Codeforces Gym 102565E OneZeroTree

发表于 | 分类于 | 9

写这个套路题只是为了练手 / 借机会整改一下大多项式模板。

直接考虑点分治。
对于当前的一棵子树，考虑求出 fi 表示这棵子树的根往下延伸的所有路径中，费用等于 i 的方案数。
考虑怎么求这个。首先 DFS 求出 gi 表示这棵子树的根往下延伸的所有路径中，长度为 i 的条数。
那么有 fi=n−1∑j=i(ji)gj

其中 n 为次数界。
稍微推一下可得 i!fi=n−i−1∑j=01j!⋅(i+j)!gi+j

容易构造卷积形式计算。

求出 fi 后，将子树按最大深度排序逐个执行 NTT 合并。

总复杂度 O(nlog2n)。

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)
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;
int to[(N << 1) + 5],pre[(N << 1) + 5],first[N + 5];
inline void add_edge(int u,int v)
{
    static int tot = 0;
    to[++tot] = v,pre[tot] = first[u],first[u] = tot;
}
namespace Poly
{
    const int N = 1 << 18;
    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],irt[N + 5];
    inline void init()
    {
        for(register int i = 2;i <= N;++i)
            lg2[i] = lg2[i >> 1] + 1;
        int w = fpow(G,(mod - 1) / N);
        rt[N >> 1] = 1;
        for(register int i = (N >> 1) + 1;i <= N;++i)
            rt[i] = (long long)rt[i - 1] * w % mod;
        for(register int i = (N >> 1) - 1;i;--i)
            rt[i] = rt[i << 1];
        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(int x = 0)
        {
            x && (a.push_back(x),1);
        }
        inline poly(const vector<int> &o)
        {
            a = o;
            shrink();
        }
        inline void shrink()
        {
            for(;!a.empty() && !a.back();a.pop_back());
        }
        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 int &operator[](int x)
        {
            return a[x];
        }
        inline void clear()
        {
            vector<int>().swap(a);
        }
        inline void ntt(int type = 1)
        {
            int n = size();
            type == -1 && (reverse(a.begin() + 1,a.end()),1);
            int lg = lg2[n] - 1;
            for(register int i = 0;i < n;++i)
                rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << lg),
                i < rev[i] && (swap(a[i],a[rev[i]]),1);
            for(register int w = 2,m = 1;w <= n;w <<= 1,m <<= 1)
                for(register int i = 0;i < n;i += w)
                    for(register int j = 0;j < m;++j)
                    {
                        int t = (long long)rt[m | j] * a[i | j | m] % mod;
                        a[i | j | m] = dec(a[i | j],t),a[i | j] = add(a[i | j],t);
                    }
            if(type == -1)
                for(register int i = 0;i < n;++i)
                    a[i] = (long long)a[i] * inv[n] % mod;
        }
        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();
            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[i] = (long long)a[i] * b[i] % mod;
            a.ntt(-1),a.shrink();
            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
        {
            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));
            for(register int k = 1;k < m;)
                k <<= 1,ret = (ret * (2 - modxn(k) * ret)).modxn(k);
            return ret.modxn(m);
        }
        inline poly log(int m) const
        {
            return (deriv() * inver(m)).integ(),modxn(m);
        }
        inline poly exp(int m) const
        {
            poly ret(1);
            for(register int k = 1;k < m;)
                k <<= 1,ret = (ret * (1 - ret.log(k) + modxn(k))).modxn(k);
            return ret.modxn(m);
        }
    };
}
using Poly::init;
using Poly::poly;
int vis[N + 5],sum,sz[N + 5],max_part[N + 5],rt;
int dep[N + 5],mxdep[N + 5];
poly f,g,h,ans;
void get_rt(int p,int fa)
{
    sz[p] = 1,max_part[p] = 0;
    for(register int i = first[p];i;i = pre[i])
        if((to[i] ^ fa) && !vis[to[i]])
            get_rt(to[i],p),
            sz[p] += sz[to[i]],
            max_part[p] = max(max_part[p],sz[to[i]]);
    max_part[p] = max(max_part[p],sum - sz[p]);
    max_part[p] < max_part[rt] && (rt = p);
}
void get_dep(int p,int fa)
{
    mxdep[p] = dep[p];
    for(register int i = first[p];i;i = pre[i])
        if((to[i] ^ fa) && !vis[to[i]])
            dep[to[i]] = dep[p] + 1,get_dep(to[i],p),mxdep[p] = max(mxdep[p],mxdep[to[i]]);
}
void insert(int p,int fa)
{
    ++h[dep[p]];
    for(register int i = first[p];i;i = pre[i])
        if((to[i] ^ fa) && !vis[to[i]])
            insert(to[i],p);
}
void solve(int p)
{
    vis[p] = 1,dep[p] = 0;
    vector<int> son;
    for(register int i = first[p];i;i = pre[i])
        if(!vis[to[i]])
            dep[to[i]] = 1,get_dep(to[i],p),son.push_back(to[i]);
    sort(son.begin(),son.end(),[](int x,int y)
    {
        return mxdep[x] < mxdep[y];
    });
    f.resize(1),f[0] = 1;
    for(int v : son)
    {
        g.resize(mxdep[v] + 1),h.resize(mxdep[v] + 1);
        for(register int i = 0;i <= mxdep[v];++i)
            h[i] = 0;
        insert(v,p);
        for(register int i = 0;i <= mxdep[v];++i)
            g[i] = (long long)Poly::fac[mxdep[v] - i] * h[mxdep[v] - i] % mod;
        for(register int i = 0;i <= mxdep[v];++i)
            h[i] = Poly::ifac[i];
        h = (g * h).modxn(mxdep[v] + 1);
        for(register int i = 0;i <= mxdep[v];++i)
            g[i] = (long long)Poly::ifac[i] * h[mxdep[v] - i] % mod;
        h = f * g;
        h.resize(h.size() + 1);
        for(register int i = h.size() - 1;i;--i)
            h[i] = add(h[i],h[i - 1]);
        ans += h,f += g;
    }
    g.resize(2),g[0] = g[1] = 1,ans += g;
    for(register int i = first[p];i;i = pre[i])
        if(!vis[to[i]])
            rt = 0,sum = sz[to[i]],get_rt(to[i],p),solve(rt);
}
int main()
{
    Poly::init(),max_part[0] = 0x3f3f3f3f;
    scanf("%d",&n);
    int u,v;
    for(register int i = 2;i <= n;++i)
        scanf("%d%d",&u,&v),add_edge(u,v),add_edge(v,u);
    sum = n,get_rt(1,0),solve(rt);
    ans.resize(n + 1);
    for(register int i = 0;i <= n;++i)
        printf("%d%c",ans[i]," \n"[i == n]);
}

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