Problem:
Score : 600 points

Problem Statement

We have a tree with N vertices, whose i-th edge connects Vertex u_i and Vertex v_i.
Vertex i has an integer a_i written on it.
For every integer k from 1 through N, solve the following problem:

We will make a sequence by lining up the integers written on the vertices along the shortest path from Vertex 1 to Vertex k, in the order they appear. Find the length of the longest increasing subsequence of this sequence.

Here, the longest increasing subsequence of a sequence A of length L is the subsequence A_{i_1} , A_{i_2} , ... , A_{i_M} with the greatest possible value of M such that 1 \leq i_1 < i_2 < ... < i_M \leq L and A_{i_1} < A_{i_2} < ... < A_{i_M}.

Constraints

2 \leq N \leq 2 \times 10^5

1 \leq a_i \leq 10^9

1 \leq u_i , v_i \leq N

u_i \neq v_i

The given graph is a tree.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 ... a_N
u_1 v_1
u_2 v_2
:
u_{N-1} v_{N-1}

Output

Print N lines. The k-th line, print the length of the longest increasing subsequence of the sequence obtained from the shortest path from Vertex 1 to Vertex k.

Sample Input 1

10
1 2 5 3 4 6 7 3 2 4
1 2
2 3
3 4
4 5
3 6
6 7
1 8
8 9
9 10

Sample Output 1

1
2
3
3
4
4
5
2
2
3

For example, the sequence A obtained from the shortest path from Vertex 1 to Vertex 5 is 1,2,5,3,4. Its longest increasing subsequence is A_1, A_2, A_4, A_5, with the length of 4.

Solution in Rust (1.42.0) (judge verdict: Time Limit Exceeded)

use proconio::{fastout, input};
use std::cmp::max;

#[derive(Debug, Default)]
struct Node {
    a: usize,
    ns: Vec<usize>,
    vs: Vec<(usize, usize)>,
}

impl Node {
    fn update(&mut self, values: &Vec<(usize, usize)>) {
        let mut max_val = 1;
        let mut old_val = 0;

        for (a, v) in values.iter() {
            if *a < self.a {
                max_val = max(*v + 1, max_val);
            }

            if *a == self.a {
                old_val = *v;
            } else {
                self.vs.push((*a, *v));
            }
        }

        self.vs.push((self.a, max(max_val, old_val)));
    }
}

#[fastout]
fn main() {
    input! {in_n: usize, in_a: [usize; in_n], in_v: [[usize; 2]; in_n - 1]};
    let mut nodes = Vec::new();

    for a in in_a.iter() {
        nodes.push(Node {
            a: *a,
            ns: vec![],
            vs: vec![],
        });
    }

    for v in in_v.iter() {
        let (s, d) = (v[0], v[1]);

        nodes[s - 1].ns.push(d - 1);
        nodes[d - 1].ns.push(s - 1);
    }

    let mut queue = Vec::new();

    nodes[0].update(&vec![]);
    queue.push(0);

    while !queue.is_empty() {
        let i = queue.pop().unwrap();

        let ns = nodes[i].ns.clone();
        let vs = nodes[i].vs.clone();

        for n in ns.iter() {
            if !nodes[*n].vs.is_empty() {
                continue;
            }

            nodes[*n].update(&vs);
            queue.push(*n);
        }
    }

    for node in nodes.iter() {
        println!("{}", node.vs.iter().map(|x| x.1).max().unwrap());
    }
}