + m - 1);
for(register int i = 0;i < k;++i)
a.a[i] = a[m - 1 + i];
for(register int i = k;i < n + m - 1;++i)
a.a[i] = 0;
a.resize(k);
}
return a;
}
poly f,seg[(N << 2) + 5][2][2];
int ans[N + 5];
void build(int p,int l,int r)
{
if(l == r)
{
seg[p][1][0] = 1,seg[p][1][1] = 0,
seg[p][0][0] = l < 2 ? !l : (long long)(l - 1) * Poly::inv[l] % mod;
if(l < 2)
seg[p][0][1] = 0;
else
seg[p][0][1] = poly(Poly::inv[l],Poly::inv[l]);
return ;
}
int mid = l + r >> 1;
build(ls,l,mid),build(rs,mid + 1,r);
for(register int k = 0;k < 2;++k)
for(register int i = 0;i < 2;++i)
for(register int j = 0;j < 2;++j)
seg[p][i][j] += seg[rs][i][k] * seg[ls][k][j];
}
void solve(int p,int l,int r,poly f[2])
{
if(l == r)
{
f[0].size()==0?f[0].resize(1):(void)1;
f[1].size()<=1?f[1].resize(2):(void)1;
if(l < 2)
ans[l] = l ? 0 : f[0][0];
else
ans[l] = ((long long)f[0][0] * (l - 1) + f[1][0] + f[1][1]) % mod * Poly::inv[l] % mod;
return ;
}
int mid = l + r >> 1;
poly g[2];
g[0] = f[0].modxn((mid - l >> 1) + 2),
g[1] = f[1].modxn((mid - l >> 1) + 2);
solve(ls,l,mid,g);
g[0] = mulT(f[0],seg[ls][0][0]) + mulT(f[1],seg[ls][0][1]),
g[1] = mulT(f[0],seg[ls][1][0]) + mulT(f[1],seg[ls][1][1]);
solve(rs,mid + 1,r,g);
}
int main()
{
init();
scanf("%d%d",&n,&k),f.resize(k);
for(register int i = 0;i < k;++i)
scanf("%d",&f.a[i]);
f = Newton_Series::calc(f);
build(1,0,n);
poly g[2];
g[0].resize((n + 1 >> 1) + 1),g[1].resize((n + 1 >> 1) + 1);
for(register int i = 0;i < g[0].size();++i)
g[0].a[i] = f[i];
solve(1,0,n,g);
for(register int i = 1;i <= n;++i)
ans[i] = (long long)ans[i] * Poly::fac[i] % mod;
for(register int i = 1;i <= n;++i)
printf("%d%c",ans[i]," \n"[i == n]);
}