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E - Subtree K-th Max

Score : $500$ points

Problem Statement

We have a rooted tree with $N$ vertices. The vertices are numbered $1$ through $N$, and the root is Vertex $1$.
The $i$-th edge connects Vertices $A_i$ and $B_i$.
Vertex $i$ has an integer $X_i$ written on it.

You are given $Q$ queries. For the $i$-th query, given a pair of integers $(V_i,K_i)$, answer the following question.

• Question: among the integers written on the vertices in the subtree rooted at Vertex $V_i$, find the $K_i$-th largest value.

Constraints

• $2 \leq N \leq 10^5$
• $0\leq X_i\leq 10^9$
• $1\leq A_i,B_i\leq N$
• $1\leq Q \leq 10^5$
• $1\leq V_i\leq N$
• $1\leq K_i\leq 20$
• The given graph is a tree.
• The subtree rooted at Vertex $V_i$ has $K_i$ or more vertices.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $Q$

$X_1$ $\ldots$ $X_N$

$A_1$ $B_1$

$\vdots$

$A_{N-1}$ $B_{N-1}$

$V_1$ $K_1$

$\vdots$

$V_Q$ $K_Q$

Output

Print $Q$ lines. The $i$-th line should contain the response to the $i$-th query.

5 2
1 2 3 4 5
1 4
2 1
2 5
3 2
1 2
2 1

4
5

The tree given in this input is shown below.

For the $1$-st query, the vertices in the subtree rooted at Vertex $1$ are Vertices $1, 2, 3, 4$, and $5$, so print the $2$-nd largest value of the numbers written on these vertices, $4$.
For the $2$-nd query, the vertices in the subtree rooted at Vertex $2$ are Vertices $2, 3$, and $5$, so print the $1$-st largest value of the numbers written on these vertices, $5$.

6 2
10 10 10 9 8 8
1 4
2 1
2 5
3 2
6 4
1 4
2 2

9
10

4 4
1 10 100 1000
1 2
2 3
3 4
1 4
2 3
3 2
4 1

1
10
100
1000