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: Wrong Answer)

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

const M: usize = 1e+9 as usize;

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

fn search(nodes: &mut Vec<Node>, index: usize, table: &mut Vec<Vec<usize>>, depth: usize) {
    let a = nodes[index].a;
    let ns = nodes[index].ns.clone();

    nodes[index].v = 0;

    for i in 1..table[depth].len() {
        if a > table[depth - 1][i - 1] {
            table[depth][i] = min(a, table[depth - 1][i]);
        }
    }

    for n in ns.iter() {
        if nodes[*n].v == 0 {
            continue;
        }

        search(nodes, *n, table, depth + 1);
    }

    for i in 1..table[depth].len() {
        if table[depth][i] == M {
            break;
        }

        nodes[index].v = i;
        table[depth][i] = M;
    }
}

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

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

    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);
    }

    for i in 0..in_n + 1 {
        table.push(vec![M; in_n + 1]);
        table[i][0] = 0;
    }

    search(&mut nodes, 0, &mut table, 1);

    for n in nodes.iter() {
        println!("{}", n.v);
    }
}