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)

#![allow(unused_imports)]
#![allow(unused_macros)]
#![allow(non_snake_case)]
#![allow(dead_code)]
#![allow(unused_variables)]
#![allow(unused_mut)]
#![allow(unused_assignments)]
use proconio::input;
use proconio::marker::Usize1;
use std::collections::*;
use std::cmp::*;
use std::f64::consts::*;

const MOD: u64 = 1000000007;
const INF: usize = std::usize::MAX / 4;

fn upper_bound(A: &Vec<usize>, x: usize) -> usize {
    let (mut l, mut r): (i64,i64) = (-1, A.len() as i64);

    while r - l > 1 {
        let mid = (l + r) / 2;
        if A[mid as usize] >= x {
            r = mid;
        } else {
            l = mid;
        }
    }
    return r as usize;
}

fn search(ans: &mut Vec<usize>, now: usize, pr: usize, edges: &Vec<Vec<usize>>, A: &Vec<usize>, dp: &mut Vec<usize>) {
    ans[now] = upper_bound(dp, INF - 1);

    for &to in &edges[now] {
        if to == pr {
            continue;
        }
        let i = upper_bound(dp, A[to]);
        let old = dp[i];
        dp[i] = A[to];
        search(ans, to, now, edges, A, dp);
        dp[i] = old;
    }
}

fn main() {
    input! {
        N: usize,
        A: [usize; N],
    }

    let mut edges = vec![vec![]; N];
    for _ in 0..N-1 {
        input! {
            fr: Usize1,
            to: Usize1,
        }
        edges[fr].push(to);
        edges[to].push(fr);
    }

    let mut dp = vec![INF; N];
    dp[0] = A[0];
    let mut ans = vec![0; N];
    search(&mut ans, 0, 0, &edges, &A, &mut dp);

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