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 std::io::Read;

const INF: i64 = 1000000000000000;

#[derive(Default)]
struct Solver {
    a: Vec<i64>,
    g: Vec<Vec<usize>>,
    ans: Vec<usize>,
    lis: Vec<i64>,
}

#[test]
fn test_lower_bound() {
    let mut s: Solver = Default::default();
    s.lis = vec![INF; 120];
    s.lis[0] = 5;
    for i in 1..99 {
        s.lis[i] = s.lis[i - 1] + (i as i64 % 3);
    }
    for i in 0..400 {
        let mut exp = 0;
        for j in 0..100 {
            if s.lis[j] >= i {
                exp = j;
                break;
            }
        }
        s.lower_bound(i);
        assert_eq!(exp, s.lower_bound(i))
    }
}

impl Solver {
    fn lower_bound(&self, target: i64) -> usize {
        let mut low = 0;
        let mut high = self.lis.len();
        if target <= self.lis[0] {
            return 0;
        }
        // target <= liw[low]
        while low + 1 < high {
            let mid = (low + high) / 2;
            if self.lis[mid] >= target {
                high = mid;
            } else {
                low = mid;
            }
        }
        return high;
    }
    fn recur(&mut self, vec: usize, parent: usize, g: &Vec<Vec<usize>>) {
        let target = self.a[vec];
        let id = self.lower_bound(target);
        let backup = self.lis[id];
        self.lis[id] = target;

        self.ans[vec] = self.lower_bound(INF);
        for &nv in &g[vec] {
            if nv == parent {
                continue;
            }
            self.recur(nv, vec, g);
        }
        self.lis[id] = backup;
    }

    fn solve(&mut self) {
        self.ans = vec![0; self.a.len()];
        self.lis = vec![INF; self.a.len() + 2];
        self.recur(0, self.a.len(), &self.g.clone());
        for ans in &self.ans {
            println!("{}", ans);
        }
    }
}

fn main() {
    let mut line = String::new();
    std::io::stdin().read_to_string(&mut line);
    let mut iter = line.split_whitespace();
    let n: usize = iter.next().unwrap().parse().unwrap();
    let a: Vec<i64> = iter.by_ref().take(n).map(|a| a.parse().unwrap()).collect();
    let mut g: Vec<Vec<usize>> = vec![Default::default(); n];
    for uv in iter.by_ref().take(2 * (n - 1)).map(|a| a.parse().unwrap()).collect::<Vec<usize>>().chunks(2) {
        g[uv[0] - 1].push(uv[1] - 1);
        g[uv[1] - 1].push(uv[0] - 1);
    }
    let mut solver = Solver { a, g, ..Default::default() };
    solver.solve();
}