codeforces 955F Cowmpany Cowmpensation 树上DP+多项式插值

给一个树,每个点的权值为正整数,且不能超过自己的父节点,根节点的最高权值不超过D

问一共有多少种分配工资的方式?

题解:

A immediate simple observation is that we can compute the answer in $O(nD) $with a simple dynamic program. How to speed it up though?

To speed it up, we need the following lemma.

Lemma 1: For a tree with nn vertices, the answer is a polynomial in $D$ of degree at most nn.

We can prove this via induction, and the fact that for any polynomial $ p(x) $ of degree dd, the sum $\sum p_i $ is a polynomial in $n$ of degree $d+1$

Now the solution is easy: compute the answerfor $ 0≤D≤n $ and use interpolation to compute the answer for $ D>n $.

The complexity is $O(n^2)$ or the initial dp and $O(n)$ for the interpolation step.

#include <bits/stdc++.h>
#define endl '\n'
#define ll long long
#define ull unsigned long long
#define fi first
#define se second
#define mp make_pair
#define pii pair<int,int>
#define ull unsigned long long
#define all(x) x.begin(),x.end()
#pragma GCC optimize("unroll-loops")
#define inline inline __attribute__(  \
(always_inline, __gnu_inline__, __artificial__)) \
__attribute__((optimize("Ofast"))) __attribute__((target("sse"))) \
__attribute__((target("sse2"))) __attribute__((target("mmx")))
#define IO ios::sync_with_stdio(false);
#define rep(ii,a,b) for(int ii=a;ii<=b;++ii)
#define per(ii,a,b) for(int ii=b;ii>=a;--ii)
#define for_node(x,i) for(int i=head[x];i;i=e[i].next)
#define show(x) cout<<#x<<"="<<x<<endl
#define showa(a,b) cout<<#a<<'['<<b<<"]="baidu<a[b]<<endl
#define show2(x,y) cout<<#x<<"="<<x<<" "<<#y<<"="<<y<<endl
#define show3(x,y,z) cout<<#x<<"="<<x<<" "<<#y<<"="<<y<<" "<<#z<<"="<<z<<endl
#define show4(w,x,y,z) cout<<#w<<"="<<w<<" "<<#x<<"="<<x<<" "<<#y<<"="<<y<<" "<<#z<<"="<<z<<endl
using namespace std;
const int maxn=4e3+10,maxm=2e6+10;
const int INF=0x3f3f3f3f;
const ll mod=1e9+7;
const double PI=acos(-1.0);
//head
ll casn,n,m,k;
ll num[maxn];
ll pow_mod(ll a,ll b,ll c=mod,ll ans=1){while(b){if(b&1) ans=(a*ans)%c;a=(a*a)%c,b>>=1;}return ans;}
namespace polysum {
	const int maxn=101000;
	const ll mod=1e9+7;
	ll a[maxn],f[maxn],g[maxn],p[maxn],p1[maxn],p2[maxn],b[maxn],h[maxn][2],C[maxn];
	ll calcn(int d,ll *a,ll n) {//d´Î¶àÏîʽ(a[0-d])ÇóµÚnÏî
		if (n<=d) return a[n];
		p1[0]=p2[0]=1;
		rep(i,0,d) {
			ll t=(n-i+mod)%mod;
			p1[i+1]=p1[i]*t%mod;
		}
		rep(i,0,d) {
			ll t=(n-d+i+mod)%mod;
			p2[i+1]=p2[i]*t%mod;
		}
		ll ans=0;
		rep(i,0,d) {
			ll t=g[i]*g[d-i]%mod*p1[i]%mod*p2[d-i]%mod*a[i]%mod;
			if ((d-i)&1) ans=(ans-t+mod)%mod;
			else ans=(ans+t)%mod;
		}
		return ans;
	}
	void init(int maxm) {//³õʼ»¯Ô¤´¦Àí½×³ËºÍÄæÔª(È¡Ä£³Ë·¨)
		f[0]=f[1]=g[0]=g[1]=1;
		rep(i,2,maxm+4) f[i]=f[i-1]*i%mod;
		g[maxm+4]=pow_mod(f[maxm+4],mod-2);
		per(i,1,maxm+3) g[i]=g[i+1]*(i+1)%mod;
	}
}
ll dp[maxn][maxn];
struct node {int to,next;}e[maxm];int head[maxn],nume;
void add(int a,int b){e[++nume]=(node){b,head[a]};head[a]=nume;}

void dfs(int now){
	rep(i,1,n) dp[now][i]=1;
	for(int i=head[now];i;i=e[i].next){
		dfs(e[i].to);
		rep(j,1,n) dp[now][j]=dp[now][j]*dp[e[i].to][j]%mod;
	}
	rep(i,1,n) dp[now][i]=(dp[now][i]+dp[now][i-1])%mod;
}

int main() {
//#define test
#ifdef test
	auto _start = chrono::high_resolution_clock::now();
	freopen("in.txt","r",stdin);freopen("out.txt","w",stdout);
#endif

	IO;
	cin>>n>>k;
	rep(i,2,n){
		int a;
		cin>>a;
		add(a,i);
	}
	dfs(1);
	polysum::init(n+1);
	rep(i,0,n+1){
		num[i]=dp[1][i];
	}
	ll ans=polysum::calcn(n,num,k);
	cout<<ans<<endl;

#ifdef test
	auto _end = chrono::high_resolution_clock::now();
  cerr << "elapsed time: " << chrono::duration<double, milli>(_end - _start).count() << " ms\n";
	fclose(stdin);fclose(stdout);system("out.txt");
#endif
	return 0;
}