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)]
fn is_whitespace(c: &char) -> bool {
    c.is_whitespace()
}

#[allow(dead_code)]
fn is_eol(c: &char) -> bool {
    ['\r', '\n'].contains(c)
}

#[allow(unused_macros)]
macro_rules! input {
    ($var:ident, $($r:tt)*) => {
        input_inner!{$var, $($r)*}
    };

    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        #[allow(unused_mut, unused_variables, bare_trait_objects)]
        let mut next = move |f: &Fn(&char)->bool| -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|f(&c))
                .take_while(|c|!f(&c))
                .collect()
        };
        input_inner!{next, $($r)*};
    };
}

#[allow(unused_macros)]
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)*}
    };
}

#[allow(unused_macros)]
macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; eol ]) => {
        $next(&is_eol).split(' ').map(|x| x.parse::<$t>().unwrap()).collect::<Vec<_>>()
    };

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

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

    ($next:expr, bytes) => {
        read_value!($next, String).into_bytes()
    };

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

    ($next:expr, parser) => {
        $next
    };

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

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

#[allow(dead_code)]
struct Edge {
    from: usize,
    to: usize,
}

#[allow(dead_code)]
type Graph = Vec<Vec<usize>>;
#[allow(dead_code)]
const M: usize = 1_000_000_007;

use std::cmp::Ordering;
trait BinarySearch<T> {
    fn upper_bound(&self, value: &T) -> Option<usize>;
    fn lower_bound(&self, value: &T) -> Option<usize>;
}

impl<T: Ord> BinarySearch<T> for [T] {
    fn upper_bound(&self, value: &T) -> Option<usize> {
        let (mut ng, mut ok) = (-1, self.len() as isize);
        while (ok - ng).abs() > 1 {
            let mid = ((ok + ng) / 2) as usize;
            match value.cmp(&self[mid]) {
                Ordering::Less => {
                    ok = mid as isize;
                }
                Ordering::Equal | Ordering::Greater => {
                    ng = mid as isize;
                }
            };
        }
        if ok >= 0 && ok < self.len() as isize {
            Some(ok as usize)
        } else {
            None
        }
    }

    fn lower_bound(&self, value: &T) -> Option<usize> {
        let (mut ng, mut ok) = (-1, self.len() as isize);
        while (ok - ng).abs() > 1 {
            let mid = ((ok + ng) / 2) as usize;
            match value.cmp(&self[mid]) {
                Ordering::Less | Ordering::Equal => {
                    ok = mid as isize;
                }
                Ordering::Greater => {
                    ng = mid as isize;
                }
            };
        }
        if ok >= 0 && ok < self.len() as isize {
            Some(ok as usize)
        } else {
            None
        }
    }
}

fn dfs(
    par: usize,
    pos: usize,
    lis: &mut Vec<usize>,
    graph: &Vec<Vec<usize>>,
    a: &Vec<usize>,
    ans: &mut Vec<usize>,
) {
    let res = lis.lower_bound(&a[pos]);
    let mut old = None;
    if let Some(index) = res {
        old = Some(lis[index]);
        lis[index] = a[pos];
    } else {
        lis.push(a[pos]);
    }
    ans[pos] = lis.len();
    for &next in &graph[pos] {
        if par == next {
            continue;
        }
        dfs(pos, next, lis, graph, a, ans);
    }
    if let Some(v) = old {
        let idx = lis.lower_bound(&a[pos]);
        lis[idx.unwrap()] = v;
    } else {
        lis.pop();
    }
}

fn main() {
    input! {
        n: usize,
        a: [usize; n],
        edges: [(usize1, usize1); n-1],
    }
    let mut graph = vec![vec![]; n];
    for &(f, t) in &edges {
        graph[f].push(t);
        graph[t].push(f);
    }
    let mut ans = vec![0; n];
    let mut lis = vec![];
    dfs(0, 0, &mut lis, &graph, &a, &mut ans);
    for x in ans {
        println!("{}", x);
    }
}