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;

fn solve(g : &Vec<Vec<usize>>, a : &Vec<u32>,
         pos : usize, prev : usize, best : usize,
         res : &mut Vec<usize>, inc : &mut Vec<u32>)
{
    let n = g.len();
    let idx = if a[pos] <= inc[0] {
        0
    } else {
        let mut low = 0;
        let mut high = n-1;
        while high - low > 1 {
            let mid = (low+high)/2;
            if a[pos] <= inc[mid] {
                high = mid;
            } else {
                low = mid;
            }
        }
        high
    };
    let cur_value = inc[idx];
    inc[idx] = a[pos];
    res[pos] = best.max(idx + 1);
    for &next in &g[pos] {
        if next == prev { continue; }
        solve(g, a, next, pos, best.max(idx+1), res, inc);
    }
    inc[idx] = cur_value;
}

fn main() {
    input! {
        n : usize,
        a : [u32; n]
    };
    let mut g = vec![Vec::new(); n];
    for _ in 0..n-1 {
        input! {
            (u, v) : (usize, usize)
        };
        g[u-1].push(v-1);
        g[v-1].push(u-1);
    }
    let mut res = vec![1; n];
    let mut inc = vec![1_000_000_007; n];
    solve(&g, &a, 0, n, 1, &mut res, &mut inc);
    for r in res { println!("{}", r); }
}