计蒜客 Red Black Tree(树形DP)

Posted 2022-06-11 taozi1115402474

You are given a rooted tree with n nodes. The nodes are numbered 1..n. The root is node 1, and m of the nodes are colored red, the rest are black.

You would like to choose a subset of nodes such that there is no node in your subset which is an ancestor of any other node in your subset. For example, if A is the parent of B and B is the parent of C, then you could have at most one of A, B or C in your subset. In addition, you would like exactly k of your chosen nodes to be red.

If exactly mm of the nodes are red, then for all k = 0..m, figure out how many ways you can choose subsets with k red nodes, and no node is an ancestor of any other node.

Input Format

Each input will consist of a single test case.

Note that your program may be run multiple times on different inputs.

Each test case will begin with a line with two integers n(1≤n≤2×10^5) and m(0≤m≤min(10^3,n)), where n is the number of nodes in the tree, and m is the number of nodes which are red. The nodes are numbered 1..n.

Each of the next n - 1 lines will contain a single integer p(1≤p≤n), which is the number of the parent of this node. The nodes are listed in order, starting with node 2, then node 3, and so on. Node 1 is skipped, since it is the root. It is guaranteed that the nodes form a single tree, with a single root at node 1 and no cycles.

Each of the next m lines will contain single integer r(1≤r≤n). These are the numbers of the red nodes. No value of r will be repeated.

Output Format

Output m + 1 lines, corresponding to the number of subsets satisfying the given criteria with a number of red nodes equal to k = 0..m, in that order. Output this number modulo 10^9 + 7.

样例输入1

4 1
1
1
1
3

样例输出1

5
4

样例输入2

4 4
1
1
1
1
2
3
4

样例输出2

1
4
3
1
0

样例输入3

14 4
1
2
1
2
3
4
5
5
13
8
10
4
4
8
3
12
13

样例输出3

100
169
90
16
0

题意

一棵树，n个点，其中有m个红色，其余为黑点，1为根，选1个点集合，集合内的任意两点不互为祖先，问集合内红色节点的个数为0-m的方案数。

题解

dp[root][m]代表根为root的子树红色节点为m的方案数。

不选root，dp[root][0]=1，考虑子树son的情况，考虑h[M]代表组成M的方案数，相当于每次一颗子树把里面的所有红色节点一个一个压进去，类似于背包。

选root，dp[root][red[root]]++，相当于下面都不能选。

代码

 1 #include<bits/stdc++.h>
 2 using namespace std;
 3 const int N=200005;
 4 const int M=1005;
 5 const int MD=1000000007;
 6 int dp[N][M],h[M],red[N],sz[N];
 7 int n,m;
 8 vector<int>G[N];
 9 void dfs(int u)
10     dp[u][0]=1;
11     sz[u]=red[u];
12     for(int i=0;i<G[u].size();i++) 
13         int v=G[u][i];
14         dfs(v);
15         int up=min(sz[u]+sz[v],m);
16         for(int j=0;j<=up;j++)h[j]=0;
17         for(int j=0;j<=sz[v];j++)
18             for(int k=0;k<=sz[u]&&j+k<=up;k++)
19                 h[j+k]=(h[j+k]+1LL*dp[v][j]*dp[u][k])%MD;
20         sz[u]+=sz[v];
21         for(int j=0;j<=sz[u];j++)dp[u][j]=h[j];
22     
23     dp[u][red[u]]=(dp[u][red[u]]+1)%MD;
24 
25 int main() 
26     scanf("%d%d",&n,&m);
27     for(int i=2,u;i<=n;i++) 
28         scanf("%d",&u);
29         G[u].push_back(i);
30     
31     for(int i=0,x;i<m;i++) 
32         scanf("%d",&x);
33         red[x]=1;
34     
35     dfs(1);
36     for(int i=0;i<=m;i++)
37         printf("%d\n",dp[1][i]);
38     return 0;
39