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: Accepted)

use proconio::{fastout, input};

const M: usize = 1e+9 as usize + 1;
// const M: usize = 99;

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

// 二分探索（配列は昇順にソートされているものとする）
fn bisect<T: PartialOrd>(a: &mut [T], v: T, lo: usize, hi: usize) -> usize {
    if hi - lo < 2 {
        if v > a[lo] {
            return lo + 1;
        } else {
            return lo;
        }
    } else {
        let m = (lo + hi) / 2;

        if v <= a[m] {
            return bisect(a, v, lo, m);
        } else {
            return bisect(a, v, m, hi);
        }
    }
}

fn dfs(nodes: &mut [Node], idx: usize, lines: &mut [usize]) {
    let i = bisect(lines, nodes[idx].a, 0, lines.len());
    let v = lines[i];

    if lines[i - 1] >= nodes[idx].a || lines[i] < nodes[idx].a {
        panic!("panic!");
    }

    lines[i] = nodes[idx].a;
    nodes[idx].v = bisect(lines, M, 0, lines.len()) - 1;

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

        dfs(nodes, *n, lines);
    }

    lines[i] = v;
}

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

    for a in in_a.iter() {
        nodes.push(Node {
            a: *a,
            v: 0,
            ns: vec![],
        })
    }

    for v in in_v.iter() {
        nodes[v[0] - 1].ns.push(v[1] - 1);
        nodes[v[1] - 1].ns.push(v[0] - 1);
    }

    lines[0] = 0;

    dfs(&mut nodes, 0, &mut lines);

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