A16351 | Doremy's Average Tree

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:

1. Choose a vertex $x$ ( $1 \leq x \leq n$ ).
2. Let $s = \frac{1}{|T|}\sum_{i \in T} w_i$ where $T$ is the set of all vertices in $x$ 's subtree.
3. 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.