Doremy's Average Tree
本题没有可用的提交语言。
Description
Doremy has a rooted tree of size $n$ whose root is vertex $r$. Initially there is a number $w_i$ written on vertex $i$. Doremy can use her power to perform this operation at most $k$ times:
- Choose a vertex $x$ ($1 \leq x \leq n$).
- Let $s = \frac{1}{|T|}\sum_{i \in T} w_i$ where $T$ is the set of all vertices in $x$'s subtree.
- For all $i \in T$, assign $w_i := s$.
Doremy wants to know what is the lexicographically smallest$^\dagger$ array $w$ after performing all the operations. Can you help her?
If there are multiple answers, you may output any one.
$^\dagger$ For arrays $a$ and $b$ both of length $n$, $a$ is lexicographically smaller than $b$ if and only if there exist an index $i$ ($1 \leq i \le n$) such that $a_i < b_i$ and for all indices $j$ such that $j<i$, $a_j=b_j$ is satisfied.
The input consists of multiple test cases. The first line contains a single integer $t$ ($1\le t\le 10^4$) — the number of test cases. The description of the test cases follows.
The first line contains three integers $n$, $r$, $k$ ($2 \le n \le 5000$, $1 \le r \le n$, $0 \le k \le \min(500,n)$).
The second line contains $n$ integers $w_1,w_2,\ldots,w_n$ ($1 \le w_i \le 10^6$).
Each of the next $n-1$ lines contains two integers $u_i$, $v_i$ ($1 \leq u_i, v_i \leq n$), representing an edge between $u_i$ and $v_i$.
It is guaranteed that the given edges form a tree.
It is guaranteed that the sum of $n$ does not exceed $50\,000$.
For each test case, In the first line, output a single integer $cnt$ ($0 \le cnt \le k$) — the number of operations you perform.
Then, in the second line output $cnt$ integers $p_1,p_2,\ldots,p_{cnt}$ — $x$ is chosen to be $p_i$ for $i$-th operation.
If there are multiple answers, you may output any one.
Input
The input consists of multiple test cases. The first line contains a single integer $t$ ($1\le t\le 10^4$) — the number of test cases. The description of the test cases follows.
The first line contains three integers $n$, $r$, $k$ ($2 \le n \le 5000$, $1 \le r \le n$, $0 \le k \le \min(500,n)$).
The second line contains $n$ integers $w_1,w_2,\ldots,w_n$ ($1 \le w_i \le 10^6$).
Each of the next $n-1$ lines contains two integers $u_i$, $v_i$ ($1 \leq u_i, v_i \leq n$), representing an edge between $u_i$ and $v_i$.
It is guaranteed that the given edges form a tree.
It is guaranteed that the sum of $n$ does not exceed $50\,000$.
Output
For each test case, In the first line, output a single integer $cnt$ ($0 \le cnt \le k$) — the number of operations you perform.
Then, in the second line output $cnt$ integers $p_1,p_2,\ldots,p_{cnt}$ — $x$ is chosen to be $p_i$ for $i$-th operation.
If there are multiple answers, you may output any one.
4
6 1 1
1 9 2 6 1 8
1 2
1 3
2 4
3 6
3 5
7 7 2
3 1 3 3 1 1 2
7 1
7 2
7 4
1 5
2 3
4 6
6 5 1
3 1 3 1 1 3
5 3
5 1
5 6
3 4
1 2
3 2 1
1000000 999999 999997
2 1
1 3
1
2
2
1 4
1
5
1
1
Note
In the first test case:
At first $w=[1,9,2,6,1,8]$. You can choose some vertex $x$ to perform at most one operation.
- If $x=1$, $w=[\frac{9}{2},\frac{9}{2},\frac{9}{2},\frac{9}{2},\frac{9}{2},\frac{9}{2}]$.
- If $x=2$, $w=[1,\frac{15}{2},2,\frac{15}{2},1,8]$.
- If $x=3$, $w=[1,9,\frac{11}{3},6,\frac{11}{3},\frac{11}{3}]$.
- If $x \in \{4, 5, 6\}$, $w=[1,9,2,6,1,8]$.
- If you don't perform any operation, $w=[1,9,2,6,1,8]$.
$w$ is lexicographically smallest when $x=2$.