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::{input, fastout};
use proconio::marker::Usize1;

#[derive(Clone, Debug)]
struct Adjacent {
    to: usize,
}

#[derive(Debug)]
struct Graph {
    n: usize,
    links: Vec<Vec<Adjacent>>,
}

impl Graph {
    fn new(n: usize) -> Graph {
        Graph {
            n,
            links: vec![vec![]; n],
        }
    }

    fn add_edge(&mut self, from: usize, to: usize) {
        self.links[from].push(Adjacent { to });
        self.links[to].push(Adjacent { to: from });
    }
}


fn dfs(v: usize, p: usize, g: &Graph, aaa: &Vec<u64>, state: &mut Vec<u64>, ans: &mut Vec<u64>) {
    let i = state.binary_search(&&aaa[v]).unwrap_or_else(|x| x);
    let pushed = i == state.len();
    let mut backup_a = 0;
    if pushed {
        state.push(aaa[v]);
    } else {
        backup_a = state[i];
        state[i] = aaa[v];
    }
    ans[v] = state.len() as u64 - 1;

    for adj in &g.links[v] {
        if adj.to == p { continue; }
        dfs(adj.to, v, g, aaa, state, ans);
    }

    if pushed {
        state.pop();
    } else {
        state[i] = backup_a;
    }
}

#[fastout]
fn main() {
    input! {
        n: usize,
        aaa: [u64; n],
        uv: [(Usize1, Usize1); n - 1],
    }

    let mut g = Graph::new(n);
    for (u, v) in uv { g.add_edge(u, v); }

    let mut state = vec![0u64];
    let mut ans = vec![0u64; n];

    dfs(0, n, &g, &aaa, &mut state, &mut ans);

    for v in ans {
        println!("{}", v);
    }
}