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(dead_code)]
use std::cmp;
use std::collections::*;

pub struct Scanner {
    bytes: Vec<u8>,
    begin: usize,
    end: usize,
}

impl Scanner {
    pub fn new() -> Self {
        use std::io::Read;
        let mut buffer: Vec<u8> = Vec::with_capacity(37_000_000);
        std::io::Read::read_to_end(&mut std::io::stdin(), &mut buffer).unwrap();
        let mut end = 0;
        while end < buffer.len() && !buffer[end].is_ascii_whitespace() {
            end += 1;
        }
        Scanner {
            bytes: buffer,
            begin: 0,
            end: end,
        }
    }
    pub fn next<T>(&mut self) -> T
    where
        T: std::str::FromStr,
        T::Err: std::fmt::Debug,
    {
        let s: &str = unsafe { std::str::from_utf8_unchecked(&self.bytes[self.begin..self.end]) };
        let ret: T = s.parse::<T>().unwrap();
        self.skip();
        ret
    }
    pub fn next_vec<T>(&mut self, n: usize) -> Vec<T>
    where
        T: std::str::FromStr,
        T::Err: std::fmt::Debug,
    {
        let mut v: Vec<T> = Vec::with_capacity(n);
        for _i in 0..n {
            v.push(self.next());
        }
        v
    }
    pub fn next_word(&mut self) -> Vec<u8> {
        let r = Vec::from(&self.bytes[self.begin..self.end]);
        self.skip();
        r
    }
    fn skip(&mut self) {
        let mut begin = self.end;
        while begin < self.bytes.len() && self.bytes[begin].is_ascii_whitespace() {
            begin += 1;
        }
        let mut end = begin;
        while end < self.bytes.len() && !self.bytes[end].is_ascii_whitespace() {
            end += 1;
        }
        self.begin = begin;
        self.end = end;
    }
}

fn lower_bound<T: std::cmp::Ord>(v: &Vec<T>, val: &T) -> usize {
    let mut lb: isize = -1;
    let mut ub: isize = v.len() as isize;
    while ub - lb > 1 {
        let mid = (lb + ub) / 2;
        if *val <= v[mid as usize] {
            ub = mid;
        } else {
            lb = mid;
        }
    }
    return ub as usize;
}

fn dfs(
    u: usize,
    p: usize,
    dp: &mut Vec<i32>,
    ans: &mut Vec<usize>,
    a: &Vec<i32>,
    edge: &Vec<Vec<usize>>,
    pans: usize,
) {
    let idx = lower_bound(dp, &a[u]);
    let prev = dp[idx];
    ans[u] = if dp[idx] == i32::max_value() {
        pans + 1
    } else {
        pans
    };
    dp[idx] = a[u];

    for v in &edge[u] {
        if *v == p {
            continue;
        }
        dfs(*v, u, dp, ans, a, edge, ans[u]);
    }
    dp[idx] = prev;
}

fn main() {
    let mut scan = Scanner::new();
    let n: usize = scan.next();
    let a: Vec<i32> = scan.next_vec(n);
    let mut edge: Vec<Vec<usize>> = vec![Vec::new(); n];
    for _ in 0..n - 1 {
        let u: usize = scan.next();
        let v: usize = scan.next();
        let (u, v) = (u - 1, v - 1);
        edge[u].push(v);
        edge[v].push(u);
    }
    let mut dp: Vec<i32> = vec![i32::max_value(); n];
    let mut ans: Vec<usize> = vec![0; n];
    dfs(0, 0, &mut dp, &mut ans, &a, &edge, 0);
    for x in ans {
        println!("{}", x);
    }
}