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)

macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String {
            bytes
                .by_ref()
                .map(|r| r.unwrap() as char)
                .skip_while(|c| c.is_whitespace())
                .take_while(|c| !c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };

    ($next:expr, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, [ $t:tt ]) => {
        {
            let len = read_value!($next, usize);
            (0..len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
        }
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

#[allow(unused_imports)]
use std::cmp::{min, max};

fn main() {
    input!{
        n: usize,
        v: [usize; n],
        es: [(usize1, usize1); n - 1]

    }
    let mut t = vec![vec![]; n];
    for &(a, b) in &es {
        t[a].push(b);
        t[b].push(a);
    }
    let mut comp: Vec<usize> = v.clone();
    comp.push(0);
    comp.sort();
    comp.dedup();
    let mut bit = BIT::new(comp.len());
    let mut ans = vec![0; n];
    dfs(0, 0, &t, &v, &comp, &mut bit, &mut ans);
    for &a in &ans {
        println!("{}", a);
    }
}

fn dfs(v: usize, p: usize, t: &Vec<Vec<usize>>, a: &Vec<usize>, comp: &Vec<usize>, bit: &mut BIT, ans: &mut Vec<usize>) {
    let iv = comp.binary_search(&a[v]).unwrap();
    let tmp = bit.max(iv - 1) + 1;
    ans[v] = max(ans[p], bit.max(iv - 1) + 1);
    bit.update(iv, tmp);
    for &c in &t[v] {
        if c == p {
            continue;
        }
        dfs(c, v, t, a, comp, bit, ans);
    }
    bit.undo();
}

#[derive(Debug)]
struct BIT {
    v: Vec<usize>,
    n: usize,
    log: Vec<Vec<(usize, usize)>>
}

impl BIT {
    fn new(n: usize) -> BIT {
        BIT{v: vec![0; n + 1], n: n, log: vec![]}
    }

    // update i th element by x
    fn update(&mut self, i: usize, x: usize) {
        let mut id = i as i64 + 1;
        let mut l = vec![];
        while id <= self.n as i64 && self.v[id as usize] < x {
            l.push((id as usize, self.v[id as usize]));
            self.v[id as usize] = x;
            id += id & -id;
        }
        self.log.push(l);
    }

    // max in 0 to i th element
    fn max(&self, i: usize) -> usize {
        let mut id = i + 1;
        let mut ret = 0;
        while id > 0 {
            ret = std::cmp::max(ret, self.v[id]);
            id &= id - 1;
        }
        ret
    }

    fn undo(&mut self) {
        for (id, x) in self.log.pop().unwrap() {
            self.v[id] = x;
        }
    }
}